Finite Group Databasebeta

← #107692

Subgroups — #107692

Order 512.

882 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34398C211yesmaximal, characteristic
2#17744C211yesmaximal, characteristic
3#17561C211yesmaximal, characteristic
4#8017C211yesmaximal, characteristic, complemented
5#7977C211yesmaximal, characteristic, complemented
6#7959C211yesmaximal, characteristic, complemented
7#7945C211yesmaximal, characteristic, complemented
8C2×#532C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12#2379—22no—
13#2361—22no—
14#2357C412yescomplemented
15#2349C412yescomplemented
16#2339—22no—
17#2331—21no—
18#2328—21no—
19#2325—22no—
20#2318—22no—
21#2283C2211yescharacteristic
22#2283C2211yescharacteristic
23C2×#382C2211yescharacteristic
24#1556—21no—
25#1555—21no—
26#1553C414yescomplemented
27#1547—22no—
28#1514—22no—
29#1510—22no—
30#1502C414yescomplemented
31#1294—22no—
32#1288—22no—
33#1286—22no—
34#1284—22no—
35C22×#41—21no—
36#532—21no—
37#532—21no—
38#532—21no—
39#532—21no—
40#532C2×C412yes—
41#532C2×C412yes—
42#532D412yes—
43#532Q812yes—
44C2×C4⋊Q8D411yescharacteristic
45C2×#127D411yescharacteristic
46C2×#124—21no—
47C2×#121—21no—
48C2×#121D412yes—
49C2×C4×D4—24no—
50C2×C4×D4—22no—
51C2×C4×D4—21no—
52C2×C4×D4C2×C412yes—
53C22×C4⋊C4—21no—
54C22×C42C2311yescharacteristic
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432C2×C412yescomplemented
63#431—21no—
64#431—21no—
65#431C2×C412yescomplemented
66#431C2×C412yescomplemented
67#423—21no—
68#423—21no—
69C2×C4⋊C8—21no—
70#407—21no—
71#407—21no—
72C2×C22⋊C8—21no—
73C4×M16—21no—
74C4×M16—21no—
75C2×#97—21no—
76#391—24no—
77#390—22no—
78#389—22no—
79#388—24no—
80#388—24no—
81#387—24no—
82#386—24no—
83#386—24no—
84#386—24no—
85#385—24no—
86#385—24no—
87#383—21no—
88#383D412yes—
89#382—21no—
90#382—21no—
91#382—21no—
92#382—21no—
93#382C2×C412yes—
94#382D412yes—
95#381—22no—
96#381—21no—
97#381D412yes—
98#380—21no—
99#380—22no—
100#380—21no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4C2×C411yescharacteristic
103C4×C4⋊C4C2×C411yescharacteristic
104C4×C4⋊C4C2×C412yescomplemented
105C4×C22⋊C4—21no—
106C4×C22⋊C4—22no—
107C4×C22⋊C4—21no—
108#376—21no—
109#376—21no—
110C2×#95—21no—
111C2×#95—22no—
112C2×#95D412yes—
113#332—24no—
114#331—24no—
115#327—28no—
116C2×#41—22no—
117C2×#41—22no—
118C2×#41—22no—
119C2×#41—22no—
120C2×#41—84no—
121C22×Q8—21no—
122C22×D4—22no—
123C22×D4—21no—
124C22×D4—21no—
125C22×D4C22⋊C412yes—
126C23×C4—22no—
127C23×C4—22no—
128C23×C4—24no—
129C23×C4C2×D411yescharacteristic
130C23×C4C2×D411yescharacteristic
131C23×C4C2×D411yescharacteristic
132C2×M16—41no—
133C2×M16—41no—
134C22×C8—41no—
135C4⋊Q8—21no—
136C4⋊Q8—21no—
137C4⋊Q8C22⋊C412yes—
138C4⋊Q8C4⋊C412yes—
139#127—21no—
140#127—21no—
141#127C22⋊C412yes—
142#127C4⋊C412yes—
143#124—22no—
144#124—22no—
145#124—22no—
146#124—22no—
147#121—22no—
148#121—22no—
149#121—22no—
150#121—22no—
151#121—22no—
152#121—22no—
153#121C22⋊C414yes—
154#121C4⋊C414yes—
155C4×D4—22no—
156C4×D4—22no—
157C4×D4—22no—
158C4×D4—22no—
159C4×D4—22no—
160C4×D4—22no—
161C4×D4—84no—
162C4×D4—88no—
163C4×D4C4214yes—
164C4×D4C4⋊C414yes—
165C2×C4⋊C4—21no—
166C2×C4⋊C4—22no—
167C2×C4⋊C4—24no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—82no—
173C2×C4⋊C4C22⋊C412yes—
174C2×C22⋊C4—22no—
175C2×C22⋊C4—22no—
176C2×C22⋊C4—22no—
177C2×C22⋊C4—24no—
178C2×C22⋊C4—24no—
179C2×C22⋊C4—24no—
180C2×C22⋊C4—41no—
181C2×C42—21no—
182C2×C42—21no—
183C2×C42—24no—
184C2×C42—22no—
185C2×C42C22×C411yescharacteristic
186C2×C42C22×C411yescharacteristic
187C2×C42C22×C411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×D411yescharacteristic
190C2×C42C2×D411yescharacteristic
191C2×C42C2×D411yescharacteristic
192C2×C42C2×Q811yescharacteristic
193C2×C42#4111yescharacteristic
194C2×C42#4111yescharacteristic
195C2×C42#4111yescharacteristic
196C2×C42#4111yescharacteristic
197C4⋊C8—22no—
198C4⋊C8—22no—
199C4⋊C8—22no—
200C4⋊C8—41no—
201C4⋊C8—41no—
202C4⋊C8—41no—
203C22⋊C8—22no—
204C22⋊C8—22no—
205C22⋊C8—22no—
206C22⋊C8—41no—
207C22⋊C8—41no—
208C22⋊C8—41no—
209#97—22no—
210#97—41no—
211#97—41no—
212#97—41no—
213C4×C8—22no—
214C4×C8—22no—
215#95—24no—
216#95—24no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—22no—
221#95—22no—
222#95—22no—
223#95—22no—
224#95—41no—
225#95—41no—
226#95—41no—
227#95C22⋊C412yes—
228#95C22⋊C414yes—
229#95C22⋊C412yes—
230C24—22no—
231C24#12011yescharacteristic
232#41—84no—
233#41—84no—
234#41—84no—
235#41—84no—
236C2×Q8—22no—
237C2×Q8—22no—
238C2×Q8—82no—
239C2×D4—24no—
240C2×D4—24no—
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—22no—
244C2×D4—22no—
245C2×D4—22no—
246C2×D4—42no—
247C2×D4—42no—
248C2×D4—48no—
249C2×D4—82no—
250C2×D4—82no—
251C2×D4#9514yes—
252C22×C4—22no—
253C22×C4—24no—
254C22×C4—24no—
255C22×C4—22no—
256C22×C4—24no—
257C22×C4—22no—
258C22×C4—41no—
259C22×C4—41no—
260C22×C4—42no—
261C22×C4—44no—
262C22×C4—42no—
263C22×C4—81no—
264C22×C4—84no—
265C22×C4—82no—
266C22×C4—84no—
267C22×C4—84no—
268C22×C4—82no—
269C22×C4—84no—
270C22×C4C2×C22⋊C411yescharacteristic
271C22×C4C2×C22⋊C411yescharacteristic
272C22×C4C2×C22⋊C411yescharacteristic
273C22×C4C2×C22⋊C411yescharacteristic
274C22×C4C2×C4⋊C411yescharacteristic
275C22×C4C4×D411yescharacteristic
276C22×C4C4×D411yescharacteristic
277C22×C4C4×D411yescharacteristic
278C22×C4C4×D411yescharacteristic
279C22×C4#12011yescharacteristic
280C22×C4#12011yescharacteristic
281C22×C4#12011yescharacteristic
282C22×C4#12111yescharacteristic
283C22×C4#12111yescharacteristic
284C22×C4C22⋊Q811yescharacteristic
285C22×C4C22⋊Q811yescharacteristic
286C22×C4#12311yescharacteristic
287C22×C4#12311yescharacteristic
288M16—81no—
289M16—81no—
290M16—82no—
291C2×C8—42no—
292C2×C8—42no—
293C2×C8—42no—
294C2×C8—82no—
295C4⋊C4—22no—
296C4⋊C4—22no—
297C4⋊C4—22no—
298C4⋊C4—44no—
299C4⋊C4—82no—
300C4⋊C4—81no—
301C4⋊C4—81no—
302C4⋊C4—84no—
303C4⋊C4—82no—
304C4⋊C4#9514yes—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—24no—
308C22⋊C4—24no—
309C22⋊C4—44no—
310C22⋊C4—84no—
311C22⋊C4—84no—
312C22⋊C4—84no—
313C22⋊C4—81no—
314C22⋊C4—81no—
315C42—42no—
316C42—42no—
317C42—82no—
318C42—84no—
319C42C2×C4211yescharacteristic
320C42C2×C22⋊C411yescharacteristic
321C42C2×C4⋊C411yescharacteristic
322C42C2×C4⋊C411yescharacteristic
323C42#11711yescharacteristic
324C42#11711yescharacteristic
325C42C4×D411yescharacteristic
326C42C4×D411yescharacteristic
327C42C4×D411yescharacteristic
328C42C4×D411yescharacteristic
329C42#12111yescharacteristic
330C42#12111yescharacteristic
331C42C22⋊Q811yescharacteristic
332C42C22⋊Q811yescharacteristic
333C42#12311yescharacteristic
334C42#12311yescharacteristic
335C23—21no—
336C23—21no—
337C23—22no—
338C23—24no—
339C23—41no—
340C23—44no—
341C23—44no—
342C23—42no—
343C23—42no—
344C23#37911yescharacteristic
345C23#38511yescharacteristic
346C23#38611yescharacteristic
347Q8—82no—
348Q8—82no—
349D4—44no—
350D4—44no—
351D4—48no—
352D4—48no—
353D4—82no—
354D4—82no—
355D4—82no—
356D4—82no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—24no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—44no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—44no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—44no—
387C2×C4—42no—
388C2×C4—84no—
389C2×C4—84no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4—164no—
394C2×C4—168no—
395C2×C4C2×#9511yescharacteristic
396C2×C4C4×C22⋊C411yescharacteristic
397C2×C4C4×C22⋊C411yescharacteristic
398C2×C4#38011yescharacteristic
399C2×C4#38011yescharacteristic
400C2×C4#38111yescharacteristic
401C2×C4#38511yescharacteristic
402C2×C4#38511yescharacteristic
403C2×C4#38511yescharacteristic
404C2×C4#38611yescharacteristic
405C2×C4#38611yescharacteristic
406C2×C4#38611yescharacteristic
407C8—82no—
408C8—82no—
409C8—82no—
410C22—21no—
411C22—21no—
412C22—21no—
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—42no—
418C22—42no—
419C22—44no—
420C22—44no—
421C22—44no—
422C22—84no—
423C22—84no—
424C22C22⋊#9511yescharacteristic
425C22#179511yescharacteristic
426C22#179611yescharacteristic
427C22#190412yes—
428C4—21no—
429C4—21no—
430C4—21no—
431C4—21no—
432C4—21no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—44no—
441C4—82no—
442C4—84no—
443C4—82no—
444C4—162no—
445C2—21no—
446C2—21no—
447C2—22no—
448C2—41no—
449C2—84no—
450C2#1076511yescharacteristic
451C2#1078511yescharacteristic
452C2#1079211yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC2×#532C22
LCSG ▷ C42 ▷ 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
4. #8017C2semidirect
5. #7977C2semidirect
6. #7959C2semidirect
7. #7945C2semidirect
14. #2357C4semidirect
15. #2349C4semidirect
26. #1553C4semidirect
30. #1502C4semidirect
62. #432C2×C4semidirect
65. #431C2×C4semidirect
66. #431C2×C4semidirect
104. C4×C4⋊C4C2×C4semidirect

Maximal subgroups

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

subgroupindex
#343982
#177442
#175612
#80172
#79772
#79592
#79452