Finite Group Databasebeta

← #108129

Subgroups — #108129

Order 512.

774 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34231C211yesmaximal, characteristic
2#17748C211yesmaximal, characteristic
3#17479C211yesmaximal, characteristic
4#8110C211yesmaximal, characteristic
5#7949C211yesmaximal, characteristic
6#7926C211yesmaximal, characteristic
7#7869C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2369—21no—
14#2365—21no—
15#2362—22no—
16#2356—22no—
17#2347—21no—
18#2346—21no—
19#2332C414yes—
20#2320—21no—
21C4×#125—22no—
22#2283C2211yescharacteristic
23#2276—21no—
24C4×#117C2211yescharacteristic
25#1558—21no—
26#1556—21no—
27#1554—22no—
28#1550C414yescomplemented
29#1497C414yescomplemented
30#1493—22no—
31#1479—22no—
32#1305—21no—
33#1305—21no—
34#1285—21no—
35#1285—21no—
36#1282—22no—
37#1276—22no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533—21no—
42#533C2×C412yes—
43#533C2×C412yes—
44#533D412yes—
45#533Q812yes—
46C2×C4⋊Q8D411yescharacteristic
47C2×C4⋊Q8D411yescharacteristic
48C2×#125—21no—
49C2×C22⋊Q8—21no—
50C2×C22⋊Q8C2×C412yes—
51C2×C4×Q8—22no—
52C2×C4×Q8—21no—
53C2×C4×Q8D412yes—
54C2×#117—21no—
55C2×#117—21no—
56C22×C42C2311yescharacteristic
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432C2×C412yes—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431C2×C412yes—
73#423—21no—
74#423C2×C412yes—
75C2×C4⋊C8—21no—
76C2×C4⋊C8C2×C412yes—
77#407—21no—
78#407—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C4×M16—21no—
82C4×M16—21no—
83C2×#97—21no—
84C2×C4×C8—21no—
85#391—22no—
86#391—24no—
87#389—22no—
88#389—22no—
89#389—22no—
90#388—24no—
91#388—24no—
92#387—24no—
93#387—24no—
94#387—24no—
95#383—22no—
96#383—21no—
97#382—22no—
98#382—21no—
99#381C2×C412yes—
100#380D411yescharacteristic
101#380D411yescharacteristic
102C4×C4⋊C4—22no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—24no—
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—24no—
108C4×C4⋊C4—21no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4D411yescharacteristic
111C4×C4⋊C4D411yescharacteristic
112C4×C22⋊C4—21no—
113C4×C22⋊C4C2×C412yes—
114C4×C22⋊C4D412yes—
115#376—22no—
116#376—21no—
117#376—21no—
118C43D412yes—
119#333—24no—
120#332—24no—
121Q8⋊C8—28no—
122C22×Q8—21no—
123C22×Q8C22⋊C412yes—
124C23×C4C22×C411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C23×C4C2×D411yescharacteristic
127C2×M16—22no—
128C2×M16—41no—
129C22×C8—22no—
130C22×C8—41no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8—21no—
134C4⋊Q8—21no—
135C4⋊Q8C22⋊C412yes—
136C4⋊Q8C22⋊C412yes—
137C4⋊Q8C4⋊C412yes—
138C4⋊Q8C4⋊C412yes—
139#125—22no—
140#125—22no—
141#125—22no—
142#125—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8—22no—
148C22⋊Q8—22no—
149C22⋊Q8C4214yes—
150C22⋊Q8C4⋊C414yes—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—22no—
156C4×Q8—22no—
157C4×Q8—48no—
158C4×Q8C22⋊C414yes—
159C4×Q8C4⋊C414yes—
160#117—22no—
161#117—22no—
162#117—22no—
163#117—22no—
164#117—44no—
165C2×C4⋊C4—22no—
166C2×C4⋊C4—21no—
167C2×C4⋊C4—21no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—24no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—24no—
176C2×C4⋊C4C22⋊C412yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—22no—
179C2×C22⋊C4C22⋊C412yes—
180C2×C22⋊C4C22⋊C412yes—
181C2×C42—22no—
182C2×C42—24no—
183C2×C42—22no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—24no—
187C2×C42C22⋊C412yes—
188C2×C42C22⋊C412yes—
189C2×C42C22×C411yescharacteristic
190C2×C42C22×C411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×D411yescharacteristic
194C2×C42C2×D411yescharacteristic
195C2×C42C2×D411yescharacteristic
196C2×C42C2×Q811yescharacteristic
197C2×C42#4111yescharacteristic
198C2×C42#4111yescharacteristic
199C2×C42#4111yescharacteristic
200C2×C42#4111yescharacteristic
201C4⋊C8—22no—
202C4⋊C8—22no—
203C4⋊C8—22no—
204C4⋊C8—22no—
205C4⋊C8—22no—
206C4⋊C8—22no—
207C4⋊C8—41no—
208C4⋊C8—41no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211C22⋊C8—41no—
212C22⋊C8—41no—
213C22⋊C8—41no—
214C22⋊C8—41no—
215C22⋊C8—41no—
216C22⋊C8—41no—
217#97—22no—
218#97—41no—
219#97—41no—
220#97—41no—
221C4×C8—22no—
222C4×C8—41no—
223C4×C8—41no—
224C4×C8—41no—
225#95—24no—
226#95—24no—
227#95—24no—
228#95—24no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—21no—
233#95—21no—
234#95—21no—
235#95—21no—
236#95—22no—
237C24C2×C22⋊C411yescharacteristic
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—22no—
241C2×Q8—44no—
242C2×Q8—42no—
243C2×Q8—42no—
244C2×Q8#9514yes—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—42no—
252C22×C4—42no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4C2×C4211yescharacteristic
256C22×C4C2×C22⋊C411yescharacteristic
257C22×C4C2×C22⋊C411yescharacteristic
258C22×C4C2×C4⋊C411yescharacteristic
259C22×C4#11711yescharacteristic
260C22×C4#11711yescharacteristic
261C22×C4C4×D411yescharacteristic
262C22×C4C4×D411yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4#12011yescharacteristic
266C22×C4#12011yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12111yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4C22⋊Q811yescharacteristic
272C22×C4C22⋊Q811yescharacteristic
273M16—82no—
274M16—82no—
275C2×C8—42no—
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—82no—
280C2×C8—82no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—24no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—24no—
289C4⋊C4—22no—
290C4⋊C4—22no—
291C4⋊C4—22no—
292C4⋊C4—24no—
293C4⋊C4—42no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—48no—
297C4⋊C4—84no—
298C4⋊C4#9514yes—
299C22⋊C4—24no—
300C22⋊C4—24no—
301C22⋊C4—44no—
302C22⋊C4—42no—
303C22⋊C4—42no—
304C42—24no—
305C42—24no—
306C42—42no—
307C42—42no—
308C42—44no—
309C42—44no—
310C42—44no—
311C42—48no—
312C42—84no—
313C42C2×C22⋊C411yescharacteristic
314C42C2×C22⋊C411yescharacteristic
315C42C2×C4⋊C411yescharacteristic
316C42C2×C4⋊C411yescharacteristic
317C42C4×D411yescharacteristic
318C42C4×D411yescharacteristic
319C42C4×D411yescharacteristic
320C42C4×D411yescharacteristic
321C42#12111yescharacteristic
322C42#12111yescharacteristic
323C42C22⋊Q811yescharacteristic
324C42C22⋊Q811yescharacteristic
325C42#12311yescharacteristic
326C42#12311yescharacteristic
327C42#12311yescharacteristic
328C42#12311yescharacteristic
329C23—21no—
330C23—21no—
331C23—41no—
332C23—41no—
333C23C4×C22⋊C411yescharacteristic
334C23#37911yescharacteristic
335C23#38011yescharacteristic
336Q8—44no—
337Q8—44no—
338Q8—44no—
339Q8—82no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—24no—
351C2×C4—24no—
352C2×C4—24no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—44no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—42no—
376C2×C4—42no—
377C2×C4C2×#9511yescharacteristic
378C2×C4C4×C22⋊C411yescharacteristic
379C2×C4#38011yescharacteristic
380C2×C4#38111yescharacteristic
381C2×C4#38511yescharacteristic
382C2×C4#38511yescharacteristic
383C2×C4#38511yescharacteristic
384C2×C4#38511yescharacteristic
385C2×C4#38611yescharacteristic
386C2×C4#38611yescharacteristic
387C2×C4#38611yescharacteristic
388C2×C4#38611yescharacteristic
389C8—82no—
390C8—82no—
391C8—82no—
392C8—82no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—81no—
402C22—81no—
403C22C22⋊#9511yescharacteristic
404C22#179111yescharacteristic
405C22#179111yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—44no—
417C4—44no—
418C4—44no—
419C4—82no—
420C4—82no—
421C4—82no—
422C4#190512yes—
423C4#190612yes—
424C2—21no—
425C2—21no—
426C2—41no—
427C2—41no—
428C2#1074711yescharacteristic
429C2#1077211yescharacteristic
430C2#1078611yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
28. #1550C4semidirect
29. #1497C4semidirect

Maximal subgroups

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

subgroupindex
#342312
#177482
#174792
#81102
#79492
#79262
#78692