Finite Group Databasebeta

← #107309

Subgroups — #107309

Order 512.

764 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34394C211yesmaximal, characteristic
2C2×#1515C211yesmaximal, characteristic
3C2×#1497C211yesmaximal, characteristic
4#8046C211yesmaximal, characteristic
5#8032C211yesmaximal, characteristic
6#8025C211yesmaximal, characteristic
7#8018C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12#2378—22no—
13#2369C412yescomplemented
14#2365C412yescomplemented
15#2356—21no—
16#2351—21no—
17#2347—21no—
18#2346—21no—
19#2330—22no—
20#2323—22no—
21#2320—22no—
22#2285C2211yescharacteristic
23#2283C2211yescharacteristic
24C2×#383C2211yescharacteristic
25#1515—22no—
26#1515C414yescomplemented
27C2×#328—22no—
28#1497—22no—
29#1497C414yescomplemented
30C2×Q8⋊C8—22no—
31#1296—22no—
32#1295—21no—
33#1295—21no—
34#1294—21no—
35#1294—21no—
36#1294—22no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41#533C2×C412yes—
42#533C2×C412yes—
43#533D412yes—
44#533Q812yes—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×#125—21no—
48C2×C22⋊Q8—21no—
49C2×C22⋊Q8D412yes—
50C2×C4×Q8—22no—
51C2×C4×Q8—21no—
52C2×C4×Q8D412yes—
53C2×#117—21no—
54C2×#117—21no—
55C22×C42C2311yescharacteristic
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432C2×C412yes—
63#432C2×C412yes—
64#431—21no—
65#431—21no—
66#431—21no—
67#431C2×C412yes—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C2×C22⋊C8—21no—
74C2×#97—21no—
75C2×#97—21no—
76C2×C4×C8—21no—
77#391—24no—
78#391—22no—
79#389—24no—
80#389—24no—
81#389—22no—
82#389—22no—
83#389—22no—
84#388—24no—
85#388—24no—
86#387—24no—
87#387—24no—
88#387—24no—
89#384D412yes—
90#383—22no—
91#383—21no—
92#383—21no—
93#383—21no—
94#383—21no—
95#383C2×C412yes—
96#382—22no—
97#382—21no—
98#382D411yescharacteristic
99#382D411yescharacteristic
100#381—21no—
101#380—21no—
102#380D412yes—
103#380D411yescharacteristic
104#380D411yescharacteristic
105C4×C4⋊C4—22no—
106C4×C4⋊C4—22no—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4—21no—
110C4×C4⋊C4C2×C412yes—
111C4×C4⋊C4C2×C412yes—
112C4×C22⋊C4—21no—
113#376—22no—
114#376—21no—
115C2×#95—21no—
116C2×#95—21no—
117C2×#95—21no—
118#328—28no—
119Q8⋊C8—28no—
120C22×Q8—21no—
121C22×Q8C22⋊C412yes—
122C23×C4—41no—
123C23×C4C2×D411yescharacteristic
124C23×C4C2×D411yescharacteristic
125C23×C4C2×D411yescharacteristic
126C22×C8—41no—
127C22×C8—41no—
128C22×C8—41no—
129C4⋊Q8—21no—
130C4⋊Q8—21no—
131C4⋊Q8—21no—
132C4⋊Q8—21no—
133C4⋊Q8C4212yes—
134C4⋊Q8C4212yes—
135C4⋊Q8C4⋊C412yes—
136C4⋊Q8C4⋊C412yes—
137#125—22no—
138#125—22no—
139#125—22no—
140#125—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C22⋊Q8—22no—
144C22⋊Q8—22no—
145C22⋊Q8—22no—
146C22⋊Q8—22no—
147C22⋊Q8C22⋊C414yes—
148C22⋊Q8C4⋊C414yes—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—22no—
152C4×Q8—22no—
153C4×Q8—22no—
154C4×Q8—22no—
155C4×Q8—84no—
156C4×Q8C22⋊C414yes—
157C4×Q8C4⋊C414yes—
158#117—22no—
159#117—22no—
160#117—22no—
161#117—22no—
162#117—82no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—21no—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—24no—
169C2×C4⋊C4—24no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C22⋊C412yes—
173C2×C4⋊C4C22⋊C412yes—
174C2×C4⋊C4C22⋊C412yes—
175C2×C4⋊C4C22⋊C412yes—
176C2×C4⋊C4C22⋊C412yes—
177C2×C22⋊C4—22no—
178C2×C22⋊C4—21no—
179C2×C22⋊C4—21no—
180C2×C22⋊C4—22no—
181C2×C42—22no—
182C2×C42—22no—
183C2×C42—22no—
184C2×C42—24no—
185C2×C42—24no—
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C22×C411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×D411yescharacteristic
193C2×C42C2×Q811yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C2×C42#4111yescharacteristic
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—22no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C4⋊C8—41no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—22no—
207C22⋊C8—41no—
208C22⋊C8—41no—
209C22⋊C8—41no—
210#97—22no—
211#97—22no—
212#97—41no—
213#97—41no—
214C4×C8—22no—
215C4×C8—41no—
216#95—24no—
217#95—24no—
218#95—24no—
219#95—24no—
220#95—22no—
221#95—22no—
222#95—21no—
223#95—21no—
224#95—22no—
225#95—21no—
226#95—21no—
227#95—21no—
228#95—21no—
229#95—22no—
230#95—22no—
231#95—22no—
232#95—41no—
233#95—41no—
234#95—41no—
235C24#12011yescharacteristic
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—22no—
239C2×Q8—42no—
240C2×Q8—42no—
241C2×Q8—44no—
242C2×Q8#9514yes—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—22no—
247C22×C4—22no—
248C22×C4—22no—
249C22×C4—41no—
250C22×C4—42no—
251C22×C4—42no—
252C22×C4—41no—
253C22×C4—42no—
254C22×C4—82no—
255C22×C4—81no—
256C22×C4—81no—
257C22×C4—82no—
258C22×C4C2×C22⋊C411yescharacteristic
259C22×C4C2×C22⋊C411yescharacteristic
260C22×C4C2×C22⋊C411yescharacteristic
261C22×C4C2×C22⋊C411yescharacteristic
262C22×C4C2×C4⋊C411yescharacteristic
263C22×C4C4×D411yescharacteristic
264C22×C4C4×D411yescharacteristic
265C22×C4C4×D411yescharacteristic
266C22×C4C4×D411yescharacteristic
267C22×C4#12011yescharacteristic
268C22×C4#12011yescharacteristic
269C22×C4#12011yescharacteristic
270C22×C4#12111yescharacteristic
271C22×C4#12111yescharacteristic
272C22×C4C22⋊Q811yescharacteristic
273C22×C4C22⋊Q811yescharacteristic
274C22×C4#12311yescharacteristic
275C22×C4#12311yescharacteristic
276C2×C8—42no—
277C2×C8—42no—
278C2×C8—42no—
279C2×C8—82no—
280C2×C8—81no—
281C2×C8—81no—
282C2×C8—82no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—22no—
290C4⋊C4—24no—
291C4⋊C4—24no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—42no—
295C4⋊C4—42no—
296C4⋊C4—42no—
297C4⋊C4—42no—
298C4⋊C4—82no—
299C4⋊C4—82no—
300C4⋊C4—84no—
301C4⋊C4—84no—
302C4⋊C4#9514yes—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—82no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C42—24no—
309C42—24no—
310C42—44no—
311C42—44no—
312C42—84no—
313C42—84no—
314C42C2×C4211yescharacteristic
315C42C2×C22⋊C411yescharacteristic
316C42C2×C4⋊C411yescharacteristic
317C42C2×C4⋊C411yescharacteristic
318C42#11711yescharacteristic
319C42#11711yescharacteristic
320C42C4×D411yescharacteristic
321C42C4×D411yescharacteristic
322C42C4×D411yescharacteristic
323C42C4×D411yescharacteristic
324C42#12111yescharacteristic
325C42#12111yescharacteristic
326C42C22⋊Q811yescharacteristic
327C42C22⋊Q811yescharacteristic
328C42#12311yescharacteristic
329C42#12311yescharacteristic
330C23—22no—
331C23—21no—
332C23—21no—
333C23—22no—
334C23#37911yescharacteristic
335C23#38511yescharacteristic
336C23#38611yescharacteristic
337Q8—44no—
338Q8—44no—
339Q8—44no—
340Q8—82no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—22no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—24no—
350C2×C4—24no—
351C2×C4—41no—
352C2×C4—41no—
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×C4—42no—
378C2×C4—82no—
379C2×C4—82no—
380C2×C4—82no—
381C2×C4—162no—
382C2×C4—162no—
383C2×C4C2×#9511yescharacteristic
384C2×C4C4×C22⋊C411yescharacteristic
385C2×C4C4×C22⋊C411yescharacteristic
386C2×C4#38011yescharacteristic
387C2×C4#38011yescharacteristic
388C2×C4#38111yescharacteristic
389C2×C4#38511yescharacteristic
390C2×C4#38511yescharacteristic
391C2×C4#38511yescharacteristic
392C2×C4#38611yescharacteristic
393C2×C4#38611yescharacteristic
394C2×C4#38611yescharacteristic
395C8—82no—
396C8—82no—
397C8—82no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—21no—
403C22—21no—
404C22—42no—
405C22—42no—
406C22C22⋊#9511yescharacteristic
407C22#179412yes—
408C22#179511yescharacteristic
409C22#179611yescharacteristic
410C22#179612yes—
411C4—21no—
412C4—21no—
413C4—21no—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—21no—
419C4—21no—
420C4—21no—
421C4—21no—
422C4—21no—
423C4—44no—
424C4—44no—
425C4—82no—
426C4—82no—
427C4—82no—
428C4—82no—
429C4—162no—
430C2—41no—
431C2—41no—
432C2#850112yes—
433C2#850212yes—
434C2#1075211yescharacteristic
435C2#1077511yescharacteristic
436C2#1077711yescharacteristic
subgroupG/H
center, socleC23#379
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
13. #2369C4semidirect
14. #2365C4semidirect
26. #1515C4semidirect
29. #1497C4semidirect

Maximal subgroups

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

subgroupindex
#343942
C2×#15152
C2×#14972
#80462
#80322
#80252
#80182