Finite Group Databasebeta

← #108117

Subgroups — #108117

Order 512.

751 conjugacy classes of subgroups.

The subgroup lattice has 423 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#34373C211yesmaximal, characteristic
2#21845C211yesmaximal, characteristic
3#17672C211yesmaximal, characteristic
4#8094C211yesmaximal, characteristic
5#7984C211yesmaximal, characteristic
6#7951C211yesmaximal, characteristic
7#7944C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12#2378—22no—
13#2362—22no—
14#2355—21no—
15#2355—22no—
16Q8×C4⋊C4—21no—
17#2336—22no—
18#2331C412yescomplemented
19#2327C412yescomplemented
20#2314—22no—
21#2285C2211yescharacteristic
22#2283C2211yescharacteristic
23#2283C2211yescharacteristic
24#1995C414yescomplemented
25#1986—22no—
26#1826—22no—
27#1537—22no—
28#1531C414yescomplemented
29#1528—22no—
30#1303—22no—
31#1289—22no—
32#1285—21no—
33#1285—21no—
34#1284—22no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533C2×C412yes—
40#533C2×C412yes—
41#533D412yes—
42#533Q812yes—
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C411yescharacteristic
45C2×#125D412yes—
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—21no—
48C2×C4×Q8—22no—
49C2×C4×Q8—21no—
50C2×C4×Q8—21no—
51C2×#117—21no—
52C2×#117D412yes—
53C22×C42C2311yescharacteristic
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432C2×C412yes—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#423—21no—
67#423C2×C412yes—
68C2×C4⋊C8—21no—
69#407—21no—
70#407—21no—
71C2×C22⋊C8—21no—
72C4×M16—21no—
73C4×M16—21no—
74C2×#97C2×C412yes—
75#391—22no—
76#391—24no—
77#389—24no—
78#389—22no—
79#388—24no—
80#388—24no—
81#387—24no—
82#387—24no—
83#387—24no—
84#387—24no—
85#387—24no—
86#384—22no—
87#384—21no—
88#383—21no—
89#383—21no—
90#382—22no—
91#382—21no—
92#382—21no—
93#382C2×C412yes—
94#381—21no—
95#381—21no—
96#380—21no—
97#380—21no—
98#380—21no—
99#380D412yes—
100C4×C4⋊C4—22no—
101C4×C4⋊C4—22no—
102C4×C4⋊C4—24no—
103C4×C4⋊C4—22no—
104C4×C4⋊C4—21no—
105C4×C4⋊C4—21no—
106C4×C4⋊C4D411yescharacteristic
107C4×C4⋊C4D411yescharacteristic
108C4×C4⋊C4D412yes—
109C4×C22⋊C4—21no—
110C4×C22⋊C4D412yes—
111#376C2×C412yes—
112#376C2×C412yes—
113#339—28no—
114#330—28no—
115C22×Q8—21no—
116C22×Q8—21no—
117C23×C4C2×D411yescharacteristic
118C23×C4C2×D411yescharacteristic
119C23×C4C2×D411yescharacteristic
120C2×M16—22no—
121C2×M16—41no—
122C22×C8—22no—
123C4⋊Q8—21no—
124C4⋊Q8—21no—
125C4⋊Q8—21no—
126C4⋊Q8—21no—
127C4⋊Q8C4212yes—
128C4⋊Q8C4212yes—
129C4⋊Q8C4⋊C412yes—
130C4⋊Q8C4⋊C412yes—
131#125—22no—
132#125—22no—
133#125C22⋊C414yes—
134#125C4⋊C414yes—
135C22⋊Q8—22no—
136C22⋊Q8—22no—
137C22⋊Q8—22no—
138C22⋊Q8—22no—
139C22⋊Q8—22no—
140C22⋊Q8—22no—
141C22⋊Q8—22no—
142C22⋊Q8—22no—
143C4×Q8—22no—
144C4×Q8—22no—
145C4×Q8—22no—
146C4×Q8—22no—
147C4×Q8—22no—
148C4×Q8—22no—
149C4×Q8—22no—
150C4×Q8—22no—
151C4×Q8—84no—
152#117—22no—
153#117—22no—
154#117—82no—
155#117C22⋊C414yes—
156#117C4⋊C414yes—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4—24no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4—41no—
169C2×C4⋊C4C22⋊C412yes—
170C2×C4⋊C4C22⋊C412yes—
171C2×C22⋊C4—22no—
172C2×C22⋊C4—22no—
173C2×C22⋊C4—22no—
174C2×C22⋊C4—41no—
175C2×C42—22no—
176C2×C42—22no—
177C2×C42—24no—
178C2×C42—24no—
179C2×C42—22no—
180C2×C42C22⋊C412yes—
181C2×C42C22⋊C412yes—
182C2×C42C22×C411yescharacteristic
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C2×D411yescharacteristic
186C2×C42C2×D411yescharacteristic
187C2×C42C2×D411yescharacteristic
188C2×C42C2×D411yescharacteristic
189C2×C42C2×Q811yescharacteristic
190C2×C42#4111yescharacteristic
191C2×C42#4111yescharacteristic
192C2×C42#4111yescharacteristic
193C2×C42#4111yescharacteristic
194C4⋊C8—22no—
195C4⋊C8—22no—
196C4⋊C8—22no—
197C4⋊C8—41no—
198C4⋊C8—41no—
199C4⋊C8—41no—
200C22⋊C8—22no—
201C22⋊C8—41no—
202C22⋊C8—41no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205C22⋊C8—41no—
206#97—22no—
207#97—22no—
208#97—22no—
209#97—41no—
210C4×C8—41no—
211C4×C8—41no—
212#95—24no—
213#95—24no—
214#95—24no—
215#95—24no—
216#95—22no—
217#95—22no—
218#95—22no—
219#95—22no—
220#95—21no—
221#95—21no—
222#95—22no—
223#95—22no—
224#95—41no—
225#95—41no—
226#95C22⋊C414yes—
227C24#12011yescharacteristic
228C2×Q8—22no—
229C2×Q8—22no—
230C2×Q8—22no—
231C2×Q8—22no—
232C2×Q8—42no—
233C2×Q8—42no—
234C2×Q8—82no—
235C22×C4—22no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—22no—
239C22×C4—24no—
240C22×C4—42no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—42no—
244C22×C4—81no—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C411yescharacteristic
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C411yescharacteristic
249C22×C4C2×C4⋊C411yescharacteristic
250C22×C4C4×D411yescharacteristic
251C22×C4C4×D411yescharacteristic
252C22×C4C4×D411yescharacteristic
253C22×C4C4×D411yescharacteristic
254C22×C4#12011yescharacteristic
255C22×C4#12011yescharacteristic
256C22×C4#12011yescharacteristic
257C22×C4#12111yescharacteristic
258C22×C4#12111yescharacteristic
259C22×C4C22⋊Q811yescharacteristic
260C22×C4C22⋊Q811yescharacteristic
261C22×C4#12311yescharacteristic
262C22×C4#12311yescharacteristic
263M16—44no—
264M16—81no—
265M16—81no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—42no—
269C2×C8—82no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—24no—
277C4⋊C4—24no—
278C4⋊C4—24no—
279C4⋊C4—24no—
280C4⋊C4—42no—
281C4⋊C4—42no—
282C4⋊C4—44no—
283C4⋊C4—44no—
284C4⋊C4—82no—
285C4⋊C4—84no—
286C4⋊C4—84no—
287C4⋊C4—82no—
288C4⋊C4#9514yes—
289C4⋊C4#9514yes—
290C22⋊C4—24no—
291C22⋊C4—24no—
292C22⋊C4—82no—
293C22⋊C4—82no—
294C22⋊C4—81no—
295C22⋊C4—81no—
296C42—24no—
297C42—24no—
298C42—42no—
299C42—42no—
300C42—44no—
301C42—82no—
302C42—84no—
303C42—84no—
304C42C2×C4211yescharacteristic
305C42C2×C22⋊C411yescharacteristic
306C42C2×C4⋊C411yescharacteristic
307C42C2×C4⋊C411yescharacteristic
308C42#11711yescharacteristic
309C42#11711yescharacteristic
310C42C4×D411yescharacteristic
311C42C4×D411yescharacteristic
312C42C4×D411yescharacteristic
313C42C4×D411yescharacteristic
314C42#12111yescharacteristic
315C42#12111yescharacteristic
316C42C22⋊Q811yescharacteristic
317C42C22⋊Q811yescharacteristic
318C42#12311yescharacteristic
319C42#12311yescharacteristic
320C23—21no—
321C23—21no—
322C23—41no—
323C23—41no—
324C23#37911yescharacteristic
325C23#38511yescharacteristic
326C23#38611yescharacteristic
327Q8—44no—
328Q8—44no—
329Q8—82no—
330Q8—82no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—24no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—42no—
357C2×C4—42no—
358C2×C4—44no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—44no—
363C2×C4—42no—
364C2×C4—44no—
365C2×C4—42no—
366C2×C4—82no—
367C2×C4—82no—
368C2×C4—82no—
369C2×C4C2×#9511yescharacteristic
370C2×C4C4×C22⋊C411yescharacteristic
371C2×C4C4×C22⋊C411yescharacteristic
372C2×C4#38011yescharacteristic
373C2×C4#38011yescharacteristic
374C2×C4#38111yescharacteristic
375C2×C4#38511yescharacteristic
376C2×C4#38511yescharacteristic
377C2×C4#38511yescharacteristic
378C2×C4#38611yescharacteristic
379C2×C4#38611yescharacteristic
380C2×C4#38611yescharacteristic
381C8—82no—
382C8—82no—
383C8—82no—
384C22—21no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—81no—
393C22—81no—
394C22C22⋊#9511yescharacteristic
395C22#179511yescharacteristic
396C22#179611yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—21no—
402C4—21no—
403C4—21no—
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—44no—
409C4—44no—
410C4—84no—
411C4—82no—
412C4—82no—
413C4—162no—
414C4#191212yes—
415C2—21no—
416C2—21no—
417C2—41no—
418C2—41no—
419C2#1074111yescharacteristic
420C2#1077511yescharacteristic
421C2#1078311yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC42C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24#120
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
18. #2331C4semidirect
19. #2327C4semidirect
24. #1995C4semidirect
28. #1531C4semidirect

Maximal subgroups

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

subgroupindex
#343732
#218452
#176722
#80942
#79842
#79512
#79442