Finite Group Databasebeta

← #375507

Subgroups — #375507

Order 512.

1475 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35349C211yesmaximal, characteristic
2#20392C211yesmaximal, characteristic
3#20354C211yesmaximal, characteristic
4#11381C212yesmaximal, complemented
5#11351C212yesmaximal, complemented
6#10918C212yesmaximal, complemented
7#10915C212yesmaximal, complemented
8#10754C212yesmaximal
9#10745C212yesmaximal
10C22×C22⋊Q8C2211yescharacteristic
11C2×#518C2212yes—
12C2×#518C2211yescharacteristic
13C22×#99C2212yes—
14#2579C412yescomplemented
15#2579C412yescomplemented
16#2487C412yescomplemented
17#2450—22no—
18#2449—22no—
19#2438—22no—
20#2437—22no—
21#2409C412yescomplemented
22#2315C2211yescharacteristic
23#2315C2211yescharacteristic
24#2308C2211yescharacteristic
25C2×#386C2211yescharacteristic
26C2×#385C2211yescharacteristic
27#2273C2212yes—
28C4×#99—22no—
29C4×#99—22no—
30#1740—22no—
31C2×#342—22no—
32C2×#342C2214yes—
33C2×#342C2212yes—
34C2×#342C2212yes—
35C2×#342C2214yes—
36C2×#342C2212yes—
37C2×#342C2214yes—
38C22⋊#95C2214yes—
39C23×Q8C2×C411yescharacteristic, complemented
40C23×D4C2311yescharacteristic
41C24×C4C2311yescharacteristic
42C2×C22⋊Q8—22no—
43C2×C22⋊Q8—22no—
44C2×C22⋊Q8—22no—
45C2×C22⋊Q8—44no—
46#518—28no—
47#518—24no—
48#518—24no—
49#518—44no—
50C2×C4×D4—44no—
51C2×C4×D4C2×C412yes—
52C2×C4×D4C2×C412yes—
53C2×C4×D4D412yes—
54C2×C4×D4D412yes—
55C2×#117—22no—
56C2×#117D412yes—
57C22×C4⋊C4—21no—
58C22×C4⋊C4C2×C411yescharacteristic, complemented
59C22×C4⋊C4C2×C411yescharacteristic, complemented
60C22×C4⋊C4C2×C411yescharacteristic, complemented
61C22×C22⋊C4—22no—
62C22×C22⋊C4—21no—
63C22×C22⋊C4D412yes—
64C22×C22⋊C4C2311yescharacteristic
65C22×C22⋊C4C2311yescharacteristic
66C22×C22⋊C4C2311yescharacteristic
67C22×C22⋊C4C2314yes—
68C22×C22⋊C4C2314yes—
69C22×C22⋊C4C2311yescharacteristic
70C22×C22⋊C4C2311yescharacteristic
71C2×#99—24no—
72C2×#99—24no—
73C2×#99Q818yescomplemented
74#400—22no—
75#399—22no—
76#398—22no—
77#398—22no—
78#396—22no—
79#396—22no—
80#395—22no—
81#393—22no—
82#391—22no—
83#389—22no—
84#387—22no—
85#387—22no—
86#386—22no—
87#386—22no—
88#386C2×C412yes—
89#386C2×C412yes—
90#386C2×C412yes—
91#386C2×C412yes—
92#385—22no—
93#385—22no—
94#385—21no—
95#385—21no—
96#385—22no—
97#385—22no—
98#385—22no—
99#385C2×C412yescomplemented
100#385C2×C412yescomplemented
101#385C2×C412yescomplemented
102#385C2×C412yescomplemented
103#381—21no—
104#380—22no—
105#380—22no—
106#380C2×C412yescomplemented
107#380C2×C412yescomplemented
108#379—21no—
109C4×C22⋊C4—22no—
110C4×C22⋊C4—22no—
111C4×C22⋊C4—44no—
112C2×#95—21no—
113C2×#95—21no—
114C2×#95—44no—
115#342—24no—
116#342—24no—
117#342—24no—
118#342—28no—
119#342—28no—
120#342—44no—
121C25C2×D411yescharacteristic
122C25C2411yescharacteristic
123C22×Q8—22no—
124C22×Q8—42no—
125C22×Q8—82no—
126C22×D4—24no—
127C22×D4—24no—
128C22×D4C22×C412yes—
129C22×D4C2×D412yes—
130C22×D4C2×Q814yes—
131C22×D4#4114yes—
132C23×C4—21no—
133C23×C4—21no—
134C23×C4—24no—
135C23×C4—21no—
136C23×C4—21no—
137C23×C4—24no—
138C23×C4—24no—
139C23×C4—42no—
140C23×C4—82no—
141C23×C4C22⋊C412yes—
142C23×C4C22⋊C412yes—
143C23×C4C22×C411yescharacteristic
144C23×C4C22×C411yescharacteristic
145C23×C4C22×C411yescharacteristic
146C23×C4C22×C411yescharacteristic
147C23×C4C22×C411yescharacteristic
148C23×C4C22×C411yescharacteristic
149C23×C4C2×D411yescharacteristic
150C23×C4C2×D411yescharacteristic
151C23×C4C2×D411yescharacteristic
152C22⋊Q8—48no—
153C22⋊Q8—48no—
154C22⋊Q8—84no—
155C4×D4—44no—
156C4×D4—44no—
157C4×D4—44no—
158C4×D4—44no—
159C4×D4—416no—
160C4×D4—416no—
161#117—28no—
162#117—44no—
163#117—44no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—22no—
169C2×C4⋊C4—22no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—22no—
173C2×C4⋊C4—42no—
174C2×C4⋊C4—42no—
175C2×C4⋊C4—42no—
176C2×C4⋊C4—42no—
177C2×C4⋊C4—82no—
178C2×C4⋊C4C22⋊C412yes—
179C2×C4⋊C4C22⋊C412yes—
180C2×C22⋊C4—24no—
181C2×C22⋊C4—24no—
182C2×C22⋊C4—28no—
183C2×C22⋊C4—24no—
184C2×C22⋊C4—24no—
185C2×C22⋊C4—24no—
186C2×C22⋊C4—24no—
187C2×C22⋊C4—24no—
188C2×C22⋊C4—24no—
189C2×C22⋊C4—24no—
190C2×C22⋊C4—24no—
191C2×C22⋊C4—24no—
192C2×C22⋊C4—24no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—21no—
195C2×C22⋊C4—21no—
196C2×C22⋊C4—21no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—24no—
199C2×C22⋊C4—24no—
200C2×C22⋊C4—24no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—22no—
203C2×C22⋊C4—22no—
204C2×C22⋊C4—44no—
205C2×C22⋊C4—42no—
206C2×C22⋊C4—44no—
207C2×C22⋊C4—82no—
208C2×C22⋊C4C22⋊C412yes—
209C2×C22⋊C4C22⋊C412yes—
210C2×C22⋊C4C22×C412yes—
211C2×C22⋊C4C22×C412yes—
212C2×C22⋊C4C22×C412yes—
213C2×C22⋊C4C2×D412yes—
214C2×C22⋊C4C2×D412yes—
215C2×C22⋊C4C2×D412yes—
216C2×C22⋊C4C2×Q818yes—
217C2×C22⋊C4#4118yes—
218C2×C42—42no—
219C2×C42—42no—
220C2×C42—42no—
221C2×C42C22⋊C412yes—
222C2×C42C22⋊C412yes—
223#99—216no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—42no—
235#95—42no—
236#95—88no—
237C24—22no—
238C24—22no—
239C24—81no—
240C24—81no—
241C24—81no—
242C24—81no—
243C24C2×C22⋊C412yes—
244C24#12412yes—
245C24C4⋊Q812yes—
246C24C23×C411yescharacteristic
247C24C22×D411yescharacteristic
248C24C22×D411yescharacteristic
249C24C22×Q812yes—
250C24C2×#4112yes—
251C24C2×#4114yes—
252C24H(5,𝔽2)14yes—
253C2×Q8—44no—
254C2×Q8—44no—
255C2×Q8—84no—
256C2×Q8—82no—
257C2×Q8—84no—
258C2×D4—24no—
259C2×D4—24no—
260C2×D4—24no—
261C2×D4—24no—
262C2×D4—28no—
263C2×D4—28no—
264C2×D4—28no—
265C2×D4—28no—
266C2×D4C4×Q818yes—
267C2×D4C22⋊Q818yes—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—22no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—28no—
285C22×C4—28no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—48no—
290C22×C4—42no—
291C22×C4—42no—
292C22×C4—42no—
293C22×C4—44no—
294C22×C4—44no—
295C22×C4—42no—
296C22×C4—42no—
297C22×C4—48no—
298C22×C4—42no—
299C22×C4—44no—
300C22×C4—44no—
301C22×C4—81no—
302C22×C4—81no—
303C22×C4—81no—
304C22×C4—81no—
305C22×C4—81no—
306C22×C4—84no—
307C22×C4—81no—
308C22×C4—81no—
309C22×C4—81no—
310C22×C4—81no—
311C22×C4—81no—
312C22×C4—81no—
313C22×C4—81no—
314C22×C4—81no—
315C22×C4—82no—
316C22×C4—82no—
317C22×C4—84no—
318C22×C4—84no—
319C22×C4—84no—
320C22×C4—84no—
321C22×C4—162no—
322C22×C4—162no—
323C22×C4C2×C22⋊C412yes—
324C22×C4C2×C22⋊C412yes—
325C22×C4C2×C22⋊C412yes—
326C22×C4C2×C22⋊C412yes—
327C22×C4C2×C22⋊C412yes—
328C22×C4#12412yes—
329C22×C4C4⋊Q812yes—
330C4⋊C4—42no—
331C4⋊C4—42no—
332C4⋊C4—44no—
333C4⋊C4—44no—
334C4⋊C4—44no—
335C4⋊C4—44no—
336C4⋊C4—48no—
337C4⋊C4—82no—
338C4⋊C4—82no—
339C4⋊C4—82no—
340C4⋊C4—82no—
341C4⋊C4—82no—
342C22⋊C4—24no—
343C22⋊C4—24no—
344C22⋊C4—24no—
345C22⋊C4—24no—
346C22⋊C4—24no—
347C22⋊C4—24no—
348C22⋊C4—28no—
349C22⋊C4—28no—
350C22⋊C4—28no—
351C22⋊C4—28no—
352C22⋊C4—42no—
353C22⋊C4—42no—
354C22⋊C4—44no—
355C22⋊C4—48no—
356C22⋊C4—42no—
357C22⋊C4—44no—
358C22⋊C4—44no—
359C22⋊C4—42no—
360C22⋊C4—42no—
361C22⋊C4—42no—
362C22⋊C4—44no—
363C22⋊C4—44no—
364C22⋊C4—82no—
365C22⋊C4—82no—
366C22⋊C4—82no—
367C42—42no—
368C42—42no—
369C42—48no—
370C42—162no—
371C42—162no—
372C23—22no—
373C23—24no—
374C23—24no—
375C23—22no—
376C23—41no—
377C23—41no—
378C23—44no—
379C23—41no—
380C23—41no—
381C23—41no—
382C23—44no—
383C23—41no—
384C23—42no—
385C23—42no—
386C23—82no—
387C23—82no—
388C23—82no—
389C23—84no—
390C23—82no—
391C23—82no—
392C23—82no—
393C23—162no—
394C23#39114yes—
395C23C22×C22⋊C411yescharacteristic
396C23C2×C4×Q812yes—
397C23C4×#4112yes—
398C23#51812yes—
399C23C2×C22⋊Q812yes—
400C23#52512yes—
401C23C2×#12411yescharacteristic
402C23C2×C4⋊Q811yescharacteristic
403C23#53212yes—
404C23#53412yes—
405C23#53412yes—
406C23#55114yes—
407C23#55114yes—
408C23#55414yes—
409C23#55414yes—
410Q8—88no—
411D4—216no—
412D4—216no—
413C2×C4—24no—
414C2×C4—22no—
415C2×C4—24no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—24no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—41no—
424C2×C4—42no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—42no—
428C2×C4—42no—
429C2×C4—42no—
430C2×C4—42no—
431C2×C4—41no—
432C2×C4—42no—
433C2×C4—42no—
434C2×C4—42no—
435C2×C4—42no—
436C2×C4—41no—
437C2×C4—42no—
438C2×C4—44no—
439C2×C4—48no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—48no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—82no—
447C2×C4—82no—
448C2×C4—82no—
449C2×C4—82no—
450C2×C4—84no—
451C2×C4—84no—
452C2×C4—84no—
453C2×C4—84no—
454C2×C4—84no—
455C2×C4—84no—
456C2×C4—84no—
457C2×C4—82no—
458C2×C4—82no—
459C2×C4—88no—
460C2×C4—88no—
461C2×C4—162no—
462C2×C4—162no—
463C2×C4—162no—
464C2×C4—164no—
465C2×C4—164no—
466C2×C4—168no—
467C2×C4#39114yes—
468C22—21no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—21no—
473C22—22no—
474C22—22no—
475C22—22no—
476C22—24no—
477C22—42no—
478C22—44no—
479C22—44no—
480C22—44no—
481C22—42no—
482C22—41no—
483C22—41no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—44no—
488C22—44no—
489C22—42no—
490C22—42no—
491C22—84no—
492C22—82no—
493C22—84no—
494C22C2×#39111yescharacteristic
495C22#229711yescharacteristic
496C22#230811yescharacteristic
497C22#236814yes—
498C22#256112yes—
499C22#256414yes—
500C22#258312yes—
501C4—24no—
502C4—42no—
503C4—42no—
504C4—42no—
505C4—42no—
506C4—42no—
507C4—82no—
508C4—82no—
509C4—88no—
510C4—88no—
511C4—81no—
512C4—81no—
513C4—81no—
514C2—22no—
515C2—21no—
516C2—24no—
517C2—21no—
518C2—22no—
519C2—24no—
520C2—44no—
521C2#1415911yescharacteristic
522C2#2177611yescharacteristic
523C2#2177711yescharacteristic
subgroupG/H
center, socleC22#2308
commutatorC24C23×C4
Frattini, cosocleC25C24
prime radicalC23×D4C23
LCSG ▷ C24 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C25 ◁ G
LECSG ▷ C25 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
4. #11381C2semidirect
5. #11351C2semidirect
6. #10918C2semidirect
7. #10915C2semidirect
14. #2579C4semidirect
15. #2579C4semidirect
16. #2487C4semidirect
21. #2409C4semidirect
39. C23×Q8C2×C4semidirect
58. C22×C4⋊C4C2×C4semidirect
59. C22×C4⋊C4C2×C4semidirect
60. C22×C4⋊C4C2×C4semidirect
73. C2×#99Q8semidirect
99. #385C2×C4semidirect
100. #385C2×C4semidirect
101. #385C2×C4semidirect
102. #385C2×C4semidirect
106. #380C2×C4semidirect
107. #380C2×C4semidirect

Maximal subgroups

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

subgroupindex
#353492
#203922
#203542
#113812
#113512
#109182
#109152
#107542
#107452