Finite Group Databasebeta

← #33009702

Subgroups — #33009702

Order 18688.

2006 conjugacy classes of subgroups.

The subgroup lattice has 696 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988462C211yesmaximal, characteristic
2C2×#17988459C211yesmaximal, characteristic
3C2×C22⋊C4×Dic73C211yesmaximal, characteristic
4C22×#14556498C211yesmaximal, characteristic
5#22608190C212yesmaximal, complemented
6#22608183C212yesmaximal, complemented
7#22608162C212yesmaximal, complemented
8#22608160C212yesmaximal, complemented
9#22608154C211yesmaximal, characteristic
10#22608153C211yesmaximal, characteristic
11C2×#385×C73C211yesmaximal, characteristic
12C24×Dic73C2211yescharacteristic
13C24×Dic73C2211yescharacteristic
14C22×C4⋊Dic73C2212yes—
15C22×#11924007C2211yescharacteristic
16C22×#11924007C2211yescharacteristic
17C22×C22⋊Dic73C2211yescharacteristic
18C22×C22⋊Dic73C2211yescharacteristic
19C22×C4×Dic73C2212yes—
20#17988462C418yescomplemented
21#17988459—24no—
22C22⋊C4×Dic73—24no—
23C2×#14556498—24no—
24C2×#14556498C2212yes—
25C2×#14556498C2212yes—
26C2×#14556498C2212yes—
27C2×#14556498C2211yescharacteristic
28C2×#14556498C2212yes—
29C2×#14556498C2211yescharacteristic
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C24×C4×C73C2211yescharacteristic
35C22×C4⋊C4×C73C2212yes—
36C22×C22⋊C4×C73C2211yescharacteristic
37C22×C22⋊C4×C73C2211yescharacteristic
38#385×C73—24no—
39C2×#95×C73C2212yes—
40C23×Dic73—24no—
41C23×Dic73—22no—
42C23×Dic73—22no—
43C23×Dic73—22no—
44C23×Dic73—22no—
45C23×Dic73—42no—
46C23×Dic73—42no—
47C23×Dic73—42no—
48C23×Dic73C2×C414yes—
49C23×Dic73D414yes—
50C23×Dic73C2312yes—
51C23×Dic73C2311yescharacteristic
52C23×Dic73C2311yescharacteristic
53C23×Dic73C2311yescharacteristic
54C23×Dic73C2312yes—
55C23×Dic73C2311yescharacteristic
56C2×C4⋊Dic73—42no—
57C2×C4⋊Dic73—42no—
58C2×C4⋊Dic73C2×C418yescomplemented
59C2×#11924007—22no—
60C2×#11924007—22no—
61C2×#11924007—24no—
62C2×#11924007—42no—
63C2×C22⋊Dic73—22no—
64C2×C22⋊Dic73—22no—
65C2×C22⋊Dic73—22no—
66C2×C22⋊Dic73—22no—
67C2×C22⋊Dic73C2×C414yes—
68C2×C22⋊Dic73Q814yes—
69C2×C4×Dic73—24no—
70C2×C4×Dic73—24no—
71C2×C4×Dic73—42no—
72#14556498—24no—
73#14556498—24no—
74#14556498—22no—
75#14556498—22no—
76#14556498—24no—
77#14556498—24no—
78#14556498—42no—
79#14556498—42no—
80#14556498—42no—
81#14556498C2×C418yescomplemented
82C25×C73C2311yescharacteristic
83C23×C4×C73—22no—
84C23×C4×C73—24no—
85C23×C4×C73—22no—
86C23×C4×C73—22no—
87C23×C4×C73—42no—
88C23×C4×C73C2311yescharacteristic
89C23×C4×C73C2311yescharacteristic
90C23×C4×C73C2312yes—
91C23×C4×C73C2312yes—
92C2×C4⋊C4×C73—24no—
93C2×C4⋊C4×C73—24no—
94C2×C4⋊C4×C73—42no—
95C2×C22⋊C4×C73—22no—
96C2×C22⋊C4×C73—22no—
97C2×C22⋊C4×C73—22no—
98C2×C22⋊C4×C73C2×C414yes—
99C2×C22⋊C4×C73D414yes—
100C2×C22⋊C4×C73Q814yes—
101#95×C73—24no—
102C22×Dic73—21no—
103C22×Dic73—21no—
104C22×Dic73—22no—
105C22×Dic73—28no—
106C22×Dic73—28no—
107C22×Dic73—22no—
108C22×Dic73—21no—
109C22×Dic73—22no—
110C22×Dic73—21no—
111C22×Dic73—21no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—21no—
116C22×Dic73—21no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—21no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—22no—
126C22×Dic73—22no—
127C22×Dic73—22no—
128C22×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73—22no—
131C22×Dic73—28no—
132C22×Dic73—48no—
133C22×Dic73—44no—
134C22×Dic73—48no—
135C22×Dic73—48no—
136C22×Dic73—44no—
137C22×Dic73—44no—
138C22×Dic73—48no—
139C22×Dic73C22×C412yes—
140C22×Dic73C22×C414yes—
141C22×Dic73C22×C412yes—
142C22×Dic73C2×D412yes—
143C22×Dic73C2×D412yes—
144C22×Dic73C2×Q814yes—
145C22×Dic73#4114yes—
146C4⋊Dic73—48no—
147#11924007—44no—
148#11924007—44no—
149C22⋊Dic73—28no—
150C22⋊Dic73—28no—
151C4×Dic73—48no—
152C24×C73—22no—
153C24×C73—22no—
154C24×C73—22no—
155C24×C73—22no—
156C24×C73C22×C412yes—
157C24×C73C2×D412yes—
158C24×C73C2×D412yes—
159C24×C73C2×Q812yes—
160C24×C73C2×Q812yes—
161C24×C73#4112yes—
162C24×C73#4112yes—
163C24×C73C2411yescharacteristic
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—22no—
168C22×C4×C73—28no—
169C22×C4×C73—22no—
170C22×C4×C73—22no—
171C22×C4×C73—22no—
172C22×C4×C73—22no—
173C22×C4×C73—22no—
174C22×C4×C73—22no—
175C22×C4×C73—21no—
176C22×C4×C73—21no—
177C22×C4×C73—21no—
178C22×C4×C73—21no—
179C22×C4×C73—21no—
180C22×C4×C73—21no—
181C22×C4×C73—22no—
182C22×C4×C73—22no—
183C22×C4×C73—22no—
184C22×C4×C73—22no—
185C22×C4×C73—48no—
186C22×C4×C73—48no—
187C22×C4×C73—44no—
188C22×C4×C73—44no—
189C22×C4×C73C22×C414yes—
190C22×C4×C73C2×D414yes—
191C22×C4×C73C2×Q814yes—
192C22×C4×C73#4114yes—
193C4⋊C4×C73—48no—
194C22⋊C4×C73—44no—
195C22⋊C4×C73C4⋊C4116yescomplemented
196C23×C73—24no—
197C23×C73—24no—
198C23×C73—22no—
199C23×C73—22no—
200C23×C73—24no—
201C23×C73—22no—
202C23×C73—22no—
203C23×C73—24no—
204C23×C73—24no—
205C23×C73—24no—
206C23×C73—24no—
207C23×C73—24no—
208C23×C73—24no—
209C23×C73—24no—
210C23×C73—24no—
211C23×C73—24no—
212C23×C73C2×C4⋊C414yes—
213C23×C73C4×D414yes—
214C23×C73C4×Q814yes—
215C23×C73#12114yes—
216C23×C73C22⋊Q814yes—
217C23×C73#12514yes—
218C23×C73C4⋊Q814yes—
219C23×C73C23×C411yescharacteristic
220C23×C73C22×D411yescharacteristic
221C23×C73C22×D411yescharacteristic
222C23×C73C22×Q811yescharacteristic
223C23×C73C22×Q811yescharacteristic
224C23×C73C2×#4112yes—
225C23×C73C2×#4111yescharacteristic
226C23×C73C2×#4112yes—
227C23×C73C2×#4111yescharacteristic
228C23×C73H(5,𝔽2)12yes—
229C23×C73H(5,𝔽2)12yes—
230C2×C4×C73—24no—
231C2×C4×C73—24no—
232C2×C4×C73—24no—
233C2×C4×C73—24no—
234C2×C4×C73—24no—
235C2×C4×C73—24no—
236C2×C4×C73—24no—
237C2×C4×C73—24no—
238C2×C4×C73—24no—
239C2×C4×C73—416no—
240C2×C4×C73—42no—
241C2×C4×C73—42no—
242C2×C4×C73—42no—
243C2×C4×C73—42no—
244C2×C4×C73—41no—
245C2×C4×C73—41no—
246C2×C4×C73—41no—
247C2×C4×C73—41no—
248C2×C4×C73—41no—
249C2×C4×C73—41no—
250C2×C4×C73—42no—
251C2×C4×C73—42no—
252C2×C4×C73—42no—
253C2×C4×C73—42no—
254C2×C4×C73C2×C4⋊C418yes—
255C2×Dic73—22no—
256C2×Dic73—22no—
257C2×Dic73—24no—
258C2×Dic73—24no—
259C2×Dic73—22no—
260C2×Dic73—22no—
261C2×Dic73—24no—
262C2×Dic73—24no—
263C2×Dic73—24no—
264C2×Dic73—24no—
265C2×Dic73—24no—
266C2×Dic73—22no—
267C2×Dic73—24no—
268C2×Dic73—22no—
269C2×Dic73—22no—
270C2×Dic73—24no—
271C2×Dic73—24no—
272C2×Dic73—416no—
273C2×Dic73—416no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—42no—
277C2×Dic73—42no—
278C2×Dic73—41no—
279C2×Dic73—42no—
280C2×Dic73—41no—
281C2×Dic73—42no—
282C2×Dic73—42no—
283C2×Dic73—42no—
284C2×Dic73—42no—
285C2×Dic73—42no—
286C2×Dic73—42no—
287C2×Dic73—42no—
288C2×Dic73—42no—
289C2×Dic73—42no—
290C2×Dic73#12314yes—
291C22×C73—24no—
292C22×C73—24no—
293C22×C73—28no—
294C22×C73—28no—
295C22×C73—24no—
296C22×C73—24no—
297C22×C73—28no—
298C22×C73—24no—
299C22×C73—28no—
300C22×C73—24no—
301C22×C73#38918yes—
302C22×C73C22×C4⋊C411yescharacteristic
303C22×C73C2×C4×D411yescharacteristic
304C22×C73C2×C4×Q811yescharacteristic
305C22×C73C4×#4112yes—
306C22×C73#51812yes—
307C22×C73C2×#12111yescharacteristic
308C22×C73C2×C22⋊Q811yescharacteristic
309C22×C73C2×#12311yescharacteristic
310C22×C73#52511yescharacteristic
311C22×C73C2×#12511yescharacteristic
312C22×C73#52912yes—
313C22×C73C2×C4⋊Q811yescharacteristic
314C22×C73#53411yescharacteristic
315C22×C73#53511yescharacteristic
316C22×C73#53812yes—
317C22×C73#54612yes—
318C22×C73#54612yes—
319C22×C73#55112yes—
320C22×C73#55112yes—
321C22×C73#55412yes—
322C22×C73#55412yes—
323C22×C73#55412yes—
324C22×C73#55412yes—
325C4×C73—28no—
326C4×C73—44no—
327C4×C73—81no—
328C4×C73—81no—
329Dic73—24no—
330Dic73—42no—
331Dic73—44no—
332Dic73—42no—
333Dic73—44no—
334Dic73—42no—
335#14044—731nomaximal, Sylow
336C2×C73—28no—
337C2×C73C2×#38911yescharacteristic
338C2×C73C4×#12311yescharacteristic
339C2×C73#232111yescharacteristic
340C2×C73#234912yes—
341C2×C73#236812yes—
342C2×C73#241111yescharacteristic
343C2×C73#241211yescharacteristic
344C2×C73#245012yes—
345C2×C73#250211yescharacteristic
346C2×C73#250512yes—
347C2×C73#258311yescharacteristic
348C2×#385—731no—
349C2×#385—731no—
350C2×#385D7311yescharacteristic
351C2×C4×C22⋊C4—731no—
352C22×#95—731no—
353#1442—732no—
354#1441—732no—
355#1436—732no—
356C4⋊#95—732no—
357C22⋊#95—731no—
358C22⋊#95—731no—
359C73#1404411yescharacteristic, complemented, Sylow
360C24×C4—731no—
361C24×C4—731no—
362C24×C4C2×D7311yescharacteristic
363C22×C4⋊C4—732no—
364C22×C4⋊C4—731no—
365C22×C4⋊C4—731no—
366C22×C4⋊C4C2×D7312yes—
367C22×C22⋊C4—731no—
368C22×C22⋊C4—731no—
369C22×C22⋊C4C2×D7311yescharacteristic
370C22×C22⋊C4C2×D7311yescharacteristic
371C22×C42—732no—
372#385—24no—
373#385—738no—
374#385—1464no—
375C4×C22⋊C4—1464no—
376C2×#95—732no—
377C2×#95—732no—
378C2×#95—732no—
379C2×#95—732no—
380C2×#95—732no—
381C2×#95—731no—
382C2×#95—732no—
383C2×#95—732no—
384C2×#95—732no—
385C2×#95—731no—
386C2×#95—1464no—
387C2×#95C2×D7312yes—
388C25C22×D7311yescharacteristic
389C23×C4—22no—
390C23×C4—24no—
391C23×C4—22no—
392C23×C4—22no—
393C23×C4—42no—
394C23×C4—2922no—
395C23×C4—2922no—
396C23×C4—2922no—
397C23×C4—734no—
398C23×C4—734no—
399C23×C4—731no—
400C23×C4—731no—
401C23×C4—731no—
402C23×C4—732no—
403C23×C4—732no—
404C23×C4—731no—
405C23×C4—1464no—
406C23×C4—1462no—
407C23×C4—1462no—
408C23×C4—1462no—
409C23×C4—1462no—
410C23×C4C22×D7311yescharacteristic
411C23×C4C22×D7311yescharacteristic
412C23×C4C22×D7312yes—
413C23×C4C22×D7312yes—
414C2×C4⋊C4—24no—
415C2×C4⋊C4—24no—
416C2×C4⋊C4—42no—
417C2×C4⋊C4—2922no—
418C2×C4⋊C4—2922no—
419C2×C4⋊C4—2922no—
420C2×C4⋊C4—738no—
421C2×C4⋊C4—1462no—
422C2×C4⋊C4—1462no—
423C2×C4⋊C4—1464no—
424C2×C22⋊C4—22no—
425C2×C22⋊C4—22no—
426C2×C22⋊C4—22no—
427C2×C22⋊C4—734no—
428C2×C22⋊C4—734no—
429C2×C22⋊C4—1462no—
430C2×C22⋊C4—1462no—
431C2×C22⋊C4—1462no—
432C2×C22⋊C4—1462no—
433C2×C22⋊C4Dic14614yes—
434C2×C22⋊C4C4×D7314yes—
435C2×C22⋊C4C22⋊D7314yes—
436C2×C42—2922no—
437C2×C42—1464no—
438C2×C42—1464no—
439#95—24no—
440#95—2922no—
441#95—2922no—
442#95—2922no—
443#95—738no—
444#95—1464no—
445#95—1464no—
446#95—1462no—
447#95—1462no—
448#95—1464no—
449#95—1464no—
450C24—22no—
451C24—22no—
452C24—22no—
453C24—22no—
454C24C2×C4×D7312yes—
455C24D4×D7312yes—
456C24C2×C22⋊D7312yes—
457C24Q8×D7312yes—
458C24C2×Dic14612yes—
459C24#1192402712yes—
460C24#1192402912yes—
461C24C23×D7311yescharacteristic
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—22no—
466C22×C4—28no—
467C22×C4—22no—
468C22×C4—22no—
469C22×C4—22no—
470C22×C4—22no—
471C22×C4—22no—
472C22×C4—22no—
473C22×C4—21no—
474C22×C4—21no—
475C22×C4—21no—
476C22×C4—21no—
477C22×C4—22no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—22no—
481C22×C4—22no—
482C22×C4—22no—
483C22×C4—48no—
484C22×C4—48no—
485C22×C4—44no—
486C22×C4—44no—
487C22×C4—2924no—
488C22×C4—2924no—
489C22×C4—2928no—
490C22×C4—2928no—
491C22×C4—2928no—
492C22×C4—2928no—
493C22×C4—2924no—
494C22×C4—732no—
495C22×C4—734no—
496C22×C4—732no—
497C22×C4—734no—
498C22×C4—732no—
499C22×C4—732no—
500C22×C4—734no—
501C22×C4—1468no—
502C22×C4—1468no—
503C22×C4—1468no—
504C22×C4—1462no—
505C22×C4—1461no—
506C22×C4—1461no—
507C22×C4—1461no—
508C22×C4—1462no—
509C22×C4—1462no—
510C22×C4—1462no—
511C22×C4—1462no—
512C22×C4—1462no—
513C22×C4—1462no—
514C22×C4—1461no—
515C22×C4—1461no—
516C22×C4—1462no—
517C22×C4—1461no—
518C22×C4—1462no—
519C22×C4—1462no—
520C22×C4—1462no—
521C22×C4—1462no—
522C22×C4—1462no—
523C22×C4—1462no—
524C22×C4—1462no—
525C22×C4—1462no—
526C22×C4—1462no—
527C22×C4—1462no—
528C22×C4—1461no—
529C22×C4—1461no—
530C22×C4—1462no—
531C22×C4C2×C4×D7314yes—
532C22×C4C2×C22⋊D7314yes—
533C22×C4C2×Dic14614yes—
534C22×C4#1192402714yes—
535C4⋊C4—48no—
536C4⋊C4—2928no—
537C4⋊C4—2924no—
538C4⋊C4—2924no—
539C22⋊C4—44no—
540C22⋊C4—1468no—
541C22⋊C4—1468no—
542C22⋊C4#11924007116yescomplemented
543C42—2928no—
544C23—24no—
545C23—22no—
546C23—22no—
547C23—24no—
548C23—24no—
549C23—24no—
550C23—24no—
551C23—24no—
552C23—24no—
553C23—24no—
554C23—24no—
555C23—24no—
556C23—24no—
557C23—24no—
558C23—22no—
559C23—22no—
560C23C2×#1192400714yes—
561C23#1455655314yes—
562C23#1455655614yes—
563C23#1455656214yes—
564C23#1455656714yes—
565C23#1455658214yes—
566C23#1455659014yes—
567C23C22×C4×D7311yescharacteristic
568C23C2×D4×D7311yescharacteristic
569C23C22×C22⋊D7311yescharacteristic
570C23C2×Q8×D7311yescharacteristic
571C23C22×Dic14611yescharacteristic
572C23#41×D7312yes—
573C23C2×#1192402712yes—
574C23C2×#1192402711yescharacteristic
575C23C2×#1192402911yescharacteristic
576C23#1455663612yes—
577C23#1455663712yes—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—24no—
582C2×C4—24no—
583C2×C4—24no—
584C2×C4—24no—
585C2×C4—24no—
586C2×C4—24no—
587C2×C4—416no—
588C2×C4—42no—
589C2×C4—42no—
590C2×C4—42no—
591C2×C4—42no—
592C2×C4—41no—
593C2×C4—41no—
594C2×C4—42no—
595C2×C4—42no—
596C2×C4—42no—
597C2×C4—42no—
598C2×C4—41no—
599C2×C4—41no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—29216no—
603C2×C4—2922no—
604C2×C4—2922no—
605C2×C4—2921no—
606C2×C4—2922no—
607C2×C4—2922no—
608C2×C4—2921no—
609C2×C4—2922no—
610C2×C4—2922no—
611C2×C4—2922no—
612C2×C4—2922no—
613C2×C4—2922no—
614C2×C4—2922no—
615C2×C4—2922no—
616C2×C4—2922no—
617C2×C4—2921no—
618C2×C4—2921no—
619C2×C4—29216no—
620C2×C4—734no—
621C2×C4—1462no—
622C2×C4—1464no—
623C2×C4—1464no—
624C2×C4—1462no—
625C2×C4—1462no—
626C2×C4—1464no—
627C2×C4—1464no—
628C2×C4—1464no—
629C2×C4—1464no—
630C2×C4—1464no—
631C2×C4—1462no—
632C2×C4—1464no—
633C2×C4—1462no—
634C2×C4—1462no—
635C2×C4—1464no—
636C2×C4—1464no—
637C2×C4—1462no—
638C2×C4C2×#1192400718yes—
639C22—24no—
640C22—28no—
641C22—24no—
642C22—24no—
643C22—28no—
644C22—24no—
645C22—24no—
646C22—28no—
647C22—24no—
648C22—28no—
649C22#1798848318yes—
650C22C22×#1192400711yescharacteristic
651C22C2×#1455655311yescharacteristic
652C22C2×#1455655611yescharacteristic
653C22C4×#1192402712yes—
654C22#1798905912yes—
655C22C2×#1455656211yescharacteristic
656C22C2×#1455656711yescharacteristic
657C22#123×D7311yescharacteristic
658C22#1798909311yescharacteristic
659C22C2×#1455658211yescharacteristic
660C22#1798912312yes—
661C22C2×#1455659011yescharacteristic
662C22#1798914811yescharacteristic
663C22#1798915211yescharacteristic
664C22#1798917712yes—
665C22#1798922512yes—
666C22#1798923112yes—
667C22#1798926012yes—
668C22#1798926812yes—
669C22#1798929112yes—
670C22#1798929412yes—
671C22#1798929512yes—
672C22#1798929612yes—
673C4—28no—
674C4—44no—
675C4—81no—
676C4—81no—
677C4—2922no—
678C4—2924no—
679C4—2922no—
680C4—2922no—
681C4—2924no—
682C4—1464no—
683C2—28no—
684C2C2×#1798848311yescharacteristic
685C2#2261245011yescharacteristic
686C2#2261258811yescharacteristic
687C2#2261283612yes—
688C2#2261301212yes—
689C2#2261331211yescharacteristic
690C2#2261332311yescharacteristic
691C2#2261375112yes—
692C2#2261440711yescharacteristic
693C2#2261443612yes—
694C2#2261530811yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC2×#385×C73C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenterC2×#385D73
hypocenterC73#14044
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ C2×#385

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
5. #22608190C2semidirect
6. #22608183C2semidirect
7. #22608162C2semidirect
8. #22608160C2semidirect
20. #17988462C4semidirect
58. C2×C4⋊Dic73C2×C4semidirect
81. #14556498C2×C4semidirect
195. C22⋊C4×C73C4⋊C4semidirect
359. C73#14044semidirect
542. C22⋊C4#11924007semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#179884622
C2×#179884592
C2×C22⋊C4×Dic732
C22×#145564982
#226081902
#226081832
#226081622
#226081602
#226081542
#226081532
C2×#385×C732
#1404473