Finite Group Databasebeta

← #33007810

Subgroups — #33007810

Order 18688.

1966 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988523C211yesmaximal, characteristic
2C2×C4⋊C4×Dic73C211yesmaximal, characteristic, complemented
3C2×#17988438C212yesmaximal, complemented
4C2×C22⋊C4×Dic73C212yesmaximal
5C2×C22⋊C4×Dic73C211yesmaximal, characteristic
6#22608198C211yesmaximal, characteristic, complemented
7#22608197C212yesmaximal, complemented
8#22608194C211yesmaximal, characteristic, complemented
9#22608174C212yesmaximal
10#95×Dic73C211yesmaximal, characteristic, complemented
11C73×#1439C211yesmaximal, characteristic
12C24×Dic73C2211yescharacteristic
13C22×C4⋊Dic73C2211yescharacteristic
14C22×#11924007C2212yes—
15C22×#11924007C2212yes—
16C22×C22⋊Dic73C2211yescharacteristic
17C22×C22⋊Dic73C2212yes—
18C22×C4×Dic73C2211yescharacteristic
19C22×C4×Dic73C2212yes—
20C22×C4×Dic73C2212yes—
21C22×C4×Dic73C2211yescharacteristic
22#17988523C418yescomplemented
23C4⋊C4×Dic73—24no—
24#17988438—28no—
25C22⋊C4×Dic73—28no—
26C22⋊C4×Dic73—24no—
27C2×#14556498C2212yes—
28C2×#14556498C2212yes—
29C2×#14556498C2211yescharacteristic
30C2×#14556498C2211yescharacteristic
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2212yes—
34C2×#14556498C2211yescharacteristic
35C22×C4⋊C4×C73C2211yescharacteristic
36C22×C22⋊C4×C73C2212yes—
37C22×C22⋊C4×C73C2211yescharacteristic
38C2×#95×C73C2211yescharacteristic
39C2×#95×C73C2212yes—
40C23×Dic73—22no—
41C23×Dic73—24no—
42C23×Dic73—24no—
43C23×Dic73—22no—
44C23×Dic73Q814yes—
45C23×Dic73C2312yes—
46C23×Dic73C2311yescharacteristic
47C23×Dic73C2311yescharacteristic
48C23×Dic73C2311yescharacteristic
49C23×Dic73C2312yes—
50C23×Dic73C2311yescharacteristic
51C2×C4⋊Dic73—22no—
52C2×C4⋊Dic73—22no—
53C2×C4⋊Dic73C2×C414yescomplemented
54C2×#11924007—24no—
55C2×#11924007—24no—
56C2×#11924007—24no—
57C2×#11924007—24no—
58C2×#11924007—24no—
59C2×#11924007C2×C418yescomplemented
60C2×C22⋊Dic73—22no—
61C2×C22⋊Dic73—24no—
62C2×C22⋊Dic73—22no—
63C2×C22⋊Dic73—24no—
64C2×C22⋊Dic73—22no—
65C2×C22⋊Dic73C2×C418yes—
66C2×C4×Dic73—24no—
67C2×C4×Dic73—24no—
68C2×C4×Dic73—24no—
69C2×C4×Dic73—22no—
70C2×C4×Dic73—24no—
71C2×C4×Dic73—24no—
72C2×C4×Dic73—24no—
73C2×C4×Dic73—24no—
74C2×C4×Dic73D418yes—
75C2×C4×Dic73D414yes—
76#14556498—22no—
77#14556498—24no—
78#14556498—24no—
79#14556498—24no—
80#14556498—24no—
81#14556498—22no—
82#14556498—24no—
83#14556498C2×C414yescomplemented
84C25×C73C2311yescharacteristic
85C23×C4×C73C2311yescharacteristic
86C23×C4×C73C2312yes—
87C23×C4×C73C2312yes—
88C23×C4×C73C2311yescharacteristic
89C2×C4⋊C4×C73—22no—
90C2×C4⋊C4×C73—24no—
91C2×C22⋊C4×C73—24no—
92C2×C22⋊C4×C73—24no—
93C2×C22⋊C4×C73—22no—
94C2×C22⋊C4×C73—24no—
95C2×C22⋊C4×C73—22no—
96C2×C22⋊C4×C73C2×C414yes—
97#95×C73—24no—
98#95×C73—22no—
99C22×Dic73—22no—
100C22×Dic73—22no—
101C22×Dic73—22no—
102C22×Dic73—22no—
103C22×Dic73—22no—
104C22×Dic73—21no—
105C22×Dic73—22no—
106C22×Dic73—21no—
107C22×Dic73—22no—
108C22×Dic73—22no—
109C22×Dic73—22no—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—21no—
114C22×Dic73—21no—
115C22×Dic73—21no—
116C22×Dic73—22no—
117C22×Dic73—22no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—21no—
121C22×Dic73—28no—
122C22×Dic73—216no—
123C22×Dic73—48no—
124C22×Dic73—44no—
125C22×Dic73—44no—
126C22×Dic73C22×C414yes—
127C22×Dic73C22×C414yes—
128C22×Dic73C2×D412yes—
129C22×Dic73C2×D414yes—
130C22×Dic73C2×D414yes—
131C22×Dic73C2×D412yes—
132C22×Dic73C2×Q812yes—
133C22×Dic73C2×Q812yes—
134C22×Dic73#4112yes—
135C22×Dic73#4114yes—
136C22×Dic73#4114yes—
137C22×Dic73#4112yes—
138C4⋊Dic73—28no—
139#11924007—216no—
140#11924007—216no—
141C22⋊Dic73—28no—
142C22⋊Dic73—216no—
143C4×Dic73—216no—
144C4×Dic73—28no—
145C4×Dic73—44no—
146C4×Dic73—48no—
147C24×C73—21no—
148C24×C73—22no—
149C24×C73—21no—
150C24×C73—22no—
151C24×C73—22no—
152C24×C73—21no—
153C24×C73—22no—
154C24×C73—21no—
155C24×C73C22×C412yes—
156C24×C73C2×Q812yes—
157C24×C73#4112yes—
158C24×C73C2411yescharacteristic
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—21no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—22no—
167C22×C4×C73—21no—
168C22×C4×C73—22no—
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—22no—
176C22×C4×C73—22no—
177C22×C4×C73C22×C412yes—
178C22×C4×C73C22×C412yes—
179C22×C4×C73C2×D414yes—
180C22×C4×C73C2×D412yes—
181C22×C4×C73#4114yes—
182C22×C4×C73#4112yes—
183C4⋊C4×C73—28no—
184C22⋊C4×C73—216no—
185C22⋊C4×C73—28no—
186C23×C73—22no—
187C23×C73—24no—
188C23×C73—24no—
189C23×C73—22no—
190C23×C73—22no—
191C23×C73—24no—
192C23×C73—24no—
193C23×C73—22no—
194C23×C73—24no—
195C23×C73—22no—
196C23×C73—22no—
197C23×C73—24no—
198C23×C73—42no—
199C23×C73—41no—
200C23×C73—41no—
201C23×C73—42no—
202C23×C73—42no—
203C23×C73—42no—
204C23×C73—42no—
205C23×C73—41no—
206C23×C73—42no—
207C23×C73—41no—
208C23×C73C4×Q814yes—
209C23×C73C23×C411yescharacteristic
210C23×C73C22×D412yes—
211C23×C73C22×D411yescharacteristic
212C23×C73C22×Q811yescharacteristic
213C23×C73C2×#4112yes—
214C23×C73C2×#4112yes—
215C23×C73C2×#4111yescharacteristic
216C23×C73C2×#4111yescharacteristic
217C23×C73C2×#4111yescharacteristic
218C23×C73C2×#4111yescharacteristic
219C23×C73C2×#4112yes—
220C2×C4×C73—24no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—24no—
224C2×C4×C73—22no—
225C2×C4×C73—24no—
226C2×C4×C73—24no—
227C2×C4×C73—24no—
228C2×C4×C73—24no—
229C2×C4×C73—24no—
230C2×C4×C73—24no—
231C2×C4×C73—24no—
232C2×C4×C73—42no—
233C2×C4×C73—42no—
234C2×C4×C73—42no—
235C2×C4×C73—42no—
236C2×C4×C73—42no—
237C2×C4×C73—42no—
238C2×C4×C73—42no—
239C2×C4×C73—41no—
240C2×C4×C73—42no—
241C2×C4×C73—42no—
242C2×C4×C73—41no—
243C2×C4×C73C4×D414yes—
244C2×Dic73—24no—
245C2×Dic73—24no—
246C2×Dic73—24no—
247C2×Dic73—24no—
248C2×Dic73—24no—
249C2×Dic73—24no—
250C2×Dic73—24no—
251C2×Dic73—24no—
252C2×Dic73—24no—
253C2×Dic73—22no—
254C2×Dic73—24no—
255C2×Dic73—22no—
256C2×Dic73—22no—
257C2×Dic73—22no—
258C2×Dic73—24no—
259C2×Dic73—22no—
260C2×Dic73—42no—
261C2×Dic73—41no—
262C2×Dic73—42no—
263C2×Dic73—41no—
264C2×Dic73—42no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—41no—
268C2×Dic73—41no—
269C2×Dic73—42no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73—416no—
273C2×Dic73C4×D418yes—
274C2×Dic73C22⋊Q814yes—
275C2×Dic73C22⋊Q818yes—
276C2×Dic73#12318yes—
277C2×Dic73#12314yes—
278C2×Dic73#12714yes—
279C22×C73—24no—
280C22×C73—28no—
281C22×C73—42no—
282C22×C73—42no—
283C22×C73—44no—
284C22×C73—42no—
285C22×C73—42no—
286C22×C73—44no—
287C22×C73—42no—
288C22×C73—42no—
289C22×C73—44no—
290C22×C73C2×C4×D412yes—
291C22×C73C2×C4×D411yescharacteristic
292C22×C73C2×C4×Q811yescharacteristic
293C22×C73C4×#4112yes—
294C22×C73C4×#4111yescharacteristic
295C22×C73C2×C22⋊Q811yescharacteristic
296C22×C73C2×C22⋊Q812yes—
297C22×C73C2×#12312yes—
298C22×C73C2×#12311yescharacteristic
299C22×C73#52512yes—
300C22×C73#52511yescharacteristic
301C22×C73#52511yescharacteristic
302C22×C73#52512yes—
303C22×C73#52912yes—
304C22×C73#52912yes—
305C22×C73#52912yes—
306C22×C73#52911yescharacteristic
307C22×C73#52912yes—
308C22×C73C2×#12711yescharacteristic
309C22×C73#53212yes—
310C22×C73#53211yescharacteristic
311C22×C73#53312yes—
312C22×C73#53311yescharacteristic
313C4×C73—24no—
314C4×C73—42no—
315C4×C73—44no—
316C4×C73—44no—
317Dic73—28no—
318Dic73—44no—
319Dic73—42no—
320Dic73—42no—
321Dic73—42no—
322Dic73#39918yescomplemented
323C4×#399—731nomaximal, Sylow
324C2×C73—44no—
325C2×C73C4×#12311yescharacteristic
326C2×C73C4×#12312yes—
327C2×C73C4×C22⋊Q811yescharacteristic
328C2×C73C4×C22⋊Q812yes—
329C2×C73C4×#12711yescharacteristic
330C2×C73C2×#39911yescharacteristic
331C2×C73#239512yes—
332C2×C73#239511yescharacteristic
333C2×C73#239611yescharacteristic
334C2×C73#239612yes—
335C2×C73#239711yescharacteristic
336C2×#399—731no—
337C2×C4×C4⋊C4—732no—
338C2×C4×C4⋊C4—731no—
339C2×C4×C22⋊C4—732no—
340C2×C4×C22⋊C4—731no—
341#1445—732no—
342#1445—731no—
343#1444—731no—
344#1439—732no—
345#1439D7311yescharacteristic
346C4×#95—731no—
347C73C4×#39911yescharacteristic, complemented, Sylow
348C24×C4—731no—
349C22×C4⋊C4—732no—
350C22×C4⋊C4—731no—
351C22×C4⋊C4—732no—
352C22×C4⋊C4C2×D7311yescharacteristic
353C22×C22⋊C4—732no—
354C22×C22⋊C4—731no—
355C22×C22⋊C4C2×D7312yes—
356C22×C22⋊C4C2×D7311yescharacteristic
357C22×C42—732no—
358C22×C42—731no—
359C22×C42—731no—
360C22×C42—732no—
361#399—738no—
362C4×C4⋊C4—1468no—
363C4×C4⋊C4—1464no—
364C4×C22⋊C4—1468no—
365C4×C22⋊C4—1464no—
366C2×#95—731no—
367C2×#95—732no—
368C2×#95—732no—
369C2×#95—732no—
370C2×#95—732no—
371C2×#95—731no—
372C2×#95—731no—
373C2×#95—732no—
374C2×#95C2×D7311yescharacteristic
375C2×#95C2×D7312yes—
376C25C22×D7311yescharacteristic
377C23×C4—731no—
378C23×C4—732no—
379C23×C4—734no—
380C23×C4—732no—
381C23×C4—731no—
382C23×C4—731no—
383C23×C4—731no—
384C23×C4—1464no—
385C23×C4—1462no—
386C23×C4—1464no—
387C23×C4—1462no—
388C23×C4C22×D7311yescharacteristic
389C23×C4C22×D7312yes—
390C23×C4C22×D7312yes—
391C23×C4C22×D7311yescharacteristic
392C2×C4⋊C4—22no—
393C2×C4⋊C4—24no—
394C2×C4⋊C4—738no—
395C2×C4⋊C4—734no—
396C2×C4⋊C4—1464no—
397C2×C4⋊C4—1464no—
398C2×C4⋊C4—1464no—
399C2×C4⋊C4—1464no—
400C2×C4⋊C4—1462no—
401C2×C4⋊C4—1464no—
402C2×C4⋊C4—1462no—
403C2×C22⋊C4—24no—
404C2×C22⋊C4—24no—
405C2×C22⋊C4—22no—
406C2×C22⋊C4—24no—
407C2×C22⋊C4—22no—
408C2×C22⋊C4—738no—
409C2×C22⋊C4—1464no—
410C2×C22⋊C4—1462no—
411C2×C22⋊C4—1464no—
412C2×C22⋊C4—1462no—
413C2×C22⋊C4—1462no—
414C2×C22⋊C4C4×D7314yes—
415C2×C42—738no—
416C2×C42—734no—
417C2×C42—1462no—
418C2×C42—1464no—
419C2×C42—1464no—
420C2×C42—1464no—
421C2×C42—1464no—
422C2×C42—1464no—
423C2×C42—1464no—
424C2×C42—1464no—
425#95—24no—
426#95—22no—
427#95—734no—
428#95—1464no—
429#95—1464no—
430#95—1462no—
431#95—1464no—
432#95—1464no—
433#95—1462no—
434#95—1464no—
435C24—22no—
436C24—21no—
437C24—22no—
438C24—21no—
439C24—22no—
440C24—21no—
441C24—21no—
442C24—22no—
443C24C2×C4×D7312yes—
444C24Q8×D7312yes—
445C24#1192402712yes—
446C24C23×D7311yescharacteristic
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—21no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—22no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—22no—
463C22×C4—22no—
464C22×C4—22no—
465C22×C4—2924no—
466C22×C4—2928no—
467C22×C4—2924no—
468C22×C4—734no—
469C22×C4—732no—
470C22×C4—732no—
471C22×C4—734no—
472C22×C4—732no—
473C22×C4—734no—
474C22×C4—734no—
475C22×C4—734no—
476C22×C4—734no—
477C22×C4—732no—
478C22×C4—732no—
479C22×C4—732no—
480C22×C4—1468no—
481C22×C4—14616no—
482C22×C4—1462no—
483C22×C4—1461no—
484C22×C4—1462no—
485C22×C4—1461no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1462no—
489C22×C4—1462no—
490C22×C4—1462no—
491C22×C4—1461no—
492C22×C4—1462no—
493C22×C4—1462no—
494C22×C4—1462no—
495C22×C4—1461no—
496C22×C4—1461no—
497C22×C4—1461no—
498C22×C4—1462no—
499C22×C4—1462no—
500C22×C4—1462no—
501C22×C4—1462no—
502C22×C4—1462no—
503C22×C4—1462no—
504C22×C4C2×C4×D7312yes—
505C22×C4C2×C4×D7312yes—
506C22×C4D4×D7314yes—
507C22×C4D4×D7312yes—
508C22×C4#1192402714yes—
509C22×C4#1192402712yes—
510C4⋊C4—28no—
511C4⋊C4—14616no—
512C4⋊C4—14616no—
513C4⋊C4—1468no—
514C22⋊C4—216no—
515C22⋊C4—28no—
516C22⋊C4—14616no—
517C22⋊C4—1468no—
518C42—2924no—
519C42—2928no—
520C42—1468no—
521C42—14616no—
522C23—22no—
523C23—24no—
524C23—24no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—24no—
533C23—22no—
534C23—42no—
535C23—41no—
536C23—42no—
537C23—41no—
538C23—42no—
539C23—41no—
540C23—41no—
541C23—42no—
542C23—42no—
543C23—42no—
544C23#1455655614yes—
545C23C22×C4×D7311yescharacteristic
546C23C2×D4×D7312yes—
547C23C2×D4×D7311yescharacteristic
548C23C2×Q8×D7311yescharacteristic
549C23#41×D7311yescharacteristic
550C23#41×D7312yes—
551C23#41×D7312yes—
552C23#41×D7311yescharacteristic
553C23C2×#1192402711yescharacteristic
554C23C2×#1192402712yes—
555C23C2×#1192402711yescharacteristic
556C2×C4—24no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—24no—
560C2×C4—24no—
561C2×C4—24no—
562C2×C4—22no—
563C2×C4—24no—
564C2×C4—24no—
565C2×C4—24no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—41no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—41no—
572C2×C4—42no—
573C2×C4—42no—
574C2×C4—42no—
575C2×C4—42no—
576C2×C4—42no—
577C2×C4—42no—
578C2×C4—42no—
579C2×C4—2922no—
580C2×C4—2922no—
581C2×C4—2921no—
582C2×C4—2921no—
583C2×C4—2922no—
584C2×C4—2922no—
585C2×C4—2922no—
586C2×C4—2922no—
587C2×C4—2921no—
588C2×C4—2922no—
589C2×C4—2921no—
590C2×C4—2922no—
591C2×C4—29216no—
592C2×C4—734no—
593C2×C4—734no—
594C2×C4—738no—
595C2×C4—738no—
596C2×C4—734no—
597C2×C4—738no—
598C2×C4—1462no—
599C2×C4—1464no—
600C2×C4—1464no—
601C2×C4—1464no—
602C2×C4—1462no—
603C2×C4—1464no—
604C2×C4—1462no—
605C2×C4—1462no—
606C2×C4—1462no—
607C2×C4—1464no—
608C2×C4—1464no—
609C2×C4—1464no—
610C2×C4—1464no—
611C2×C4—1464no—
612C2×C4—1464no—
613C2×C4—1464no—
614C2×C4#1455655114yes—
615C22—24no—
616C22—28no—
617C22—42no—
618C22—42no—
619C22—44no—
620C22—42no—
621C22—42no—
622C22—44no—
623C22—42no—
624C22—42no—
625C22—44no—
626C22C4×D4×D7312yes—
627C22C2×#1455655111yescharacteristic
628C22C2×#1455655611yescharacteristic
629C22C4×#1192402712yes—
630C22#1798905511yescharacteristic
631C22C22⋊Q8×D7311yescharacteristic
632C22C22⋊Q8×D7312yes—
633C22#123×D7312yes—
634C22#123×D7311yescharacteristic
635C22#1798909312yes—
636C22#1798909311yescharacteristic
637C22#1798909711yescharacteristic
638C22#1798909712yes—
639C22#1798911412yes—
640C22#1798911612yes—
641C22#1798911811yescharacteristic
642C22#1798911812yes—
643C22#1798912012yes—
644C22#127×D7311yescharacteristic
645C22#1798913212yes—
646C22#1798913611yescharacteristic
647C22#1798913912yes—
648C22#1798913911yescharacteristic
649C4—24no—
650C4—42no—
651C4—44no—
652C4—44no—
653C4—2924no—
654C4—2922no—
655C4—2922no—
656C4—2922no—
657C4—738no—
658C4—1468no—
659C2—44no—
660C2#2261245611yescharacteristic
661C2#2261245812yes—
662C2#2261245911yescharacteristic
663C2#2261246312yes—
664C2#2261249511yescharacteristic
665C2#399×D7311yescharacteristic
666C2#2261318712yes—
667C2#2261319311yescharacteristic
668C2#2261319811yescharacteristic
669C2#2261320612yes—
670C2#2261320911yescharacteristic
subgroupG/H
center, FrattiniC24C23×D73
commutatorC23×C73C23×C4
FittingC73×#1439C2
socle, cosocleC24×C73C24
prime radicalC25×C73C23
hypercenter#1439D73
hypocenterC73C4×#399
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C24 ◁ #1439

Complements

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

NH ≅ G/Ntype
2. C2×C4⋊C4×Dic73C2semidirect
3. C2×#17988438C2semidirect
6. #22608198C2semidirect
7. #22608197C2semidirect
8. #22608194C2semidirect
10. #95×Dic73C2semidirect
22. #17988523C4semidirect
53. C2×C4⋊Dic73C2×C4semidirect
59. C2×#11924007C2×C4semidirect
83. #14556498C2×C4semidirect
322. Dic73#399semidirect
347. C73C4×#399semidirect

Maximal subgroups

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

subgroupindex
C2×#179885232
C2×C4⋊C4×Dic732
C2×#179884382
C2×C22⋊C4×Dic732
#226081982
#226081972
#226081942
#226081742
#95×Dic732
C73×#14392
C4×#39973