Finite Group Databasebeta

← #122104

Subgroups — #122104

Order 512.

550 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#37566C211yesmaximal, characteristic
2#8050C211yesmaximal, characteristic
3#7980C211yesmaximal, characteristic
4#7976C211yesmaximal, characteristic
5#7958C211yesmaximal, characteristic
6#7945C211yesmaximal, characteristic
7#7858C211yesmaximal, characteristic
8#2920C2211yescharacteristic
9#2920C2211yescharacteristic
10C2×#431C2211yescharacteristic
11#2841—22no—
12#2667—21no—
13#2666—22no—
14#2666—22no—
15#2665—21no—
16#2612—22no—
17#2608—21no—
18#2607—21no—
19#2606C412yescomplemented
20#2605C412yescomplemented
21#2283C2211yescharacteristic
22#2283C2211yescharacteristic
23#2283C2211yescharacteristic
24#2283C2211yescharacteristic
25#1297—21no—
26#1297—21no—
27#1289—22no—
28#1288—22no—
29#1286—22no—
30#1284—21no—
31#1284—21no—
32#1274—22no—
33C22×C42C2311yescharacteristic
34#432—21no—
35#432—21no—
36#432—21no—
37#432—21no—
38#431—21no—
39#431—21no—
40#431—21no—
41#431—21no—
42#431—21no—
43#431—21no—
44#431—21no—
45#431Q812yescomplemented
46#423—21no—
47#423—21no—
48C2×C4⋊C8—21no—
49#407—21no—
50#407—21no—
51C2×C22⋊C8—21no—
52C4×M16—21no—
53C4×M16Q812yescomplemented
54C2×C4×C8Q812yescomplemented
55#400—22no—
56#400—22no—
57#400—22no—
58#399—22no—
59#398—22no—
60#398—22no—
61#398—22no—
62#398—22no—
63#398—22no—
64#398—22no—
65#396—22no—
66#396—22no—
67#395—22no—
68#395—22no—
69#395—22no—
70#393—22no—
71#391—22no—
72#391—22no—
73#391—22no—
74#389—22no—
75#389—22no—
76#389—22no—
77#387—22no—
78#387—22no—
79#387—22no—
80#387—22no—
81#387—22no—
82#387—22no—
83#383—21no—
84#383C2×C412yes—
85#383D412yes—
86#383Q812yescomplemented
87#382—21no—
88#382—21no—
89#382—21no—
90#382D412yes—
91#381—21no—
92#381—21no—
93#381—21no—
94#381—21no—
95#380—21no—
96#380—21no—
97#380—21no—
98#380D412yes—
99C4×C4⋊C4—21no—
100C4×C4⋊C4—21no—
101C4×C4⋊C4—21no—
102C4×C4⋊C4C2×C412yes—
103C4×C22⋊C4—21no—
104C4×C22⋊C4—21no—
105C4×C22⋊C4D411yescharacteristic
106C4×C22⋊C4D411yescharacteristic
107C4×C22⋊C4D411yescharacteristic
108C4×C22⋊C4D411yescharacteristic
109#376—21no—
110#376—21no—
111#376C2×C411yescharacteristic
112#376C2×C411yescharacteristic
113#376D412yes—
114C22×Q8—22no—
115C23×C4C2×D411yescharacteristic
116C23×C4C2×D411yescharacteristic
117C23×C4C2×D411yescharacteristic
118C2×M16—22no—
119C2×M16—41no—
120C22×C8—22no—
121C2×C4⋊C4—22no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—41no—
134C2×C22⋊C4—22no—
135C2×C22⋊C4—21no—
136C2×C22⋊C4—21no—
137C2×C22⋊C4—21no—
138C2×C22⋊C4—21no—
139C2×C22⋊C4—41no—
140C2×C42—21no—
141C2×C42—21no—
142C2×C42—21no—
143C2×C42—21no—
144C2×C42—41no—
145C2×C42C22⋊C412yes—
146C2×C42C22⋊C412yes—
147C2×C42C22×C411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×Q811yescharacteristic
152C2×C42C2×Q811yescharacteristic
153C2×C42C2×Q811yescharacteristic
154C2×C42C2×Q811yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C4⋊C8—22no—
160C4⋊C8—41no—
161C4⋊C8—41no—
162C4⋊C8—41no—
163C4⋊C8—41no—
164C4⋊C8—41no—
165C22⋊C8—22no—
166C22⋊C8—41no—
167C22⋊C8—41no—
168C22⋊C8—41no—
169C22⋊C8—41no—
170C22⋊C8—41no—
171#97—22no—
172#97—41no—
173C4×C8—22no—
174C4×C8—22no—
175C4×C8—22no—
176C4×C8—22no—
177#95—22no—
178#95—22no—
179#95—22no—
180#95—22no—
181#95—22no—
182#95—22no—
183#95—22no—
184#95—21no—
185#95—21no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—21no—
193#95—21no—
194#95—22no—
195#95—22no—
196#95—22no—
197#95—22no—
198#95—41no—
199#95C22⋊C412yes—
200#95C22⋊C412yes—
201#95C22⋊C412yes—
202#95C22⋊C412yes—
203C24#12011yescharacteristic
204C2×Q8—82no—
205C2×Q8—82no—
206C2×Q8—82no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4C2×C22⋊C411yescharacteristic
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4#12011yescharacteristic
221C22×C4#12011yescharacteristic
222C22×C4#12011yescharacteristic
223C22×C4C22⋊Q811yescharacteristic
224C22×C4C22⋊Q811yescharacteristic
225C22×C4C22⋊Q811yescharacteristic
226C22×C4C22⋊Q811yescharacteristic
227C22×C4C22⋊Q811yescharacteristic
228C22×C4C22⋊Q811yescharacteristic
229C22×C4#12411yescharacteristic
230C22×C4#12411yescharacteristic
231C22×C4#12411yescharacteristic
232C22×C4C4⋊Q811yescharacteristic
233C22×C4C4⋊Q811yescharacteristic
234C22×C4C4⋊Q811yescharacteristic
235M16—82no—
236M16—82no—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—82no—
241C4⋊C4—82no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C4⋊C4—82no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C4⋊C4—82no—
248C4⋊C4—82no—
249C4⋊C4—82no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C22⋊C4—82no—
255C22⋊C4—81no—
256C22⋊C4—81no—
257C22⋊C4—81no—
258C22⋊C4—81no—
259C22⋊C4—82no—
260C42—42no—
261C42—42no—
262C42—81no—
263C42—81no—
264C42—81no—
265C42—81no—
266C42—82no—
267C42C4×Q811yescharacteristic
268C42C4×Q811yescharacteristic
269C42C4×Q811yescharacteristic
270C42C4×Q811yescharacteristic
271C42C22⋊Q811yescharacteristic
272C42C22⋊Q811yescharacteristic
273C42C22⋊Q811yescharacteristic
274C42C22⋊Q811yescharacteristic
275C42C22⋊Q811yescharacteristic
276C42C22⋊Q811yescharacteristic
277C42#12411yescharacteristic
278C42#12411yescharacteristic
279C42#12411yescharacteristic
280C42C4⋊Q811yescharacteristic
281C42C4⋊Q811yescharacteristic
282C42C4⋊Q811yescharacteristic
283C23—22no—
284C23—41no—
285C23—41no—
286C23#37911yescharacteristic
287C23#39311yescharacteristic
288C23#39511yescharacteristic
289Q8—164no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—42no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—42no—
313C2×C4—42no—
314C2×C4—42no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4#39111yescharacteristic
330C2×C4#39111yescharacteristic
331C2×C4#39111yescharacteristic
332C2×C4#39111yescharacteristic
333C2×C4#39111yescharacteristic
334C2×C4#39111yescharacteristic
335C2×C4#39311yescharacteristic
336C2×C4#39311yescharacteristic
337C2×C4#39311yescharacteristic
338C2×C4#39511yescharacteristic
339C2×C4#39511yescharacteristic
340C2×C4#39511yescharacteristic
341C8—82no—
342C8—82no—
343C8—82no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—82no—
353C22#144611yescharacteristic
354C22#179412yes—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—82no—
368C4—82no—
369C4—82no—
370C4—82no—
371C4—82no—
372C4—82no—
373C4—162no—
374C2—22no—
375C2—41no—
376C2—41no—
377C2#1138911yescharacteristic
378C2#1138911yescharacteristic
379C2#1139311yescharacteristic
subgroupG/H
center, socleC22#1446
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#120
LCSG ▷ C2×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
19. #2606C4semidirect
20. #2605C4semidirect
45. #431Q8semidirect
53. C4×M16Q8semidirect
54. C2×C4×C8Q8semidirect
86. #383Q8semidirect

Maximal subgroups

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

subgroupindex
#375662
#80502
#79802
#79762
#79582
#79452
#78582