Finite Group Databasebeta

← #33240457

Subgroups — #33240457

Order 18688.

738 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989023C211yesmaximal, characteristic, complemented
2C22×#14556531C211yesmaximal, characteristic, complemented
3#22619512C211yesmaximal, characteristic, complemented
4#22612042C212yesmaximal, complemented
5#22612032C212yesmaximal, complemented
6#22612025C212yesmaximal, complemented
7#22612015C212yesmaximal, complemented
8#22611717C211yesmaximal, characteristic, complemented
9#22611647C211yesmaximal, characteristic, complemented
10#22611641C211yesmaximal, characteristic, complemented
11C2×#466×C73C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13#17989098—21no—
14#17989023—24no—
15C2×#14556535C2212yescomplemented
16C2×#14556532C2212yescomplemented
17C2×#14556531—24no—
18C2×#14556531C2211yescharacteristic
19C2×#14556531C2211yescharacteristic
20C2×#14556531C2212yescomplemented
21C22×#11924003C2212yescomplemented
22#17988828C2211yescharacteristic, complemented
23#17988828C2211yescharacteristic, complemented
24#17988823C2211yescharacteristic, complemented
25#17988823C2211yescharacteristic, complemented
26#17988816—21no—
27#17988816—21no—
28#17988625C2211yescharacteristic
29#17988584—21no—
30#17988414C2212yescomplemented
31#17988412C2212yescomplemented
32#17988390C2212yescomplemented
33#17988388C2212yescomplemented
34#17988388C2212yescomplemented
35#17988387C2212yescomplemented
36C22×D8×C73C2211yescharacteristic
37C23×C8×C73C2211yescharacteristic
38C2×#121×C73—21no—
39#466×C73—21no—
40#466×C73—21no—
41#466×C73C2212yes—
42#466×C73C2212yes—
43C2×#107×C73C2211yescharacteristic
44C2×#102×C73—21no—
45C2×#11924028—21no—
46#14556613—22no—
47#14556613—22no—
48C2×Dic292C2311yescharacteristic
49C2×#11924017—21no—
50C2×D584C2311yescharacteristic
51#14556574—41no—
52#14556570—21no—
53#14556570—21no—
54#14556565—21no—
55#14556565—21no—
56#14556560—41no—
57C4×C22⋊D73—41no—
58C4×C22⋊D73—41no—
59#14556550—21no—
60#14556535—24no—
61#14556532—24no—
62#14556531—24no—
63#14556531—22no—
64#14556531D414yescomplemented
65C2×#11924003—24no—
66C2×#11924003C2312yes—
67C2×#11924003C2312yes—
68#14556525C2311yescharacteristic
69#14556525C2311yescharacteristic
70#14556523—21no—
71#14556516—21no—
72#14556516—21no—
73#14556513—21no—
74#14556513—21no—
75C22×D4×C73—21no—
76C22×D4×C73—41no—
77C23×C4×C73D411yescharacteristic
78C2×D8×C73—22no—
79C2×D8×C73—21no—
80C2×D8×C73—22no—
81C2×D8×C73C2312yes—
82C22×C8×C73—22no—
83C22×C8×C73—22no—
84C22×C8×C73D411yescharacteristic
85C22×C8×C73D411yescharacteristic
86C22×C8×C73D411yescharacteristic
87C22×C8×C73C2311yescharacteristic
88C22×C8×C73C2311yescharacteristic
89C22×C8×C73C2311yescharacteristic
90#121×C73—22no—
91#121×C73—22no—
92#121×C73—41no—
93#121×C73—41no—
94C2×C4⋊C4×C73—21no—
95C2×C22⋊C4×C73—41no—
96#107×C73—21no—
97#107×C73C2312yes—
98#102×C73—22no—
99#102×C73—22no—
100#11924028—42no—
101#11924028—42no—
102C2×Dic146—21no—
103C2×D292—21no—
104C2×C22⋊D73—41no—
105C2×C22⋊D73—41no—
106C2×C22⋊D73—41no—
107C2×C4×D73—41no—
108Dic292—21no—
109Dic292—21no—
110#11924017—42no—
111D584—21no—
112D584—21no—
113C4⋊Dic73—21no—
114C4⋊Dic73—21no—
115#11924007—41no—
116#11924007—41no—
117C22⋊Dic73—41no—
118C22⋊Dic73—41no—
119C22⋊Dic73—41no—
120#11924005—41no—
121#11924005—41no—
122C4×Dic73—41no—
123#11924003—22no—
124#11924003C2×D414yescomplemented
125C24×C73—41no—
126C24×C73D811yescharacteristic
127C2×D4×C73—22no—
128C2×D4×C73—42no—
129C2×D4×C73—42no—
130C2×D4×C73—42no—
131C2×D4×C73—41no—
132C2×D4×C73—42no—
133C2×D4×C73—84no—
134C22×C4×C73—22no—
135C22×C4×C73—22no—
136C22×C4×C73—41no—
137C22×C4×C73D811yescharacteristic
138C22×C4×C73D811yescharacteristic
139C22×C4×C73D811yescharacteristic
140C22×C4×C73C2×D411yescharacteristic
141C22×C4×C73C2×D411yescharacteristic
142C22×C4×C73C2×D411yescharacteristic
143D8×C73—22no—
144D8×C73—22no—
145D8×C73—42no—
146C2×C8×C73—22no—
147C2×C8×C73—22no—
148C2×C8×C73—22no—
149C2×C8×C73—21no—
150C2×C8×C73—21no—
151C2×C8×C73—21no—
152C2×C8×C73—21no—
153C2×C8×C73C2×D412yes—
154C2×C8×C73C2×D411yescharacteristic
155C2×C8×C73C2×D411yescharacteristic
156C2×C8×C73C2×D411yescharacteristic
157C2×C8×C73#4112yes—
158C2×C8×C73C2411yescharacteristic
159C4⋊C4×C73—22no—
160C4⋊C4×C73—41no—
161C22⋊C4×C73—42no—
162C22⋊C4×C73—42no—
163C23×C73—22no—
164C23×C73—22no—
165C23×C73—41no—
166C23×C73—42no—
167C23×C73—82no—
168C23×C73—82no—
169C23×C73—82no—
170C23×C73C2×D811yescharacteristic
171C23×C73C2×D811yescharacteristic
172C23×C73C2×D811yescharacteristic
173C22×D73—41no—
174D4×C73—42no—
175D4×C73—42no—
176D4×C73—84no—
177D4×C73—84no—
178D4×C73—82no—
179C2×C4×C73—22no—
180C2×C4×C73—21no—
181C2×C4×C73—21no—
182C2×C4×C73—21no—
183C2×C4×C73—21no—
184C2×C4×C73—22no—
185C2×C4×C73—22no—
186C2×C4×C73—42no—
187C2×C4×C73—82no—
188C2×C4×C73#12112yes—
189C2×C4×C73C2×D811yescharacteristic
190C2×C4×C73C2×D811yescharacteristic
191C2×C4×C73C2×D811yescharacteristic
192C2×C4×C73#13512yes—
193C2×C4×C73C22×D411yescharacteristic
194C22⋊D73—82no—
195C22⋊D73—82no—
196C22⋊D73—82no—
197C2×Dic73—41no—
198C2×Dic73—41no—
199C2×Dic73—41no—
200D292—41no—
201D292—41no—
202C4×D73—82no—
203Dic146—41no—
204Dic146—41no—
205C8×C73—21no—
206C8×C73—21no—
207C8×C73#12112yes—
208C8×C73C22×D411yescharacteristic
209C8×C73C2×#4111yescharacteristic
210C22×C73—21no—
211C22×C73—21no—
212C22×C73—21no—
213C22×C73—21no—
214C22×C73—21no—
215C22×C73—21no—
216C22×C73—21no—
217C22×C73—44no—
218C22×C73—82no—
219C22×C73—82no—
220C22×C73—82no—
221C22×C73—164no—
222C22×C73#46612yes—
223C22×C73#50912yescomplemented
224C22×C73C22×D811yescharacteristic
225C2×D73—81no—
226C2×D73—81no—
227C2×D73—81no—
228C4×C73—21no—
229C4×C73—21no—
230C4×C73—82no—
231C4×C73#46612yes—
232C4×C73C2×#12111yescharacteristic
233C4×C73C2×#13511yescharacteristic
234Dic73—81no—
235Dic73—81no—
236Dic73—81no—
237#33605—731nomaximal, Sylow
238C2×C73—22no—
239C2×C73—22no—
240C2×C73—41no—
241C2×C73—162no—
242C2×C73C2×#46611yescharacteristic
243C2×C73C2×#50911yescharacteristic
244C2×C73#341411yescharacteristic
245D73—161no—
246C2×#509—731no—
247C22×M32—731no—
248#3050—731no—
249C2×#466D7311yescharacteristic, complemented
250#2220—732no—
251#2218—732no—
252#2217—732no—
253#2215—732no—
254#2159—731no—
255#2140—731no—
256#2139—731no—
257C73#3360511yescharacteristic, complemented, Sylow
258C2×#135—731no—
259C22×D8C2×D7311yescharacteristic
260C23×C8C2×D7311yescharacteristic
261#525—1461no—
262C2×#121—21no—
263#509—1464no—
264C2×SD32—732no—
265C2×D16—732no—
266C2×M32—731no—
267C2×M32—731no—
268C2×M32—732no—
269C2×M32—1464no—
270C22×C16—732no—
271#467—731no—
272#467—731no—
273#466—21no—
274#466—21no—
275#466—731no—
276#466—731no—
277#466C2×D7312yes—
278#466C2×D7312yes—
279#465—1461no—
280#465—1461no—
281#427—731no—
282C2×#107C2×D7311yescharacteristic
283#416—1461no—
284C2×#102—21no—
285#367—732no—
286#366—732no—
287#358—732no—
288#357—732no—
289#357—732no—
290#357—732no—
291C2×#41—1461no—
292C22×D4—21no—
293C22×D4—41no—
294C23×C4C22⋊D7311yescharacteristic
295#135—1462no—
296#135—1462no—
297C2×Q16—731no—
298C2×SD16—1461no—
299C2×D8—22no—
300C2×D8—21no—
301C2×D8—22no—
302C2×D8—731no—
303C2×D8C22×D7312yes—
304C22×C8—22no—
305C22×C8—22no—
306C22×C8C22⋊D7311yescharacteristic
307C22×C8C22⋊D7311yescharacteristic
308C22×C8C22⋊D7311yescharacteristic
309C22×C8C22×D7311yescharacteristic
310C22×C8C22×D7311yescharacteristic
311C22×C8C22×D7311yescharacteristic
312#123—2921no—
313C22⋊Q8—1461no—
314C22⋊Q8—1461no—
315#121—22no—
316#121—22no—
317#121—41no—
318#121—41no—
319#121—1461no—
320#121—1461no—
321#120—2921no—
322C4×D4—2921no—
323C4×D4—2921no—
324#117—1461no—
325C2×C4⋊C4—21no—
326C2×C22⋊C4—41no—
327SD32—1464no—
328D16—1464no—
329M32—734no—
330M32—1464no—
331M32—1462no—
332C2×C16—732no—
333C2×C16—732no—
334C2×C16—1464no—
335#107—21no—
336#107—731no—
337#107—731no—
338#107C22×D7312yes—
339#106—1461no—
340#103—1461no—
341#103—1461no—
342#102—22no—
343#102—22no—
344#102—1461no—
345#102—1461no—
346C24—41no—
347C24#1192401311yescharacteristic
348#41—2922no—
349#41—2922no—
350C2×Q8—1461no—
351C2×D4—22no—
352C2×D4—42no—
353C2×D4—42no—
354C2×D4—42no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—84no—
358C2×D4—2921no—
359C2×D4—2921no—
360C2×D4—2921no—
361C2×D4—1461no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—2921no—
366C22×C4#1192401311yescharacteristic
367C22×C4#1192401311yescharacteristic
368C22×C4#1192401311yescharacteristic
369C22×C4C2×C22⋊D7311yescharacteristic
370C22×C4C2×C22⋊D7311yescharacteristic
371C22×C4C2×C22⋊D7311yescharacteristic
372Q16—1461no—
373Q16—1461no—
374SD16—2922no—
375D8—22no—
376D8—22no—
377D8—42no—
378D8—1461no—
379D8—1461no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8D4×D7312yes—
388C2×C8C2×C22⋊D7311yescharacteristic
389C2×C8C2×C22⋊D7311yescharacteristic
390C2×C8C2×C22⋊D7311yescharacteristic
391C2×C8#1192402712yes—
392C2×C8C23×D7311yescharacteristic
393C4⋊C4—22no—
394C4⋊C4—41no—
395C4⋊C4—2921no—
396C4⋊C4—2921no—
397C4⋊C4—1461no—
398C4⋊C4—1461no—
399C22⋊C4—42no—
400C22⋊C4—42no—
401C22⋊C4—2921no—
402C22⋊C4—2921no—
403C22⋊C4—2921no—
404C22⋊C4—2921no—
405C22⋊C4—2921no—
406C42—2921no—
407C16—734no—
408C16—1462no—
409C23—22no—
410C23—22no—
411C23—41no—
412C23—42no—
413C23—82no—
414C23—82no—
415C23—82no—
416C23—2921no—
417C23C2×#1192401311yescharacteristic
418C23C2×#1192401311yescharacteristic
419C23C2×#1192401311yescharacteristic
420Q8—2921no—
421Q8—2921no—
422D4—42no—
423D4—42no—
424D4—84no—
425D4—84no—
426D4—82no—
427D4—2921no—
428D4—2921no—
429D4—5842no—
430D4—5842no—
431D4—5842no—
432C2×C4—22no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—42no—
440C2×C4—82no—
441C2×C4—2921no—
442C2×C4—2921no—
443C2×C4—2921no—
444C2×C4—5842no—
445C2×C4#1455656312yes—
446C2×C4C2×#1192401311yescharacteristic
447C2×C4C2×#1192401311yescharacteristic
448C2×C4C2×#1192401311yescharacteristic
449C2×C4#1455661112yes—
450C2×C4C22×C22⋊D7311yescharacteristic
451C8—21no—
452C8—21no—
453C8#1455656312yes—
454C8C2×D4×D7311yescharacteristic
455C8C2×#1192402711yescharacteristic
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—82no—
465C22—82no—
466C22—82no—
467C22—164no—
468C22—5841no—
469C22—5841no—
470C22—5841no—
471C22#1798882112yes—
472C22#1798902312yescomplemented
473C22C22×#1192401311yescharacteristic
474C4—21no—
475C4—21no—
476C4—82no—
477C4—5841no—
478C4—5841no—
479C4—5841no—
480C4#1798882112yes—
481C4C2×#1455656311yescharacteristic
482C4C2×#1455661111yescharacteristic
483C2—22no—
484C2—22no—
485C2—41no—
486C2—162no—
487C2—11681no—
488C2C2×#1798882111yescharacteristic
489C2C2×#1798902311yescharacteristic
490C2#2262306511yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC2×#466×C73C2
socleC22×C73C22×D8
hypercenterC2×#466D73
hypocenterC73#33605
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C2×#466

Complements

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

NH ≅ G/Ntype
1. C2×#17989023C2semidirect
2. C22×#14556531C2semidirect
3. #22619512C2semidirect
4. #22612042C2semidirect
5. #22612032C2semidirect
6. #22612025C2semidirect
7. #22612015C2semidirect
8. #22611717C2semidirect
9. #22611647C2semidirect
10. #22611641C2semidirect
11. C2×#466×C73C2semidirect
15. C2×#14556535C22semidirect
16. C2×#14556532C22semidirect
20. C2×#14556531C22semidirect
21. C22×#11924003C22semidirect
22. #17988828C22semidirect
23. #17988828C22semidirect
24. #17988823C22semidirect
25. #17988823C22semidirect
30. #17988414C22semidirect
31. #17988412C22semidirect
32. #17988390C22semidirect
33. #17988388C22semidirect
34. #17988388C22semidirect
35. #17988387C22semidirect
64. #14556531D4semidirect
124. #11924003C2×D4semidirect
223. C22×C73#509semidirect
249. C2×#466D73semidirect
257. C73#33605semidirect
472. C22#17989023semidirect

Maximal subgroups

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

subgroupindex
C2×#179890232
C22×#145565312
#226195122
#226120422
#226120322
#226120252
#226120152
#226117172
#226116472
#226116412
C2×#466×C732
#3360573