Finite Group Databasebeta

← #33122371

Subgroups — #33122371

Order 18688.

1070 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988864C211yesmaximal, characteristic
2C2×C4×#11924015C211yesmaximal, characteristic
3#22614849C211yesmaximal, characteristic
4#22611266C212yesmaximal, complemented
5#22611074C212yesmaximal
6#22610893C211yesmaximal, characteristic, complemented
7#22610868C211yesmaximal, characteristic, complemented
8#22610286C212yesmaximal
9#22610185C212yesmaximal, complemented
10#22609936C211yesmaximal, characteristic, complemented
11C2×#557×C73C211yesmaximal, characteristic, complemented
12C22×#11924015C2211yescharacteristic
13C2×#14556567—21no—
14C2×C4×D292C2211yescharacteristic
15#17988864—24no—
16C4×#11924015—24no—
17C2×#14556524C2212yes—
18C2×#14556522C2211yescharacteristic
19C2×C4⋊#10596831C2211yescharacteristic
20C2×#14556515C2211yescharacteristic
21C2×#14556515C2212yes—
22C2×#14556511C2212yes—
23C2×#14556511C2211yescharacteristic
24C2×C4×#10596831C2211yescharacteristic
25#17988509—22no—
26#17988489C2212yes—
27#17988485C2212yes—
28#17988457C2211yescharacteristic
29#17988444—21no—
30#17988349C2212yes—
31#17988349C2212yes—
32#17988349C2212yes—
33#17988342C2212yes—
34#17988342C2212yes—
35#557×C73—24no—
36#557×C73—24no—
37C2×C4⋊Q8×C73—21no—
38C2×C4⋊Q8×C73C2211yescharacteristic
39C2×#125×C73—22no—
40C2×#125×C73C2212yes—
41C2×C4×Q8×C73—21no—
42C2×C4×Q8×C73C2212yes—
43C2×C4×Q8×C73C2211yescharacteristic
44C2×C4×Q8×C73C2211yescharacteristic
45C22×D292C2311yescharacteristic
46C22×C4×D73—21no—
47C2×#11924015—22no—
48C2×#11924015—22no—
49C2×#11924015Q814yescomplemented
50C22×#10596831C2312yes—
51C22×#10596831C2311yescharacteristic
52C22×#10596831C2311yescharacteristic
53#14556567—44no—
54C4×D292—22no—
55C4×D292—22no—
56C2×C4⋊Dic73—21no—
57C2×C4⋊Dic73—21no—
58C2×C4⋊Dic73C2311yescharacteristic
59C2×C4⋊Dic73C2312yes—
60C2×#11924007—22no—
61C2×#11924005—22no—
62C2×#11924005—21no—
63C2×#11924005—22no—
64#14556524—24no—
65#14556522—22no—
66C4⋊#10596831—22no—
67#14556515—22no—
68#14556515—24no—
69#14556511—24no—
70#14556511—22no—
71C4×#10596831—22no—
72#14556498—22no—
73C22×Q8×C73—21no—
74C22×Q8×C73D411yescharacteristic
75C22×Q8×C73C2311yescharacteristic
76C4⋊Q8×C73—28no—
77C4⋊Q8×C73—22no—
78C4⋊Q8×C73D414yes—
79#125×C73—216no—
80#125×C73—24no—
81#125×C73—24no—
82C4×Q8×C73—28no—
83C4×Q8×C73—24no—
84C4×Q8×C73—24no—
85C4×Q8×C73—22no—
86C4×Q8×C73—22no—
87C4×Q8×C73—22no—
88C4×Q8×C73D414yes—
89C2×C4⋊C4×C73—22no—
90C2×C4⋊C4×C73—21no—
91C2×C4⋊C4×C73—22no—
92C2×C4⋊C4×C73—21no—
93C2×C4⋊C4×C73—22no—
94C2×C4⋊C4×C73D411yescharacteristic
95C2×C4⋊C4×C73D412yes—
96C2×C4⋊C4×C73C2311yescharacteristic
97C2×C4⋊C4×C73C2312yes—
98C2×C42×C73—22no—
99C2×C42×C73—21no—
100C2×C42×C73C2312yes—
101C2×C42×C73C2311yescharacteristic
102C23×D73—21no—
103C2×D292—21no—
104C2×D292—21no—
105C2×D292—21no—
106C2×D292—21no—
107C2×D292—21no—
108C2×D292C2×Q812yes—
109C22×Dic73—21no—
110C22×Dic73—22no—
111C2×C4×D73—42no—
112C2×C4×D73—42no—
113C2×C4×D73—42no—
114#11924015—28no—
115C2×#10596831—21no—
116C2×#10596831—21no—
117C2×#10596831—21no—
118C2×#10596831—22no—
119C2×#10596831—22no—
120C2×#10596831—21no—
121C2×#10596831—21no—
122C2×#10596831—22no—
123C2×#10596831C2×Q812yes—
124C4⋊Dic73—21no—
125C4⋊Dic73—21no—
126C4⋊Dic73—22no—
127C4⋊Dic73—22no—
128C4⋊Dic73—42no—
129C4⋊Dic73—81no—
130#11924007—44no—
131#11924005—44no—
132#11924005—42no—
133#11924005—82no—
134C2×Q8×C73—22no—
135C2×Q8×C73—24no—
136C2×Q8×C73—22no—
137C2×Q8×C73—22no—
138C2×Q8×C73—21no—
139C2×Q8×C73—21no—
140C2×Q8×C73—21no—
141C2×Q8×C73—42no—
142C2×Q8×C73—42no—
143C2×Q8×C73C2×D412yes—
144C2×Q8×C73C2×Q812yes—
145C2×Q8×C73#4112yes—
146C22×C4×C73—21no—
147C22×C4×C73—21no—
148C22×C4×C73—22no—
149C22×C4×C73C2×D412yes—
150C22×C4×C73C2×D412yes—
151C22×C4×C73C2×D411yescharacteristic
152C22×C4×C73C2×D411yescharacteristic
153C22×C4×C73C2411yescharacteristic
154C4⋊C4×C73—28no—
155C4⋊C4×C73—28no—
156C4⋊C4×C73—28no—
157C4⋊C4×C73—24no—
158C4⋊C4×C73—24no—
159C4⋊C4×C73—22no—
160C4⋊C4×C73—24no—
161C4⋊C4×C73—21no—
162C4⋊C4×C73—22no—
163C4⋊C4×C73C2×D414yes—
164C4⋊C4×C73C2×D412yes—
165C42×C73—28no—
166C42×C73—24no—
167C42×C73—22no—
168C42×C73—22no—
169C42×C73C2×D412yes—
170C42×C73C2×D412yes—
171C23×C73C22×D411yescharacteristic
172C22×D73—22no—
173C22×D73—41no—
174C22×D73—41no—
175C22×D73—41no—
176C22×D73—41no—
177C22×D73—41no—
178C22×D73—41no—
179Q8×C73—22no—
180Q8×C73—22no—
181Q8×C73—22no—
182Q8×C73—44no—
183Q8×C73—48no—
184Q8×C73C22⋊Q814yescomplemented
185C2×C4×C73—22no—
186C2×C4×C73—22no—
187C2×C4×C73—22no—
188C2×C4×C73—21no—
189C2×C4×C73—21no—
190C2×C4×C73—22no—
191C2×C4×C73—22no—
192C2×C4×C73—22no—
193C2×C4×C73—24no—
194C2×C4×C73—22no—
195C2×C4×C73—22no—
196C2×C4×C73—22no—
197C2×C4×C73—22no—
198C2×C4×C73—42no—
199C2×C4×C73—42no—
200C2×C4×C73—41no—
201C2×C4×C73—41no—
202C2×C4×C73#12012yes—
203C2×C4×C73#12012yes—
204C2×C4×C73#13512yes—
205C2×C4×C73AGL(1,ℤ/8)12yes—
206C2×C4×C73C22×D411yescharacteristic
207C2×C4×C73C22×D411yescharacteristic
208C2×C4×C73C22×Q811yescharacteristic
209C2×C4×C73C2×#4111yescharacteristic
210C2×C4×C732−1+411yescharacteristic
211C2×C4×C732−1+411yescharacteristic
212C2×Dic73—41no—
213C2×Dic73—42no—
214C2×Dic73—41no—
215C2×Dic73—42no—
216C2×Dic73—41no—
217C2×Dic73—42no—
218D292—22no—
219D292—22no—
220D292—41no—
221D292—41no—
222C4×D73—84no—
223#10596831—22no—
224#10596831—41no—
225#10596831—42no—
226C22×C73C2×#12011yescharacteristic
227C22×C73#54811yescharacteristic
228C22×C73D4×Q811yescharacteristic
229C22×C73C2×#13511yescharacteristic
230C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
231C22×C73#57711yescharacteristic
232C22×C73#57811yescharacteristic
233C2×D73—42no—
234C2×D73—42no—
235C2×D73—42no—
236C2×D73—81no—
237C2×D73—81no—
238C2×D73—81no—
239C2×D73—81no—
240C4×C73—24no—
241C4×C73—22no—
242C4×C73—42no—
243C4×C73—44no—
244C4×C73—42no—
245C4×C73—42no—
246C4×C73#44914yes—
247C4×C73C2×C22⋊Q811yescharacteristic
248C4×C73#53611yescharacteristic
249C4×C73#54811yescharacteristic
250C4×C73D4×Q811yescharacteristic
251Dic73—82no—
252Dic73—81no—
253#24328—731nomaximal, Sylow
254C2×C73#240911yescharacteristic
255C2×C73C2×#44911yescharacteristic
256C2×C73#301311yescharacteristic
257C2×C73#336411yescharacteristic
258C2×C73#336811yescharacteristic
259C2×C73#338111yescharacteristic
260C2×C73#338611yescharacteristic
261D73—82no—
262C2×#557D7311yescharacteristic, complemented
263C2×#474—731no—
264C2×C4×SD16—731no—
265#2538—731no—
266#2059—732no—
267#2025—732no—
268#1990—731no—
269#1984—731no—
270#1880—732no—
271#1863—732no—
272#1803—731no—
273C73#2432811yescharacteristic, complemented, Sylow
274C22×SD16—731no—
275#557—24no—
276#557—24no—
277C2×C4⋊Q8—21no—
278C2×C4⋊Q8C2×D7311yescharacteristic
279C2×#125—22no—
280C2×#125C2×D7312yes—
281C2×C22⋊Q8—1461no—
282C2×C4×Q8—21no—
283C2×C4×Q8C2×D7312yes—
284C2×C4×Q8C2×D7311yescharacteristic
285C2×C4×Q8C2×D7311yescharacteristic
286C2×C4×D4—731no—
287#474—1464no—
288C4×SD16—1464no—
289C2×#107—732no—
290C2×#106—731no—
291C2×C4⋊C8—731no—
292C2×#103—732no—
293C2×#103—731no—
294C2×#102—732no—
295C2×#102—731no—
296C2×C4×C8—731no—
297#396—1462no—
298#390—732no—
299#389—732no—
300#384—731no—
301#380—1461no—
302#340—732no—
303#340—732no—
304#340—732no—
305#336—732no—
306#336—732no—
307C22×Q8—21no—
308C22×Q8C22⋊D7311yescharacteristic
309C22×Q8C22×D7311yescharacteristic
310C22×D4—731no—
311C23×C4—1461no—
312C2×SD16—734no—
313C2×SD16—1462no—
314C2×SD16—1462no—
315C22×C8—731no—
316C22×C8—731no—
317C22×C8—732no—
318C4⋊Q8—28no—
319C4⋊Q8—22no—
320C4⋊Q8C22⋊D7314yes—
321#125—216no—
322#125—24no—
323#125—24no—
324C22⋊Q8—2924no—
325C4×Q8—28no—
326C4×Q8—24no—
327C4×Q8—24no—
328C4×Q8—22no—
329C4×Q8—22no—
330C4×Q8—22no—
331C4×Q8C22⋊D7314yes—
332C4×D4—1462no—
333C4×D4—1462no—
334C2×C4⋊C4—22no—
335C2×C4⋊C4—21no—
336C2×C4⋊C4—22no—
337C2×C4⋊C4—21no—
338C2×C4⋊C4—22no—
339C2×C4⋊C4—731no—
340C2×C4⋊C4—732no—
341C2×C4⋊C4—1461no—
342C2×C4⋊C4—1462no—
343C2×C4⋊C4—1461no—
344C2×C4⋊C4C22⋊D7311yescharacteristic
345C2×C4⋊C4C22⋊D7312yes—
346C2×C4⋊C4C22×D7311yescharacteristic
347C2×C4⋊C4C22×D7312yes—
348C2×C22⋊C4—1461no—
349C2×C22⋊C4—1462no—
350C2×C22⋊C4—1462no—
351C2×C42—22no—
352C2×C42—21no—
353C2×C42C22×D7312yes—
354C2×C42C22×D7311yescharacteristic
355#107—1464no—
356#106—1462no—
357C4⋊C8—1462no—
358#103—1462no—
359#103—1464no—
360#102—1462no—
361#102—1464no—
362C4×C8—1462no—
363#95—1462no—
364C24—1461no—
365C2×Q8—22no—
366C2×Q8—24no—
367C2×Q8—22no—
368C2×Q8—22no—
369C2×Q8—21no—
370C2×Q8—21no—
371C2×Q8—21no—
372C2×Q8—42no—
373C2×Q8—42no—
374C2×Q8C2×C22⋊D7312yes—
375C2×Q8Q8×D7312yes—
376C2×Q8#1192402912yes—
377C2×D4—732no—
378C2×D4—1461no—
379C2×D4—1461no—
380C2×D4—1461no—
381C2×D4—1461no—
382C2×D4—1461no—
383C22×C4—21no—
384C22×C4—21no—
385C22×C4—22no—
386C22×C4—2922no—
387C22×C4—2922no—
388C22×C4—2922no—
389C22×C4—1461no—
390C22×C4—1462no—
391C22×C4C2×C22⋊D7312yes—
392C22×C4C2×C22⋊D7312yes—
393C22×C4C2×C22⋊D7311yescharacteristic
394C22×C4C2×C22⋊D7311yescharacteristic
395C22×C4C23×D7311yescharacteristic
396SD16—1468no—
397C2×C8—732no—
398C2×C8—1461no—
399C2×C8—1462no—
400C2×C8—1462no—
401C2×C8—1461no—
402C2×C8—1461no—
403C2×C8—1462no—
404C2×C8—1461no—
405C2×C8—1461no—
406C4⋊C4—28no—
407C4⋊C4—28no—
408C4⋊C4—28no—
409C4⋊C4—24no—
410C4⋊C4—24no—
411C4⋊C4—22no—
412C4⋊C4—24no—
413C4⋊C4—21no—
414C4⋊C4—22no—
415C4⋊C4—2924no—
416C4⋊C4—2922no—
417C4⋊C4—5841no—
418C4⋊C4—1461no—
419C4⋊C4—1461no—
420C4⋊C4—1462no—
421C4⋊C4—1462no—
422C4⋊C4C2×C22⋊D7312yes—
423C4⋊C4C2×C22⋊D7314yes—
424C22⋊C4—2924no—
425C22⋊C4—2922no—
426C22⋊C4—5842no—
427C42—28no—
428C42—24no—
429C42—22no—
430C42—22no—
431C42C2×C22⋊D7312yes—
432C42C2×C22⋊D7312yes—
433C23—2921no—
434C23—2921no—
435C23—2921no—
436C23—2921no—
437C23—2921no—
438C23—2921no—
439C23—1462no—
440C23C22×C22⋊D7311yescharacteristic
441Q8—22no—
442Q8—22no—
443Q8—22no—
444Q8—44no—
445Q8—48no—
446Q8#1455656714yescomplemented
447D4—2921no—
448D4—2921no—
449D4—1462no—
450D4—1462no—
451C2×C4—21no—
452C2×C4—21no—
453C2×C4—22no—
454C2×C4—22no—
455C2×C4—22no—
456C2×C4—22no—
457C2×C4—22no—
458C2×C4—22no—
459C2×C4—22no—
460C2×C4—22no—
461C2×C4—24no—
462C2×C4—22no—
463C2×C4—22no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—2922no—
469C2×C4—2922no—
470C2×C4—2921no—
471C2×C4—2921no—
472C2×C4—2922no—
473C2×C4—2921no—
474C2×C4—5844no—
475C2×C4#1455656012yes—
476C2×C4#1455656012yes—
477C2×C4#1455661112yes—
478C2×C4#1455661612yes—
479C2×C4C22×C22⋊D7311yescharacteristic
480C2×C4C22×C22⋊D7311yescharacteristic
481C2×C4C2×Q8×D7311yescharacteristic
482C2×C4C2×#1192402911yescharacteristic
483C2×C4#1455663911yescharacteristic
484C2×C4#1455663911yescharacteristic
485C8—2921no—
486C8—2922no—
487C8—1462no—
488C22—2922no—
489C22—2922no—
490C22—2922no—
491C22—5841no—
492C22—5841no—
493C22—5841no—
494C22—5841no—
495C22C2×#1455656011yescharacteristic
496C22#1798924811yescharacteristic
497C22Q8×C22⋊D7311yescharacteristic
498C22C2×#1455661111yescharacteristic
499C22C2×#1455661611yescharacteristic
500C22#1798939511yescharacteristic
501C22#1798940211yescharacteristic
502C4—24no—
503C4—22no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—44no—
508C4—5842no—
509C4—5841no—
510C4#1798873214yes—
511C4C2×#1455656711yescharacteristic
512C4#1798916111yescharacteristic
513C4#1798924811yescharacteristic
514C4Q8×C22⋊D7311yescharacteristic
515C2—5842no—
516C2#2261329011yescharacteristic
517C2C2×#1798873211yescharacteristic
518C2#2261924611yescharacteristic
519C2#2262261611yescharacteristic
520C2#2262266011yescharacteristic
521C2#2262280211yescharacteristic
522C2#2262284811yescharacteristic
subgroupG/H
centerC23C22×C22⋊D73
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC2×#557×C73C2
socleC23×C73C22×D4
prime radicalC22×D292C23
hypercenterC2×#557D73
hypocenterC73#24328
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
4. #22611266C2semidirect
6. #22610893C2semidirect
7. #22610868C2semidirect
9. #22610185C2semidirect
10. #22609936C2semidirect
11. C2×#557×C73C2semidirect
49. C2×#11924015Q8semidirect
184. Q8×C73C22⋊Q8semidirect
262. C2×#557D73semidirect
273. C73#24328semidirect
446. Q8#14556567semidirect

Maximal subgroups

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

subgroupindex
C2×#179888642
C2×C4×#119240152
#226148492
#226112662
#226110742
#226108932
#226108682
#226102862
#226101852
#226099362
C2×#557×C732
#2432873