Finite Group Databasebeta

← #223149

Subgroups — #223149

Order 512.

794 conjugacy classes of subgroups.

The subgroup lattice has 409 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36191C211yesmaximal, characteristic
2#21271C212yesmaximal
3#9890C212yesmaximal, complemented
4#9126C212yesmaximal, complemented
5#9125C212yesmaximal, complemented
6#8722C211yesmaximal, characteristic, complemented
7#8719C211yesmaximal, characteristic, complemented
8#8584C211yesmaximal, characteristic, complemented
9C4×#330C211yesmaximal, characteristic, complemented
10#8031C211yesmaximal, characteristic, complemented
11#8030C211yesmaximal, characteristic, complemented
12C2×#432C2212yes—
13C2×#432C2212yes—
14#2862C2211yescharacteristic
15#2851C2211yescharacteristic
16#2851C2211yescharacteristic
17#2574—22no—
18#2570C412yes—
19#2520—22no—
20#2450—22no—
21#2440C412yes—
22#2439C412yes—
23#2368—22no—
24#2352C412yes—
25C4×#125C2211yescharacteristic
26#2285C2211yescharacteristic
27C2×#382C2211yescharacteristic
28C42×D4C2211yescharacteristic
29#1955—22no—
30#1940—22no—
31#1923—22no—
32#1923—22no—
33#1767C2212yes—
34#1749C2212yes—
35#1749C2212yes—
36C4×#97C2212yes—
37C2×#330C2212yes—
38C2×#330C2212yes—
39#1295C2211yescharacteristic
40#1295C2211yescharacteristic
41#1295C2212yes—
42#1295C2211yescharacteristic
43#1295C2212yes—
44#1295C2211yescharacteristic
45#1295C2212yes—
46#1295C2212yes—
47C2×C4⋊Q8C2×C412yes—
48C2×#125C2311yescharacteristic
49C2×C22⋊Q8—22no—
50C2×C4×D4C2×C411yescharacteristic
51C2×C4×D4D412yes—
52C2×#117C2×C412yes—
53C22×C4⋊C4C2×C412yes—
54C22×C42C2311yescharacteristic
55C22×C42C2311yescharacteristic
56#432—24no—
57#432—24no—
58#432—42no—
59#432—42no—
60C2×C4⋊C8—22no—
61C2×C4⋊C8—22no—
62C2×C4⋊C8—22no—
63C2×C22⋊C8—22no—
64C2×C22⋊C8—22no—
65C2×C22⋊C8—22no—
66C2×C22⋊C8—22no—
67C2×#97—22no—
68C2×#97C2312yes—
69C2×#97C2312yes—
70C2×#97C2312yes—
71C2×#97C2312yes—
72C2×C4×C8—22no—
73#400—22no—
74#398—22no—
75#398—22no—
76#396—22no—
77#395—22no—
78#393—22no—
79#391—22no—
80#391—22no—
81#391—22no—
82#389C2×C412yes—
83#389C2×C412yes—
84#389C2×C412yes—
85#388—22no—
86#387—22no—
87#387—22no—
88#387—22no—
89#387—22no—
90#387—22no—
91#387—22no—
92#387—22no—
93#386—22no—
94#385—22no—
95#385—22no—
96#385—22no—
97#384C2×C412yes—
98#382—22no—
99#382C2×C412yes—
100#382C2×C412yes—
101#382C2311yescharacteristic
102#382C2311yescharacteristic
103#382C2311yescharacteristic
104#380—22no—
105#380C2×C412yes—
106#380C2×C412yes—
107C4×C4⋊C4—22no—
108C4×C4⋊C4—22no—
109C4×C4⋊C4C2×C412yes—
110C4×C4⋊C4C2×C411yescharacteristic
111C4×C4⋊C4D412yes—
112C4×C22⋊C4—22no—
113C4×C22⋊C4C2×C411yescharacteristic
114C4×C22⋊C4C2×C411yescharacteristic
115C4×C22⋊C4D412yes—
116C4×C22⋊C4D412yes—
117C2×#95—22no—
118C43C2311yescharacteristic
119#336—22no—
120#336—22no—
121#336—22no—
122#330—24no—
123#330—42no—
124C22×Q8—22no—
125C22×D4C22⋊C411yescharacteristic
126C23×C4—22no—
127C23×C4C22×C411yescharacteristic
128C23×C4C22×C411yescharacteristic
129C23×C4C2×D412yes—
130C23×C4C2×D412yes—
131C22×C8—22no—
132C22×C8—22no—
133C22×C8—22no—
134C22×C8—22no—
135C4⋊Q8—82no—
136#125—22no—
137#125—21no—
138#125—21no—
139C22⋊Q8—48no—
140C4×D4—24no—
141C4×D4—44no—
142#117—82no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—22no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—22no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—82no—
152C2×C4⋊C4C22⋊C411yescharacteristic
153C2×C4⋊C4C22⋊C412yes—
154C2×C4⋊C4C22×C412yes—
155C2×C4⋊C4C22×C412yes—
156C2×C4⋊C4C22×C412yes—
157C2×C4⋊C4C22×C412yes—
158C2×C22⋊C4—22no—
159C2×C22⋊C4—22no—
160C2×C22⋊C4—22no—
161C2×C22⋊C4C22⋊C411yescharacteristic
162C2×C22⋊C4C22⋊C412yes—
163C2×C22⋊C4C22⋊C412yes—
164C2×C22⋊C4C22⋊C411yescharacteristic
165C2×C42—22no—
166C2×C42—22no—
167C2×C42—21no—
168C2×C42—21no—
169C2×C42—21no—
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42C22×C411yescharacteristic
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C2×D412yes—
178C2×C42C2×D412yes—
179C2×C42C2×D412yes—
180C2×C42C2×D412yes—
181C2×C42C2411yescharacteristic
182C4⋊C8—44no—
183C4⋊C8—44no—
184C4⋊C8—44no—
185C22⋊C8—44no—
186C22⋊C8—44no—
187C22⋊C8—44no—
188C22⋊C8—44no—
189#97—22no—
190#97—22no—
191#97—22no—
192#97—22no—
193#97—22no—
194#97—22no—
195#97—22no—
196#97—44no—
197#97#4114yes—
198C4×C8—44no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—22no—
203#95—22no—
204#95—22no—
205#95—22no—
206#95—22no—
207#95—22no—
208#95—22no—
209#95—22no—
210#95—22no—
211#95—22no—
212#95C22⋊C412yes—
213#95C22⋊C412yes—
214#95C22⋊C412yes—
215C24C2×C22⋊C411yescharacteristic
216C24C2×C22⋊C411yescharacteristic
217C2×Q8—44no—
218C2×Q8—82no—
219C2×Q8—82no—
220C2×D4—22no—
221C2×D4—42no—
222C22×C4—22no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—42no—
239C22×C4—42no—
240C22×C4—42no—
241C22×C4—42no—
242C22×C4—44no—
243C22×C4—82no—
244C22×C4—82no—
245C22×C4C2×C22⋊C411yescharacteristic
246C22×C4C2×C22⋊C412yes—
247C22×C4C2×C22⋊C411yescharacteristic
248C22×C4C2×C22⋊C412yes—
249C22×C4C2×C22⋊C411yescharacteristic
250C22×C4C2×C22⋊C411yescharacteristic
251C22×C4C2×C22⋊C412yes—
252C22×C4C23×C411yescharacteristic
253C22×C4C22×D412yes—
254C2×C8—24no—
255C2×C8—24no—
256C2×C8—42no—
257C2×C8—42no—
258C2×C8—42no—
259C2×C8—42no—
260C2×C8—42no—
261C2×C8—42no—
262C2×C8—42no—
263C2×C8—42no—
264C2×C8—42no—
265C2×C8—42no—
266C4⋊C4—22no—
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—42no—
270C4⋊C4—42no—
271C4⋊C4—42no—
272C4⋊C4—42no—
273C4⋊C4—44no—
274C4⋊C4—44no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4—44no—
278C4⋊C4—82no—
279C4⋊C4—82no—
280C4⋊C4—82no—
281C4⋊C4—82no—
282C22⋊C4—22no—
283C22⋊C4—24no—
284C22⋊C4—24no—
285C22⋊C4—44no—
286C22⋊C4—82no—
287C22⋊C4—82no—
288C22⋊C4#10114yes—
289C42—21no—
290C42—21no—
291C42—21no—
292C42—21no—
293C42—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—24no—
299C42—42no—
300C42—41no—
301C42—82no—
302C42—82no—
303C42C2×#4112yes—
304C42H(5,𝔽2)11yescharacteristic
305C422−1+411yescharacteristic
306C23—21no—
307C23—21no—
308C23—22no—
309C23—22no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—22no—
314C23—22no—
315C23C2×#10112yes—
316C23C22×C22⋊C411yescharacteristic
317Q8—164no—
318D4—44no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—22no—
323C2×C4—21no—
324C2×C4—22no—
325C2×C4—22no—
326C2×C4—22no—
327C2×C4—22no—
328C2×C4—21no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4—82no—
358C2×C4—82no—
359C2×C4—164no—
360C2×C4C2×#10112yes—
361C2×C4C2×#10112yes—
362C2×C4#42114yes—
363C2×C4C4×#4111yescharacteristic
364C2×C4#52011yescharacteristic
365C2×C4#54612yes—
366C2×C4#54712yes—
367C8—44no—
368C8—44no—
369C8—82no—
370C8—82no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—42no—
375C22—42no—
376C22—41no—
377C22—42no—
378C22—41no—
379C22—41no—
380C22—42no—
381C22#233911yescharacteristic
382C22C22×#10111yescharacteristic
383C22#288811yescharacteristic
384C22C2×#42112yes—
385C22#290112yes—
386C4—24no—
387C4—22no—
388C4—21no—
389C4—21no—
390C4—22no—
391C4—22no—
392C4—22no—
393C4—41no—
394C4—41no—
395C4—42no—
396C4—42no—
397C4—82no—
398C4—82no—
399C4—82no—
400C4—82no—
401C2—42no—
402C2—81no—
403C2#1766411yescharacteristic
404C2#1767912yes—
405C2#1768911yescharacteristic
406C2#2248111yescharacteristic
407C2#2250712yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×D4 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9890C2semidirect
4. #9126C2semidirect
5. #9125C2semidirect
6. #8722C2semidirect
7. #8719C2semidirect
8. #8584C2semidirect
9. C4×#330C2semidirect
10. #8031C2semidirect
11. #8030C2semidirect

Maximal subgroups

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

subgroupindex
#361912
#212712
#98902
#91262
#91252
#87222
#87192
#85842
C4×#3302
#80312
#80302