Finite Group Databasebeta

← C4×#17988522 #33007811

Subgroups — C4×#17988522 #33007811

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×#17988522C211yesmaximal, characteristic
2C2×C4×#11924007C212yesmaximal, complemented
3C2×C4×#11924005C212yesmaximal
4C2×C4×#11924005C211yesmaximal, characteristic
5#22608199C211yesmaximal, characteristic, complemented
6#22608197C212yesmaximal, complemented
7#22608194C211yesmaximal, characteristic, complemented
8#22608175C212yesmaximal
9#22608173C211yesmaximal, characteristic
10C4×#14556498C211yesmaximal, characteristic, complemented
11C2×C4×C4⋊C4×C73C211yesmaximal, characteristic, complemented
12C23×C4×D73C2211yescharacteristic
13C22×#11924007C2212yes—
14C22×#11924007C2212yes—
15C22×#11924005C2212yes—
16C22×#11924005C2211yescharacteristic
17C22×#11924005C2211yescharacteristic
18C22×#11924005C2212yes—
19C22×C4×Dic73C2211yescharacteristic
20C22×C4×Dic73C2212yes—
21#17988522C418yesdirect product factor
22C4×#11924007—28no—
23C4×#11924005—28no—
24C4×#11924005—24no—
25C2×#14556498C2211yescharacteristic
26C2×#14556498C2212yes—
27C2×#14556498C2212yes—
28C2×#14556498C2211yescharacteristic
29C2×#14556498C2211yescharacteristic
30C2×#14556498C2212yes—
31C2×#14556498C2212yes—
32C2×#14556498C2212yes—
33C2×#14556498C2211yescharacteristic
34C22×C4⋊C4×C73C2211yescharacteristic
35C22×C4⋊C4×C73C2211yescharacteristic
36C22×C42×C73C2212yes—
37C22×C42×C73C2211yescharacteristic
38C4×C4⋊C4×C73—24no—
39C2×#95×C73C2212yes—
40C24×D73C2311yescharacteristic
41C23×Dic73C2311yescharacteristic
42C23×Dic73C2312yes—
43C23×Dic73C2311yescharacteristic
44C23×Dic73C2311yescharacteristic
45C23×Dic73C2312yes—
46C22×C4×D73—24no—
47C22×C4×D73—22no—
48C22×C4×D73—24no—
49C22×C4×D73—22no—
50C22×C4×D73Q814yes—
51C2×#11924007—24no—
52C2×#11924007—24no—
53C2×#11924007—24no—
54C2×#11924007—24no—
55C2×#11924007—24no—
56C2×#11924007C2×C418yescomplemented
57C2×#11924005—24no—
58C2×#11924005—22no—
59C2×#11924005—24no—
60C2×#11924005—24no—
61C2×#11924005—22no—
62C2×#11924005—24no—
63C2×#11924005—22no—
64C2×#11924005—22no—
65C2×#11924005—24no—
66C2×#11924005—22no—
67C2×#11924005C2×C418yes—
68C2×#11924005C2×C414yes—
69C2×C4×Dic73—22no—
70C2×C4×Dic73—24no—
71C2×C4×Dic73—24no—
72C2×C4×Dic73—24no—
73C2×C4×Dic73D418yes—
74#14556498—24no—
75#14556498—24no—
76#14556498—24no—
77#14556498—24no—
78#14556498—24no—
79#14556498—22no—
80#14556498—22no—
81#14556498—22no—
82#14556498C2×C414yescomplemented
83C23×C4×C73C2312yes—
84C23×C4×C73C2311yescharacteristic
85C23×C4×C73C2312yes—
86C23×C4×C73C2311yescharacteristic
87C23×C4×C73C2311yescharacteristic
88C2×C4⋊C4×C73—22no—
89C2×C4⋊C4×C73—22no—
90C2×C4⋊C4×C73—24no—
91C2×C4⋊C4×C73—22no—
92C2×C4⋊C4×C73C2×C414yescomplemented
93C2×C42×C73—24no—
94C2×C42×C73—24no—
95C2×C42×C73—24no—
96C2×C42×C73—24no—
97C2×C42×C73D414yes—
98#95×C73—24no—
99C23×D73—22no—
100C23×D73—21no—
101C23×D73—22no—
102C23×D73—21no—
103C23×D73—21no—
104C23×D73—22no—
105C23×D73—22no—
106C23×D73—21no—
107C23×D73C22×C412yes—
108C23×D73C2×Q812yes—
109C23×D73#4112yes—
110C22×Dic73—22no—
111C22×Dic73—22no—
112C22×Dic73—22no—
113C22×Dic73—22no—
114C22×Dic73—22no—
115C22×Dic73—21no—
116C22×Dic73—21no—
117C22×Dic73—21no—
118C22×Dic73—22no—
119C22×Dic73—22no—
120C22×Dic73—22no—
121C22×Dic73—22no—
122C22×Dic73—22no—
123C22×Dic73—22no—
124C22×Dic73—22no—
125C22×Dic73—21no—
126C22×Dic73—21no—
127C22×Dic73—22no—
128C22×Dic73—22no—
129C22×Dic73—22no—
130C22×Dic73—22no—
131C22×Dic73—21no—
132C22×Dic73C22×C412yes—
133C22×Dic73C22×C414yes—
134C22×Dic73C2×D414yes—
135C22×Dic73C2×D414yes—
136C22×Dic73C2×Q812yes—
137C22×Dic73#4112yes—
138C22×Dic73#4114yes—
139C2×C4×D73—28no—
140C2×C4×D73—216no—
141C2×C4×D73—44no—
142C2×C4×D73—48no—
143C2×C4×D73—44no—
144#11924007—216no—
145#11924007—216no—
146#11924005—216no—
147#11924005—28no—
148#11924005—28no—
149#11924005—216no—
150C4×Dic73—216no—
151C4×Dic73—44no—
152C24×C73C2411yescharacteristic
153C22×C4×C73—22no—
154C22×C4×C73—22no—
155C22×C4×C73—22no—
156C22×C4×C73—22no—
157C22×C4×C73—22no—
158C22×C4×C73—22no—
159C22×C4×C73—22no—
160C22×C4×C73—22no—
161C22×C4×C73—22no—
162C22×C4×C73—22no—
163C22×C4×C73—22no—
164C22×C4×C73—22no—
165C22×C4×C73—22no—
166C22×C4×C73—21no—
167C22×C4×C73—21no—
168C22×C4×C73—22no—
169C22×C4×C73—22no—
170C22×C4×C73—22no—
171C22×C4×C73C22×C414yes—
172C22×C4×C73C22×C412yes—
173C22×C4×C73C2×D414yes—
174C22×C4×C73C2×D412yes—
175C22×C4×C73C2×D412yes—
176C22×C4×C73C2×D412yes—
177C22×C4×C73C2×Q812yes—
178C22×C4×C73#4114yes—
179C22×C4×C73#4112yes—
180C22×C4×C73#4114yes—
181C22×C4×C73#4112yes—
182C4⋊C4×C73—28no—
183C4⋊C4×C73—28no—
184C42×C73—28no—
185C42×C73—48no—
186C23×C73C23×C411yescharacteristic
187C23×C73C22×D412yes—
188C23×C73C22×D411yescharacteristic
189C23×C73C22×Q811yescharacteristic
190C23×C73C2×#4112yes—
191C23×C73C2×#4111yescharacteristic
192C23×C73C2×#4111yescharacteristic
193C23×C73C2×#4111yescharacteristic
194C23×C73C2×#4112yes—
195C23×C73C2×#4112yes—
196C23×C73C2×#4111yescharacteristic
197C22×D73—24no—
198C22×D73—22no—
199C22×D73—24no—
200C22×D73—22no—
201C22×D73—22no—
202C22×D73—24no—
203C22×D73—22no—
204C22×D73—24no—
205C22×D73—22no—
206C22×D73—24no—
207C22×D73—24no—
208C22×D73—22no—
209C22×D73—41no—
210C22×D73—42no—
211C22×D73—42no—
212C22×D73—41no—
213C22×D73—42no—
214C22×D73—42no—
215C22×D73—42no—
216C22×D73—41no—
217C22×D73—41no—
218C22×D73—42no—
219C22×D73C4×Q814yes—
220C2×C4×C73—24no—
221C2×C4×C73—22no—
222C2×C4×C73—22no—
223C2×C4×C73—24no—
224C2×C4×C73—24no—
225C2×C4×C73—24no—
226C2×C4×C73—24no—
227C2×C4×C73—22no—
228C2×C4×C73—24no—
229C2×C4×C73—22no—
230C2×C4×C73—24no—
231C2×C4×C73—24no—
232C2×C4×C73—22no—
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—42no—
240C2×C4×C73—42no—
241C2×C4×C73—42no—
242C2×C4×C73—42no—
243C2×C4×C73C4×D414yes—
244C2×C4×C73C22⋊Q814yes—
245C2×C4×C73C22⋊Q818yes—
246C2×C4×C73#12314yes—
247C2×C4×C73#12318yes—
248C2×C4×C73#12714yes—
249C2×Dic73—24no—
250C2×Dic73—24no—
251C2×Dic73—24no—
252C2×Dic73—24no—
253C2×Dic73—24no—
254C2×Dic73—24no—
255C2×Dic73—22no—
256C2×Dic73—24no—
257C2×Dic73—22no—
258C2×Dic73—24no—
259C2×Dic73—24no—
260C2×Dic73—24no—
261C2×Dic73—22no—
262C2×Dic73—24no—
263C2×Dic73—24no—
264C2×Dic73—41no—
265C2×Dic73—42no—
266C2×Dic73—42no—
267C2×Dic73—41no—
268C2×Dic73—42no—
269C2×Dic73—42no—
270C2×Dic73—42no—
271C2×Dic73—42no—
272C2×Dic73—41no—
273C2×Dic73—41no—
274C2×Dic73—41no—
275C2×Dic73—41no—
276C2×Dic73—42no—
277C2×Dic73C4×D418yes—
278C4×D73—416no—
279C22×C73C2×C4×D412yes—
280C22×C73C2×C4×D411yescharacteristic
281C22×C73C2×C4×Q811yescharacteristic
282C22×C73C4×#4112yes—
283C22×C73C4×#4111yescharacteristic
284C22×C73C2×C22⋊Q811yescharacteristic
285C22×C73C2×C22⋊Q812yes—
286C22×C73C2×#12311yescharacteristic
287C22×C73C2×#12312yes—
288C22×C73#52511yescharacteristic
289C22×C73#52512yes—
290C22×C73#52512yes—
291C22×C73#52511yescharacteristic
292C22×C73#52912yes—
293C22×C73#52911yescharacteristic
294C22×C73#52912yes—
295C22×C73#52912yes—
296C22×C73#52912yes—
297C22×C73C2×#12711yescharacteristic
298C22×C73#53211yescharacteristic
299C22×C73#53212yes—
300C22×C73#53312yes—
301C22×C73#53311yescharacteristic
302C2×D73—24no—
303C2×D73—28no—
304C2×D73—42no—
305C2×D73—42no—
306C2×D73—42no—
307C2×D73—44no—
308C2×D73—42no—
309C2×D73—44no—
310C2×D73—42no—
311C2×D73—42no—
312C2×D73—44no—
313C4×C73—24no—
314C4×C73—44no—
315C4×C73—42no—
316C4×C73—44no—
317C4×C73#39918yescomplemented
318Dic73—28no—
319Dic73—42no—
320Dic73—42no—
321Dic73—44no—
322Dic73—42no—
323C4×#399—731nomaximal, Sylow
324C2×C73C4×#12311yescharacteristic
325C2×C73C4×#12312yes—
326C2×C73C4×C22⋊Q811yescharacteristic
327C2×C73C4×C22⋊Q812yes—
328C2×C73C4×#12711yescharacteristic
329C2×C73C2×#39911yescharacteristic
330C2×C73#239511yescharacteristic
331C2×C73#239512yes—
332C2×C73#239611yescharacteristic
333C2×C73#239612yes—
334C2×C73#239711yescharacteristic
335D73—44no—
336C2×#399—731no—
337C2×C4×C4⋊C4—732no—
338C2×C4×C4⋊C4D7311yescharacteristic, complemented
339C2×C4×C22⋊C4—732no—
340C2×C4×C22⋊C4—731no—
341#1445—732no—
342#1445—731no—
343#1444—731no—
344#1439—732no—
345#1439—731no—
346C4×#95—731no—
347C73C4×#39911yescharacteristic, complemented, Sylow
348C24×C4—731no—
349C22×C4⋊C4—732no—
350C22×C4⋊C4—732no—
351C22×C4⋊C4C2×D7311yescharacteristic
352C22×C4⋊C4C2×D7311yescharacteristic
353C22×C22⋊C4—732no—
354C22×C22⋊C4—731no—
355C22×C22⋊C4—732no—
356C22×C22⋊C4—731no—
357C22×C42—732no—
358C22×C42—731no—
359C22×C42C2×D7312yes—
360C22×C42C2×D7311yescharacteristic
361#399—738no—
362C4×C4⋊C4—24no—
363C4×C4⋊C4—1468no—
364C4×C22⋊C4—1468no—
365C4×C22⋊C4—1464no—
366C2×#95—731no—
367C2×#95—732no—
368C2×#95—731no—
369C2×#95—732no—
370C2×#95—732no—
371C2×#95—732no—
372C2×#95—731no—
373C2×#95—732no—
374C2×#95—731no—
375C2×#95C2×D7312yes—
376C25—731no—
377C23×C4—732no—
378C23×C4—731no—
379C23×C4—731no—
380C23×C4—732no—
381C23×C4—731no—
382C23×C4—734no—
383C23×C4—1464no—
384C23×C4—1462no—
385C23×C4—1464no—
386C23×C4—1462no—
387C23×C4C22×D7312yes—
388C23×C4C22×D7311yescharacteristic
389C23×C4C22×D7312yes—
390C23×C4C22×D7311yescharacteristic
391C23×C4C22×D7311yescharacteristic
392C2×C4⋊C4—22no—
393C2×C4⋊C4—22no—
394C2×C4⋊C4—24no—
395C2×C4⋊C4—22no—
396C2×C4⋊C4—738no—
397C2×C4⋊C4—1464no—
398C2×C4⋊C4—1464no—
399C2×C4⋊C4—1464no—
400C2×C4⋊C4—1464no—
401C2×C4⋊C4—1464no—
402C2×C4⋊C4C4×D7314yescomplemented
403C2×C22⋊C4—738no—
404C2×C22⋊C4—734no—
405C2×C22⋊C4—1464no—
406C2×C22⋊C4—1464no—
407C2×C22⋊C4—1464no—
408C2×C22⋊C4—1462no—
409C2×C22⋊C4—1462no—
410C2×C22⋊C4—1464no—
411C2×C22⋊C4—1462no—
412C2×C22⋊C4—1462no—
413C2×C22⋊C4—1464no—
414C2×C22⋊C4—1462no—
415C2×C42—24no—
416C2×C42—24no—
417C2×C42—24no—
418C2×C42—24no—
419C2×C42—738no—
420C2×C42—1464no—
421C2×C42—1462no—
422C2×C42—1464no—
423C2×C42—1464no—
424C2×C42C22⋊D7314yes—
425#95—24no—
426#95—734no—
427#95—1464no—
428#95—1464no—
429#95—1464no—
430#95—1464no—
431#95—1464no—
432#95—1462no—
433#95—1462no—
434#95—1462no—
435C24—732no—
436C24—732no—
437C24—732no—
438C24—1462no—
439C24—1461no—
440C24—1462no—
441C24—1461no—
442C24—1461no—
443C24—1462no—
444C24—1462no—
445C24—1461no—
446C24C23×D7311yescharacteristic
447C22×C4—22no—
448C22×C4—22no—
449C22×C4—22no—
450C22×C4—22no—
451C22×C4—22no—
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—22no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—22no—
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—734no—
470C22×C4—734no—
471C22×C4—732no—
472C22×C4—734no—
473C22×C4—732no—
474C22×C4—732no—
475C22×C4—1462no—
476C22×C4—1462no—
477C22×C4—1462no—
478C22×C4—1461no—
479C22×C4—1461no—
480C22×C4—1461no—
481C22×C4—1462no—
482C22×C4—1462no—
483C22×C4—1461no—
484C22×C4—1462no—
485C22×C4—1462no—
486C22×C4—1462no—
487C22×C4—1462no—
488C22×C4—1461no—
489C22×C4—1462no—
490C22×C4—1462no—
491C22×C4—1462no—
492C22×C4—1461no—
493C22×C4—1468no—
494C22×C4—14616no—
495C22×C4—1462no—
496C22×C4—1462no—
497C22×C4—1462no—
498C22×C4—1462no—
499C22×C4C2×C4×D7314yes—
500C22×C4C2×C4×D7312yes—
501C22×C4D4×D7314yes—
502C22×C4C2×C22⋊D7312yes—
503C22×C4C2×C22⋊D7312yes—
504C22×C4C2×C22⋊D7312yes—
505C22×C4Q8×D7312yes—
506C22×C4#1192402814yes—
507C22×C4#1192402812yes—
508C22×C4#1192402814yes—
509C22×C4#1192402912yes—
510C4⋊C4—28no—
511C4⋊C4—28no—
512C4⋊C4—14616no—
513C4⋊C4—14616no—
514C22⋊C4—14616no—
515C22⋊C4—1468no—
516C22⋊C4—14616no—
517C22⋊C4—1468no—
518C42—28no—
519C42—48no—
520C42—2924no—
521C42—14616no—
522C23—2922no—
523C23—2922no—
524C23—2921no—
525C23—2921no—
526C23—2922no—
527C23—2921no—
528C23—2922no—
529C23—2922no—
530C23—2921no—
531C23—2922no—
532C23—734no—
533C23—1462no—
534C23—1464no—
535C23—1462no—
536C23—1464no—
537C23—1462no—
538C23—1464no—
539C23—1462no—
540C23—1462no—
541C23—1464no—
542C23—1462no—
543C23—1464no—
544C23—1464no—
545C23C22×C4×D7311yescharacteristic
546C23C2×D4×D7312yes—
547C23C22×C22⋊D7311yescharacteristic
548C23C2×Q8×D7311yescharacteristic
549C23#41×D7312yes—
550C23#41×D7311yescharacteristic
551C23#41×D7311yescharacteristic
552C23C2×#1192402811yescharacteristic
553C23C2×#1192402812yes—
554C23C2×#1192402812yes—
555C23C2×#1192402911yescharacteristic
556C2×C4—24no—
557C2×C4—24no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—22no—
561C2×C4—24no—
562C2×C4—24no—
563C2×C4—24no—
564C2×C4—22no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—42no—
570C2×C4—42no—
571C2×C4—42no—
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—2922no—
582C2×C4—2921no—
583C2×C4—2921no—
584C2×C4—2921no—
585C2×C4—2921no—
586C2×C4—2922no—
587C2×C4—2921no—
588C2×C4—2922no—
589C2×C4—2922no—
590C2×C4—2921no—
591C2×C4—2922no—
592C2×C4—29216no—
593C2×C4—738no—
594C2×C4—1464no—
595C2×C4—1464no—
596C2×C4—1462no—
597C2×C4—1462no—
598C2×C4—1464no—
599C2×C4—1464no—
600C2×C4—1462no—
601C2×C4—1464no—
602C2×C4—1464no—
603C2×C4—1464no—
604C2×C4—1464no—
605C2×C4—1464no—
606C2×C4—1464no—
607C2×C4—1464no—
608C2×C4—1464no—
609C2×C4C4×C22⋊D7314yes—
610C2×C4#1455656714yes—
611C2×C4#1455656918yes—
612C2×C4#1455657414yes—
613C2×C4#1455657618yes—
614C2×C4#1455658714yes—
615C22—2922no—
616C22—2922no—
617C22—2922no—
618C22—2924no—
619C22—2922no—
620C22—2924no—
621C22—2922no—
622C22—2922no—
623C22—2924no—
624C22—1464no—
625C22—1468no—
626C22C4×D4×D7312yes—
627C22C2×C4×C22⋊D7311yescharacteristic
628C22C4×Q8×D7311yescharacteristic
629C22C4×#1192402812yes—
630C22C4×#1192402911yescharacteristic
631C22C2×#1455656711yescharacteristic
632C22C2×#1455656912yes—
633C22C2×#1455657411yescharacteristic
634C22C2×#1455657612yes—
635C22#1798909411yescharacteristic
636C22#1798909612yes—
637C22#1798909612yes—
638C22#1798909811yescharacteristic
639C22#1798911512yes—
640C22#1798911512yes—
641C22#1798912112yes—
642C22#1798912411yescharacteristic
643C22#1798912412yes—
644C22C2×#1455658711yescharacteristic
645C22#1798913311yescharacteristic
646C22#1798913412yes—
647C22#1798914112yes—
648C22#1798914111yescharacteristic
649C4—24no—
650C4—44no—
651C4—42no—
652C4—44no—
653C4—2922no—
654C4—2922no—
655C4—2924no—
656C4—2922no—
657C4—1468no—
658C4#1798852218yesdirect product factor
659C2—2924no—
660C2C4×#1455657411yescharacteristic
661C2C4×#1455657612yes—
662C2C4×#1455656711yescharacteristic
663C2C4×#1455656912yes—
664C2C4×#1455658711yescharacteristic
665C2C2×#1798852211yescharacteristic
666C2#2261318811yescharacteristic
667C2#2261319412yes—
668C2#2261320111yescharacteristic
669C2#2261320712yes—
670C2#2261321011yescharacteristic
subgroupG/H
centerC23×C4C22×D73
commutatorC23×C73C23×C4
FrattiniC24C23×D73
FittingC2×C4×C4⋊C4×C73C2
socle, cosocleC24×C73C24
prime radicalC24×D73C23
hypercenterC2×C4×C4⋊C4D73
hypocenterC73C4×#399
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23×C4 ◁ C2×C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C4×#11924007C2semidirect
5. #22608199C2semidirect
6. #22608197C2semidirect
7. #22608194C2semidirect
10. C4×#14556498C2semidirect
11. C2×C4×C4⋊C4×C73C2semidirect
21. #17988522C4direct (N × H)
56. C2×#11924007C2×C4semidirect
82. #14556498C2×C4semidirect
92. C2×C4⋊C4×C73C2×C4semidirect
317. C4×C73#399semidirect
338. C2×C4×C4⋊C4D73semidirect
347. C73C4×#399semidirect
402. C2×C4⋊C4C4×D73semidirect
658. C4#17988522direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179885222
C2×C4×#119240072
C2×C4×#119240052
#226081992
#226081972
#226081942
#226081752
#226081732
C4×#145564982
C2×C4×C4⋊C4×C732
C4×#39973