Finite Group Databasebeta

← #498146

Subgroups — #498146

Order 512.

794 conjugacy classes of subgroups.

The subgroup lattice has 528 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2189C211yesmaximal, characteristic
2C2×#2175C211yesmaximal, characteristic, complemented
3C2×#2138C211yesmaximal, characteristic
4#23026C211yesmaximal, characteristic, complemented
5#22448C211yesmaximal, characteristic
6#13240C211yesmaximal, characteristic, complemented
7#13150C211yesmaximal, characteristic, complemented
8#13133C211yesmaximal, characteristic, complemented
9#13060C211yesmaximal, characteristic, complemented
10#13015C211yesmaximal, characteristic, complemented
11#13014C211yesmaximal, characteristic, complemented
12#12937C211yesmaximal, characteristic, complemented
13#12921C211yesmaximal, characteristic, complemented
14#12637C211yesmaximal, characteristic, complemented
15#12611C211yesmaximal, characteristic, complemented
16C22×#135C2211yescharacteristic
17C22×Q32C2211yescharacteristic
18C22×SD32C2211yescharacteristic
19C23×C16C2211yescharacteristic
20C2×#467C2211yescharacteristic
21C2×C4×Q16C2211yescharacteristic
22C2×C4×D8C2211yescharacteristic
23C2×C4×D8C2211yescharacteristic
24C2×#427C2211yescharacteristic
25#2362—21no—
26#2339—21no—
27#2189C418yescomplemented
28#2175—24no—
29C2×#367C2211yescharacteristic
30C2×#366C2211yescharacteristic
31#2138—24no—
32C2×#358C2211yescharacteristic
33C2×#358C2211yescharacteristic
34C2×#358C2211yescharacteristic
35C2×#358C2211yescharacteristic
36C2×#358C2211yescharacteristic
37C2×#357C2211yescharacteristic
38C2×#357C2211yescharacteristic
39C2×#357C2211yescharacteristic
40C2×C22⋊C16C2211yescharacteristic
41C2×#346C2211yescharacteristic
42C2×#346C2211yescharacteristic
43#1969C2211yescharacteristic
44#1969C2211yescharacteristic
45#1969C2211yescharacteristic
46#1942C2211yescharacteristic
47#1927—21no—
48#1877—21no—
49#1877—21no—
50#1873—21no—
51#1865—21no—
52#1808C2211yescharacteristic
53#1788—21no—
54C4×#107C2211yescharacteristic
55C8×C22⋊C4C2211yescharacteristic
56#1380C2211yescharacteristic
57#1380C2211yescharacteristic
58#1380C2211yescharacteristic
59#1380C2211yescharacteristic
60#1374C2211yescharacteristic
61#1374C2211yescharacteristic
62C22×#41—21no—
63C2×#135—22no—
64C2×#135—22no—
65C2×#135—22no—
66C2×#135—41no—
67C2×#135—41no—
68C2×#135C2×C414yes—
69C22×Q16C2311yescharacteristic
70C22×Q16C2311yescharacteristic
71C22×SD16—21no—
72C22×D8C2311yescharacteristic
73C23×C8C2311yescharacteristic
74C2×C22⋊Q8—21no—
75C2×C4×Q8—21no—
76C2×C4×D4—21no—
77C2×C4×D4—21no—
78C2×#117—21no—
79C2×Q32—41no—
80C2×Q32—41no—
81C2×Q32C2×C414yescomplemented
82C2×SD32—22no—
83C2×SD32—22no—
84C2×SD32C2×C414yes—
85C22×C16—22no—
86C22×C16—22no—
87C22×C16—41no—
88C22×C16C2311yescharacteristic
89C22×C16C2311yescharacteristic
90C22×C16C2311yescharacteristic
91C22×C16C2311yescharacteristic
92#467—22no—
93#467C2×C414yescomplemented
94C4×Q16—22no—
95C4×Q16—41no—
96C4×D8—22no—
97C4×D8—22no—
98C4×D8—41no—
99C4×D8—41no—
100#427—22no—
101#427—41no—
102C2×#107C2311yescharacteristic
103C2×#107C2311yescharacteristic
104C2×#107C2311yescharacteristic
105C2×#107C2311yescharacteristic
106C2×#107C2311yescharacteristic
107C2×#106—21no—
108C2×#103—21no—
109C2×#103—21no—
110C2×#103—21no—
111C2×#103—21no—
112C2×#103—21no—
113C2×#102—21no—
114C2×#102—21no—
115C2×C22⋊C8D411yescharacteristic
116C2×C4×C8C2311yescharacteristic
117C2×C4×C8C2311yescharacteristic
118#391—21no—
119#391—21no—
120#391—21no—
121#389—21no—
122#388—41no—
123#388—41no—
124#387—41no—
125#386—41no—
126#386—41no—
127#381—41no—
128#380—21no—
129C4×C4⋊C4—21no—
130C4×C4⋊C4—41no—
131C4×C22⋊C4D411yescharacteristic
132C2×#95—41no—
133#367—41no—
134#366—22no—
135#358—22no—
136#358—22no—
137#358—22no—
138#358—41no—
139#358C2×C414yescomplemented
140#357—22no—
141#357C2×C414yes—
142#357D414yes—
143C22⋊C16C2×C414yescomplemented
144#346—22no—
145#346—41no—
146#340—21no—
147#340—21no—
148#340—21no—
149#336D411yescharacteristic
150#336D411yescharacteristic
151C2×#41—24no—
152C2×#41—42no—
153C2×#41—42no—
154C2×#41—42no—
155C2×#41—81no—
156C2×#41—81no—
157C22×Q8—21no—
158C22×Q8—21no—
159C22×D4—21no—
160C22×D4—41no—
161C23×C4—41no—
162C23×C4—41no—
163C23×C4—41no—
164C23×C4C2×D411yescharacteristic
165#135—28no—
166#135—44no—
167#135—44no—
168#135—44no—
169C2×Q16—21no—
170C2×Q16—21no—
171C2×Q16—21no—
172C2×Q16—21no—
173C2×Q16—21no—
174C2×Q16—21no—
175C2×Q16—21no—
176C2×Q16—21no—
177C2×Q16—21no—
178C2×Q16—21no—
179C2×Q16C22×C412yes—
180C2×Q16C22×C412yes—
181C2×SD16—24no—
182C2×SD16—42no—
183C2×SD16—42no—
184C2×D8—21no—
185C2×D8—21no—
186C2×D8—21no—
187C2×D8C22×C412yes—
188C2×D8C2×D412yes—
189C2×D8#4112yes—
190C22×C8—21no—
191C22×C8—21no—
192C22×C8—21no—
193C22×C8—21no—
194C22×C8C22×C412yes—
195C22×C8C2×D411yescharacteristic
196C22×C8C2×D411yescharacteristic
197C22×C8C2×D411yescharacteristic
198C22×C8#4112yes—
199C22×C8C2411yescharacteristic
200C22⋊Q8—24no—
201C22⋊Q8—42no—
202C4×Q8—42no—
203C4×Q8—81no—
204C4×D4—42no—
205C4×D4—42no—
206C4×D4—81no—
207C4×D4—81no—
208#117—42no—
209#117—81no—
210C2×C4⋊C4—21no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—21no—
213C2×C4⋊C4—21no—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—41no—
216C2×C4⋊C4—41no—
217C2×C4⋊C4—41no—
218C2×C22⋊C4—41no—
219C2×C22⋊C4—41no—
220C2×C22⋊C4—41no—
221C2×C22⋊C4—41no—
222C2×C22⋊C4—41no—
223C2×C22⋊C4D811yescharacteristic
224C2×C22⋊C4D811yescharacteristic
225C2×C42—41no—
226C2×C42—41no—
227C2×C42—41no—
228C2×C42C2×D411yescharacteristic
229C2×C42C2×D411yescharacteristic
230Q32—44no—
231SD32—28no—
232C2×C16—21no—
233C2×C16—21no—
234C2×C16—21no—
235C2×C16—21no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—21no—
241C2×C16—44no—
242C2×C16C22×C412yes—
243C2×C16C22×C412yes—
244C2×C16C2×D412yes—
245#107—21no—
246#107—21no—
247#107—21no—
248#107—21no—
249#107—21no—
250#107—21no—
251#107—21no—
252#107—21no—
253#107C22×C412yes—
254#107C2×D412yes—
255#106—42no—
256#103—24no—
257#103—42no—
258#103—81no—
259#103—81no—
260#103—81no—
261#102—42no—
262#102—42no—
263C22⋊C8—22no—
264C4×C8—21no—
265C4×C8—21no—
266C4×C8—21no—
267C4×C8—21no—
268#95—41no—
269#95—41no—
270#95—41no—
271#95—41no—
272#95—41no—
273#95—41no—
274#95—41no—
275#95—41no—
276#95—41no—
277#95D811yescharacteristic
278#95D811yescharacteristic
279C24—41no—
280C24C2×D811yescharacteristic
281#41—48no—
282#41—84no—
283#41—84no—
284#41—84no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×D4—22no—
298C2×D4—22no—
299C2×D4—22no—
300C2×D4—44no—
301C2×D4—41no—
302C2×D4—41no—
303C2×D4—41no—
304C2×D4—82no—
305C2×D4—82no—
306C22×C4—21no—
307C22×C4—21no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—41no—
311C22×C4—41no—
312C22×C4—41no—
313C22×C4—41no—
314C22×C4—44no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—82no—
319C22×C4—82no—
320C22×C4—82no—
321C22×C4—82no—
322C22×C4—82no—
323C22×C4—82no—
324C22×C4—161no—
325C22×C4—161no—
326C22×C4C4×D412yes—
327C22×C4C2×D811yescharacteristic
328C22×C4C2×D811yescharacteristic
329C22×C4C2×D811yescharacteristic
330C22×C4C2×D811yescharacteristic
331C22×C4C2×D811yescharacteristic
332C22×C4#13512yes—
333C22×C4C22×D411yescharacteristic
334Q16—22no—
335Q16—22no—
336Q16—22no—
337Q16—22no—
338Q16—41no—
339Q16—41no—
340Q16—41no—
341Q16—41no—
342SD16—48no—
343D8—22no—
344D8—22no—
345D8—22no—
346D8C4×D414yes—
347C2×C8—21no—
348C2×C8—21no—
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—22no—
352C2×C8—22no—
353C2×C8—21no—
354C2×C8—21no—
355C2×C8—21no—
356C2×C8—21no—
357C2×C8—21no—
358C2×C8—21no—
359C2×C8C4×D412yes—
360C2×C8C23×C411yescharacteristic
361C2×C8C22×D411yescharacteristic
362C2×C8C2×#4111yescharacteristic
363C2×C8C2×#4111yescharacteristic
364C2×C8H(5,𝔽2)11yescharacteristic
365C2×C8H(5,𝔽2)11yescharacteristic
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—41no—
369C4⋊C4—41no—
370C4⋊C4—44no—
371C4⋊C4—41no—
372C4⋊C4—41no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4—41no—
376C4⋊C4—41no—
377C4⋊C4—82no—
378C4⋊C4—82no—
379C22⋊C4—41no—
380C22⋊C4—41no—
381C22⋊C4—44no—
382C22⋊C4—82no—
383C22⋊C4—82no—
384C22⋊C4—82no—
385C22⋊C4—161no—
386C42—21no—
387C42—21no—
388C42—21no—
389C42—21no—
390C42—82no—
391C42—82no—
392C42—82no—
393C16—22no—
394C16—22no—
395C16—22no—
396C16—41no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—21no—
403C23—42no—
404C23—42no—
405C23—42no—
406C23—81no—
407C23—81no—
408C23—81no—
409C23—81no—
410C23C4×D812yes—
411C23C22×D811yescharacteristic
412Q8—42no—
413Q8—42no—
414Q8—42no—
415Q8—42no—
416Q8—81no—
417Q8—81no—
418Q8—81no—
419Q8—81no—
420D4—24no—
421D4—42no—
422D4—42no—
423D4—42no—
424D4—88no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—22no—
430C2×C4—22no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—42no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—88no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—81no—
453C2×C4—81no—
454C2×C4—81no—
455C2×C4—81no—
456C2×C4—81no—
457C2×C4—81no—
458C2×C4—81no—
459C2×C4—81no—
460C2×C4—81no—
461C2×C4—81no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—164no—
466C2×C4—164no—
467C2×C4C4×D812yes—
468C2×C4C2×C4×D411yescharacteristic
469C2×C4D4211yescharacteristic
470C2×C4#54611yescharacteristic
471C2×C4C2×#13511yescharacteristic
472C2×C4#57611yescharacteristic
473C2×C4#57711yescharacteristic
474C8—22no—
475C8—41no—
476C8—41no—
477C8C2×C4×D411yescharacteristic
478C8#51811yescharacteristic
479C8#54611yescharacteristic
480C8#55111yescharacteristic
481C22—22no—
482C22—22no—
483C22—22no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—44no—
489C22—82no—
490C22—82no—
491C22—82no—
492C22—82no—
493C22—82no—
494C22—82no—
495C22C2×C4×D811yescharacteristic
496C22D4×D811yescharacteristic
497C22#329411yescharacteristic
498C22#341411yescharacteristic
499C22#341411yescharacteristic
500C22#341611yescharacteristic
501C22#341711yescharacteristic
502C4—22no—
503C4—41no—
504C4—41no—
505C4—41no—
506C4—41no—
507C4—82no—
508C4—82no—
509C4—82no—
510C4—82no—
511C4—161no—
512C4—161no—
513C4—161no—
514C4#235811yescharacteristic
515C4#294611yescharacteristic
516C4#328211yescharacteristic
517C4#329711yescharacteristic
518C2—42no—
519C2—84no—
520C2#2287011yescharacteristic
521C2#3345111yescharacteristic
522C2#3345511yescharacteristic
523C2#3383311yescharacteristic
524C2#3383511yescharacteristic
525C2#3387811yescharacteristic
526C2#3388211yescharacteristic
subgroupG/H
center, socleC23C22×D8
commutatorC2×C8C23×C4
Frattini, cosocleC22×C8C24
prime radicalC22×#135C22
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C4×C22⋊C4 ◁ C8×C22⋊C4 ◁ G
LECSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
2. C2×#2175C2semidirect
4. #23026C2semidirect
6. #13240C2semidirect
7. #13150C2semidirect
8. #13133C2semidirect
9. #13060C2semidirect
10. #13015C2semidirect
11. #13014C2semidirect
12. #12937C2semidirect
13. #12921C2semidirect
14. #12637C2semidirect
15. #12611C2semidirect
27. #2189C4semidirect
81. C2×Q32C2×C4semidirect
93. #467C2×C4semidirect
139. #358C2×C4semidirect
143. C22⋊C16C2×C4semidirect

Maximal subgroups

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

subgroupindex
C2×#21892
C2×#21752
C2×#21382
#230262
#224482
#132402
#131502
#131332
#130602
#130152
#130142
#129372
#129212
#126372
#126112