Finite Group Databasebeta

← #32993280

Subgroups — #32993280

Order 18688.

910 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#14556522C211yesmaximal, characteristic, complemented
2C2×#17988589C211yesmaximal, characteristic
3C2×#17988586C211yesmaximal, characteristic
4C2×#17988555C211yesmaximal, characteristic
5C2×#17988355C211yesmaximal, characteristic, complemented
6C2×#17988351C211yesmaximal, characteristic, complemented
7C2×#343×C73C211yesmaximal, characteristic, complemented
8C22×#11924028C2211yescharacteristic
9C22×#11924012C2211yescharacteristic
10C23×#10596831C2211yescharacteristic
11C22×C4⋊Dic73C2211yescharacteristic
12C2×#14556522—22no—
13C2×#14556522—22no—
14C2×#14556522—22no—
15#17988589—22no—
16#17988589—22no—
17#17988586—22no—
18#17988586—22no—
19C2×#14556516—21no—
20C2×#14556514—21no—
21C2×#14556513—21no—
22C2×#14556511—21no—
23#17988555—22no—
24#17988555C414yescomplemented
25C2×#14556503—22no—
26#17988355—22no—
27#17988351—22no—
28C22×M16×C73C2211yescharacteristic
29C22×M16×C73C2211yescharacteristic
30C22×C4⋊C4×C73C2211yescharacteristic
31#343×C73—22no—
32C2×#11924028—21no—
33C2×#11924028—21no—
34C2×#11924028—21no—
35C2×#11924028—21no—
36C2×#11924028—21no—
37C2×#11924028—21no—
38C2×#11924028—21no—
39C2×#11924028—21no—
40C2×#11924028—21no—
41C2×#11924028—21no—
42C2×#11924028C2×C412yes—
43C2×#11924028Q812yes—
44C22×Dic146D411yescharacteristic
45C22×D292D411yescharacteristic
46C22×C22⋊D73—21no—
47C23×Dic73—21no—
48C22×C4×D73—21no—
49C2×#11924012—21no—
50C2×#11924012—21no—
51C2×#11924012—21no—
52C2×#11924012—21no—
53C2×#11924012—21no—
54C2×#11924012D412yes—
55C22×#10596831—21no—
56C22×#10596831—21no—
57C22×#10596831—21no—
58C22×#10596831—21no—
59C22×#10596831—21no—
60C22×#10596831C2×C412yes—
61C22×#10596831D412yes—
62C22×#10596831D412yes—
63C2×C4⋊Dic73—21no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73—21no—
68C2×C4⋊Dic73D412yes—
69#14556522—28no—
70#14556516—24no—
71#14556514—24no—
72#14556513—24no—
73#14556511—24no—
74#14556503—28no—
75C23×C4×C73—21no—
76C23×C4×C73C2311yescharacteristic
77C2×M16×C73—21no—
78C2×M16×C73—21no—
79C2×M16×C73—21no—
80C2×M16×C73—21no—
81C2×M16×C73—21no—
82C2×M16×C73—21no—
83C2×M16×C73—21no—
84C2×M16×C73—21no—
85C2×M16×C73—21no—
86C2×M16×C73—21no—
87C2×M16×C73C2×C412yes—
88C2×M16×C73D412yes—
89C22×C8×C73—21no—
90C22×C8×C73—21no—
91C2×C4⋊C4×C73—21no—
92C2×C4⋊C4×C73—21no—
93C2×C4⋊C4×C73—21no—
94C2×C4⋊C4×C73—21no—
95C2×C4⋊C4×C73—21no—
96C2×C4⋊C4×C73D412yes—
97C23×D73—21no—
98#11924028—22no—
99#11924028—22no—
100#11924028—22no—
101#11924028—22no—
102#11924028—22no—
103#11924028—22no—
104#11924028—22no—
105#11924028—22no—
106#11924028—41no—
107#11924028—41no—
108#11924028—41no—
109#11924028—41no—
110#11924028—41no—
111#11924028—41no—
112#11924028—41no—
113#11924028—41no—
114C2×Dic146—21no—
115C2×Dic146—21no—
116C2×Dic146—21no—
117C2×Dic146—21no—
118C2×Dic146—21no—
119C2×Dic146C4⋊C412yes—
120C2×D292—21no—
121C2×D292—21no—
122C2×D292—21no—
123C2×D292—21no—
124C2×D292—21no—
125C2×D292C4⋊C412yes—
126C2×C22⋊D73—22no—
127C2×C22⋊D73—22no—
128C2×C22⋊D73—41no—
129C2×C22⋊D73—41no—
130C2×C22⋊D73—41no—
131C2×C22⋊D73—41no—
132C22×Dic73—21no—
133C22×Dic73—22no—
134C22×Dic73—41no—
135C22×Dic73—41no—
136C22×Dic73—82no—
137C2×C4×D73—22no—
138C2×C4×D73—22no—
139C2×C4×D73—41no—
140C2×C4×D73—41no—
141C2×C4×D73—41no—
142C2×C4×D73—41no—
143#11924012—22no—
144#11924012—22no—
145#11924012—41no—
146#11924012—41no—
147C2×#10596831—22no—
148C2×#10596831—22no—
149C2×#10596831—22no—
150C2×#10596831—22no—
151C2×#10596831—22no—
152C2×#10596831—22no—
153C2×#10596831—22no—
154C2×#10596831—41no—
155C2×#10596831—41no—
156C2×#10596831C22⋊C414yes—
157C4⋊Dic73—22no—
158C4⋊Dic73—22no—
159C4⋊Dic73—41no—
160C4⋊Dic73—41no—
161C24×C73#4111yescharacteristic
162C22×C4×C73—22no—
163C22×C4×C73—41no—
164C22×C4×C73—41no—
165C22×C4×C73—82no—
166C22×C4×C73C22×C411yescharacteristic
167C22×C4×C73C2×D411yescharacteristic
168C22×C4×C73C2×D411yescharacteristic
169C22×C4×C73C2×D411yescharacteristic
170C22×C4×C73C2×D411yescharacteristic
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×Q811yescharacteristic
175C22×C4×C73#4111yescharacteristic
176C22×C4×C73#4111yescharacteristic
177C22×C4×C73#4111yescharacteristic
178C22×C4×C73#4111yescharacteristic
179C22×C4×C73#4111yescharacteristic
180M16×C73—22no—
181M16×C73—22no—
182M16×C73—22no—
183M16×C73—22no—
184M16×C73—41no—
185M16×C73—41no—
186M16×C73—41no—
187M16×C73—41no—
188C2×C8×C73—22no—
189C2×C8×C73—22no—
190C2×C8×C73—41no—
191C2×C8×C73—41no—
192C2×C8×C73—41no—
193C2×C8×C73—41no—
194C4⋊C4×C73—22no—
195C4⋊C4×C73—22no—
196C4⋊C4×C73—41no—
197C4⋊C4×C73—41no—
198C23×C73—22no—
199C23×C73—22no—
200C23×C73#11711yescharacteristic
201C23×C73#12111yescharacteristic
202C23×C73#12111yescharacteristic
203C23×C73#12111yescharacteristic
204C23×C73C22⋊Q811yescharacteristic
205C23×C73#12611yescharacteristic
206C23×C73#12611yescharacteristic
207C22×D73—22no—
208C22×D73—41no—
209C22×D73—41no—
210C22×D73—42no—
211C22×D73—42no—
212C2×C4×C73—42no—
213C2×C4×C73—82no—
214C2×C4×C73—82no—
215C2×C4×C73—82no—
216C2×C4×C73C2×C22⋊C411yescharacteristic
217C2×C4×C73C2×C4⋊C411yescharacteristic
218C2×C4×C73C4×D411yescharacteristic
219C2×C4×C73C4×D411yescharacteristic
220C2×C4×C73C4×D411yescharacteristic
221C2×C4×C73C4×D411yescharacteristic
222C2×C4×C73#12011yescharacteristic
223C2×C4×C73#12011yescharacteristic
224C2×C4×C73#12011yescharacteristic
225C2×C4×C73#12111yescharacteristic
226C2×C4×C73#12111yescharacteristic
227C2×C4×C73#12111yescharacteristic
228C2×C4×C73#12111yescharacteristic
229C2×C4×C73#12111yescharacteristic
230C2×C4×C73C22⋊Q811yescharacteristic
231C2×C4×C73C22⋊Q811yescharacteristic
232C2×C4×C73C22⋊Q811yescharacteristic
233C2×C4×C73C22⋊Q811yescharacteristic
234C2×C4×C73C22⋊Q811yescharacteristic
235C2×C4×C73#12311yescharacteristic
236C2×C4×C73#12311yescharacteristic
237C2×C4×C73#12311yescharacteristic
238C2×C4×C73#12311yescharacteristic
239C2×C4×C73#12311yescharacteristic
240C2×C4×C73#12411yescharacteristic
241C2×C4×C73#12411yescharacteristic
242C2×C4×C73#12411yescharacteristic
243C2×C4×C73#12711yescharacteristic
244C22⋊D73—42no—
245C22⋊D73—42no—
246C22⋊D73—42no—
247C22⋊D73—42no—
248C2×Dic73—22no—
249C2×Dic73—41no—
250C2×Dic73—41no—
251C2×Dic73—42no—
252C2×Dic73—82no—
253C2×Dic73—82no—
254C2×Dic73—82no—
255D292—22no—
256D292—22no—
257D292—41no—
258D292—41no—
259C4×D73—42no—
260C4×D73—42no—
261C4×D73—42no—
262C4×D73—42no—
263Dic146—22no—
264Dic146—22no—
265Dic146—41no—
266Dic146—41no—
267C8×C73—42no—
268C8×C73—42no—
269#10596831—24no—
270#10596831—42no—
271C22×C73—22no—
272C22×C73—22no—
273C22×C73—22no—
274C22×C73—22no—
275C22×C73—21no—
276C22×C73—21no—
277C22×C73—21no—
278C22×C73—21no—
279C22×C73#38011yescharacteristic
280C22×C73#38811yescharacteristic
281C22×C73#38811yescharacteristic
282C22×C73#39211yescharacteristic
283C22×C73#39611yescharacteristic
284C22×C73#39611yescharacteristic
285C22×C73#39711yescharacteristic
286C22×C73#47112yes—
287C22×C73#47112yes—
288C2×D73—44no—
289C2×D73—42no—
290C2×D73—82no—
291C2×D73—81no—
292C2×D73—81no—
293C4×C73—82no—
294C4×C73#38511yescharacteristic
295C4×C73#38511yescharacteristic
296C4×C73#38611yescharacteristic
297C4×C73#39011yescharacteristic
298C4×C73#39311yescharacteristic
299C4×C73#39411yescharacteristic
300C4×C73#39411yescharacteristic
301C4×C73#39911yescharacteristic
302Dic73—42no—
303Dic73—82no—
304#11106—731nomaximal, Sylow
305C2×C73—21no—
306C2×C73—21no—
307C2×C73—21no—
308C2×C73—21no—
309C2×C73#143911yescharacteristic
310C2×C73#181512yes—
311C2×C73#201911yescharacteristic
312C2×C73#201911yescharacteristic
313C2×C73#207912yes—
314D73—82no—
315C22×#106—731no—
316C2×#417—731no—
317C2×#417—731no—
318C2×#408—731no—
319C2×#343—731no—
320C2×#343D7311yescharacteristic, complemented
321C2×#341—731no—
322C73#1110611yescharacteristic, complemented, Sylow
323C22×#41—731no—
324C22×M16—731no—
325C22×M16C2×D7311yescharacteristic
326C22×M16C2×D7311yescharacteristic
327C23×C8—731no—
328C22×C4⋊C4—731no—
329C22×C4⋊C4C2×D7311yescharacteristic
330C2×#106—1462no—
331C2×#106—1462no—
332C2×#106—1462no—
333#417—1462no—
334#417—1462no—
335#417—1462no—
336#417—1462no—
337C2×#103—1461no—
338C2×#103—1461no—
339C2×#102—1461no—
340C2×#102—1461no—
341#408—734no—
342#408—1462no—
343C2×C22⋊C8—1462no—
344#343—22no—
345#343—1462no—
346#341—1462no—
347C2×#41—732no—
348C2×#41—732no—
349C2×#41—1461no—
350C2×#41—1461no—
351C2×#41—1461no—
352C2×#41—1461no—
353C2×#41—1461no—
354C2×#41—1461no—
355C2×#41—1461no—
356C2×#41—1461no—
357C2×#41—1461no—
358C2×#41—1461no—
359C22×Q8—731no—
360C22×D4—731no—
361C22×D4—1461no—
362C23×C4—21no—
363C23×C4—1461no—
364C23×C4—1461no—
365C23×C4C22×D7311yescharacteristic
366C2×M16—21no—
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—21no—
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—21no—
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—732no—
377C2×M16—1461no—
378C2×M16—1461no—
379C2×M16—1461no—
380C2×M16—1461no—
381C2×M16—1461no—
382C2×M16C4×D7312yes—
383C2×M16C22⋊D7312yes—
384C22×C8—21no—
385C22×C8—21no—
386C22×C8—732no—
387C22×C8—732no—
388C22×C8—732no—
389C22×C8—1461no—
390C22×C8—1461no—
391C22×C8—1461no—
392C22×C8—1461no—
393C22×C8—1461no—
394C2×C4⋊C4—21no—
395C2×C4⋊C4—21no—
396C2×C4⋊C4—21no—
397C2×C4⋊C4—21no—
398C2×C4⋊C4—21no—
399C2×C4⋊C4—732no—
400C2×C4⋊C4—1461no—
401C2×C4⋊C4—1461no—
402C2×C4⋊C4—1461no—
403C2×C4⋊C4—1461no—
404C2×C4⋊C4—1461no—
405C2×C4⋊C4D29212yes—
406#106—1468no—
407#103—1464no—
408#103—1464no—
409#102—1464no—
410#102—1464no—
411C22⋊C8—1468no—
412C24—1461no—
413C24#1192402811yescharacteristic
414#41—2921no—
415#41—2921no—
416#41—2921no—
417#41—2921no—
418#41—2921no—
419#41—2921no—
420#41—2921no—
421#41—2921no—
422#41—1462no—
423#41—1462no—
424#41—1462no—
425#41—1462no—
426#41—1462no—
427#41—1462no—
428#41—1462no—
429#41—1462no—
430C2×Q8—732no—
431C2×Q8—1461no—
432C2×Q8—1461no—
433C2×Q8—1461no—
434C2×Q8—1461no—
435C2×Q8—1461no—
436C2×D4—2921no—
437C2×D4—2921no—
438C2×D4—2921no—
439C2×D4—2921no—
440C2×D4—732no—
441C2×D4—1462no—
442C2×D4—1462no—
443C2×D4—1461no—
444C2×D4—1461no—
445C2×D4—1461no—
446C2×D4—1461no—
447C2×D4—1461no—
448C22×C4—22no—
449C22×C4—41no—
450C22×C4—41no—
451C22×C4—82no—
452C22×C4—2921no—
453C22×C4—2921no—
454C22×C4—2921no—
455C22×C4—2921no—
456C22×C4—2921no—
457C22×C4—2921no—
458C22×C4—5842no—
459C22×C4—1461no—
460C22×C4—1462no—
461C22×C4—1462no—
462C22×C4—1462no—
463C22×C4C2×C4×D7311yescharacteristic
464C22×C4D4×D7311yescharacteristic
465C22×C4D4×D7311yescharacteristic
466C22×C4D4×D7311yescharacteristic
467C22×C4D4×D7311yescharacteristic
468C22×C4D4×D7311yescharacteristic
469C22×C4C2×C22⋊D7311yescharacteristic
470C22×C4C2×D29211yescharacteristic
471C22×C4Q8×D7311yescharacteristic
472C22×C4#1192402711yescharacteristic
473C22×C4#1192402811yescharacteristic
474C22×C4#1192402811yescharacteristic
475C22×C4#1192402811yescharacteristic
476C22×C4#1192402911yescharacteristic
477M16—22no—
478M16—22no—
479M16—22no—
480M16—22no—
481M16—41no—
482M16—41no—
483M16—41no—
484M16—41no—
485M16—2921no—
486M16—2921no—
487M16—1462no—
488M16—1462no—
489C2×C8—22no—
490C2×C8—22no—
491C2×C8—41no—
492C2×C8—41no—
493C2×C8—41no—
494C2×C8—41no—
495C2×C8—2921no—
496C2×C8—2921no—
497C2×C8—734no—
498C2×C8—1462no—
499C2×C8—1462no—
500C2×C8—1462no—
501C2×C8—1462no—
502C2×C8—1462no—
503C2×C8—1462no—
504C2×C8—1462no—
505C4⋊C4—22no—
506C4⋊C4—22no—
507C4⋊C4—41no—
508C4⋊C4—41no—
509C4⋊C4—2921no—
510C4⋊C4—2921no—
511C4⋊C4—1462no—
512C4⋊C4—1462no—
513C23—22no—
514C23—22no—
515C23—2921no—
516C23—2921no—
517C23—2922no—
518C23—2922no—
519C23—1462no—
520C23#1455654811yescharacteristic
521C23#1455656411yescharacteristic
522C23#1455656411yescharacteristic
523C23#1455656511yescharacteristic
524C23#1455656911yescharacteristic
525C23#1455658511yescharacteristic
526C23#1455658611yescharacteristic
527Q8—2921no—
528Q8—2921no—
529Q8—1462no—
530Q8—1462no—
531D4—2922no—
532D4—2922no—
533D4—2922no—
534D4—2922no—
535D4—2921no—
536D4—2921no—
537D4—1462no—
538D4—1462no—
539C2×C4—42no—
540C2×C4—82no—
541C2×C4—82no—
542C2×C4—82no—
543C2×C4—2921no—
544C2×C4—2921no—
545C2×C4—2922no—
546C2×C4—2922no—
547C2×C4—2922no—
548C2×C4—2922no—
549C2×C4—2922no—
550C2×C4—5842no—
551C2×C4—5842no—
552C2×C4—5842no—
553C2×C4—1462no—
554C2×C4C22⋊C4×D7311yescharacteristic
555C2×C4C4⋊C4×D7311yescharacteristic
556C2×C4#1455655111yescharacteristic
557C2×C4C4×C22⋊D7311yescharacteristic
558C2×C4#1455655311yescharacteristic
559C2×C4C4×D29211yescharacteristic
560C2×C4#1455655911yescharacteristic
561C2×C4#1455656111yescharacteristic
562C2×C4#1455656111yescharacteristic
563C2×C4#1455656311yescharacteristic
564C2×C4#1455656411yescharacteristic
565C2×C4#1455656411yescharacteristic
566C2×C4#1455656411yescharacteristic
567C2×C4#1455656611yescharacteristic
568C2×C4#1455656711yescharacteristic
569C2×C4#1455656911yescharacteristic
570C2×C4#1455656911yescharacteristic
571C2×C4#1455656911yescharacteristic
572C2×C4#1455657111yescharacteristic
573C2×C4#1455657311yescharacteristic
574C2×C4#1455657411yescharacteristic
575C2×C4#1455657611yescharacteristic
576C2×C4#1455657611yescharacteristic
577C2×C4#1455657611yescharacteristic
578C2×C4#1455657711yescharacteristic
579C2×C4#1455657711yescharacteristic
580C2×C4#1455657911yescharacteristic
581C2×C4#1455658711yescharacteristic
582C8—42no—
583C8—42no—
584C8—2922no—
585C8—1464no—
586C22—22no—
587C22—22no—
588C22—21no—
589C22—21no—
590C22—21no—
591C22—21no—
592C22—22no—
593C22—22no—
594C22—2922no—
595C22—2924no—
596C22—5841no—
597C22—5841no—
598C22—5842no—
599C22#1798844611yescharacteristic
600C22#1798847611yescharacteristic
601C22#1798847811yescharacteristic
602C22#1798849611yescharacteristic
603C22#1798850911yescharacteristic
604C22#1798851111yescharacteristic
605C22#1798851611yescharacteristic
606C22#1798884912yes—
607C22#1798885312yes—
608C4—82no—
609C4—2922no—
610C4—5842no—
611C4#1798845811yescharacteristic
612C4#1798846111yescharacteristic
613C4#1798846511yescharacteristic
614C4#1798848711yescharacteristic
615C4#1798850011yescharacteristic
616C4#1798850211yescharacteristic
617C4#1798850511yescharacteristic
618C4#1798852211yescharacteristic
619C2—21no—
620C2—21no—
621C2—21no—
622C2—21no—
623C2—5842no—
624C2#2260817511yescharacteristic
625C2#2260998912yes—
626C2#2261104411yescharacteristic
627C2#2261104711yescharacteristic
628C2#2261138012yes—
subgroupG/H
centerC23#14556564
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC2×#343×C73C2
socleC23×C73#121
prime radicalC22×#11924028C22
cosocleC23×C4×C73C23
hypercenterC2×#343D73
hypocenterC73#11106
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#343

Complements

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

NH ≅ G/Ntype
1. C22×#14556522C2semidirect
5. C2×#17988355C2semidirect
6. C2×#17988351C2semidirect
7. C2×#343×C73C2semidirect
24. #17988555C4semidirect
320. C2×#343D73semidirect
322. C73#11106semidirect

Maximal subgroups

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

subgroupindex
C22×#145565222
C2×#179885892
C2×#179885862
C2×#179885552
C2×#179883552
C2×#179883512
C2×#343×C732
#1110673