Finite Group Databasebeta

← #107277

Subgroups — #107277

Order 512.

686 conjugacy classes of subgroups.

The subgroup lattice has 377 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60250C211yesmaximal, characteristic
2#17590C211yesmaximal, characteristic
3#17491C211yesmaximal, characteristic
4#8122C211yesmaximal, characteristic
5#8106C211yesmaximal, characteristic
6#8096C211yesmaximal, characteristic
7#8066C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2986—22no—
10#2985—22no—
11#2984—22no—
12#2983—21no—
13#2982—21no—
14#2962C414yescomplemented
15#2960—22no—
16#2924—22no—
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20#2920C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#2283C2211yescharacteristic
23#1518—22no—
24#1511C414yes—
25#1506—22no—
26#1497C414yescomplemented
27#1487—22no—
28#1485—22no—
29#1307—22no—
30#1305—22no—
31#1303—22no—
32#1299—21no—
33#1299—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533C2×C412yes—
40#533D412yes—
41#533D412yes—
42C2×C4⋊Q8Q811yescharacteristic
43C2×C4⋊Q8Q811yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8C2×C412yes—
47C2×C4×Q8—21no—
48C2×C4×Q8D412yes—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#446—24no—
52#446—24no—
53#446—28no—
54#446—24no—
55C8×Q8—24no—
56C8×Q8—24no—
57C8×Q8—24no—
58#433—28no—
59#432—24no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432C2×C412yes—
68#431—24no—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431C2×C412yes—
79#431D411yescharacteristic
80#431D411yescharacteristic
81#431D412yes—
82#424—24no—
83#423—21no—
84#423—21no—
85#423—21no—
86#423D412yes—
87C2×C4⋊C8C2×C412yes—
88#407—21no—
89#407—21no—
90#407C2×C412yes—
91#407D412yes—
92C2×C22⋊C8—21no—
93#405—24no—
94C4×M16—21no—
95C4×M16—21no—
96C4×M16—21no—
97C4×M16C2×C412yes—
98C2×C4×C8—21no—
99#383—21no—
100#382—21no—
101#381C2×C412yes—
102#380—21no—
103C4×C4⋊C4—21no—
104C4×C22⋊C4—21no—
105#376—21no—
106#328—28no—
107Q8⋊C8—28no—
108C22×Q8—21no—
109C22×Q8C4⋊C412yes—
110C23×C4C22×C411yescharacteristic
111C23×C4C2×D411yescharacteristic
112C23×C4#4111yescharacteristic
113C2×M16—22no—
114C2×M16—41no—
115C2×M16—41no—
116C2×M16C22⋊C414yes—
117C22×C8—22no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8C4⋊C412yes—
123C4⋊Q8C4⋊C412yes—
124C4⋊Q8C4⋊C412yes—
125C4⋊Q8C4⋊C412yes—
126#125—22no—
127#125—22no—
128#125—22no—
129#125—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C22⋊Q8—22no—
135C22⋊Q8—22no—
136C22⋊Q8C4214yes—
137C22⋊Q8C22⋊C414yes—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144C4×Q8C22⋊C414yes—
145C4×Q8C22⋊C414yes—
146#117—22no—
147#117—22no—
148#117—22no—
149#117—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—21no—
153C2×C4⋊C4—21no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—41no—
156C2×C4⋊C4C4⋊C412yes—
157C2×C22⋊C4—22no—
158C2×C22⋊C4—22no—
159C2×C42—22no—
160C2×C42—41no—
161C2×C42C22×C411yescharacteristic
162C2×C42C22×C411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×Q811yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C4⋊C8—28no—
174C4⋊C8—28no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184C4⋊C8C22⋊C412yes—
185C4⋊C8C22⋊C412yes—
186C22⋊C8—21no—
187C22⋊C8—21no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191C22⋊C8—22no—
192C22⋊C8—22no—
193C22⋊C8—22no—
194C22⋊C8—22no—
195C22⋊C8—41no—
196C22⋊C8—41no—
197#97—28no—
198#97—22no—
199#97—22no—
200#97—41no—
201#97—41no—
202C4×C8—28no—
203C4×C8—22no—
204C4×C8—22no—
205C4×C8—21no—
206C4×C8—22no—
207C4×C8—21no—
208C4×C8—41no—
209C4×C8—41no—
210#95—22no—
211#95—22no—
212#95—41no—
213#95—41no—
214C24C4×D411yescharacteristic
215C2×Q8—22no—
216C2×Q8—22no—
217C2×Q8—22no—
218C2×Q8—44no—
219C2×Q8—82no—
220C2×Q8#9514yes—
221C22×C4—22no—
222C22×C4—22no—
223C22×C4—42no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—81no—
227C22×C4C2×C4211yescharacteristic
228C22×C4C2×C22⋊C411yescharacteristic
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C4⋊C411yescharacteristic
231C22×C4#11711yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236C22×C4C4×D411yescharacteristic
237C22×C4#12011yescharacteristic
238C22×C4#12011yescharacteristic
239C22×C4#12111yescharacteristic
240C22×C4#12111yescharacteristic
241C22×C4C22⋊Q811yescharacteristic
242C22×C4C22⋊Q811yescharacteristic
243C22×C4#12311yescharacteristic
244C22×C4#12311yescharacteristic
245M16—44no—
246M16—82no—
247M16—82no—
248M16—82no—
249C2×C8—22no—
250C2×C8—24no—
251C2×C8—22no—
252C2×C8—44no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—42no—
256C4⋊C4—24no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—24no—
261C4⋊C4—22no—
262C4⋊C4—22no—
263C4⋊C4—22no—
264C4⋊C4—22no—
265C4⋊C4—24no—
266C4⋊C4—24no—
267C4⋊C4—82no—
268C4⋊C4#9514yes—
269C22⋊C4—24no—
270C22⋊C4—24no—
271C22⋊C4—82no—
272C42—24no—
273C42—24no—
274C42—82no—
275C42C2×C22⋊C411yescharacteristic
276C42C2×C22⋊C411yescharacteristic
277C42C2×C22⋊C411yescharacteristic
278C42C2×C4⋊C411yescharacteristic
279C42C2×C4⋊C411yescharacteristic
280C42#11711yescharacteristic
281C42C4×D411yescharacteristic
282C42C4×D411yescharacteristic
283C42#12011yescharacteristic
284C42#12011yescharacteristic
285C42#12111yescharacteristic
286C42#12111yescharacteristic
287C42C22⋊Q811yescharacteristic
288C42C22⋊Q811yescharacteristic
289C42#12311yescharacteristic
290C42#12311yescharacteristic
291C23—41no—
292C23—41no—
293C23—41no—
294C23C4×C22⋊C411yescharacteristic
295C23#38511yescharacteristic
296C23#38611yescharacteristic
297Q8—44no—
298Q8—44no—
299Q8—44no—
300Q8—44no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—24no—
308C2×C4—24no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—44no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4C2×#9511yescharacteristic
327C2×C4C4×C22⋊C411yescharacteristic
328C2×C4#37911yescharacteristic
329C2×C4#38011yescharacteristic
330C2×C4#38011yescharacteristic
331C2×C4#38111yescharacteristic
332C2×C4#38511yescharacteristic
333C2×C4#38511yescharacteristic
334C2×C4#38511yescharacteristic
335C2×C4#38611yescharacteristic
336C2×C4#38611yescharacteristic
337C2×C4#38611yescharacteristic
338C8—42no—
339C8—42no—
340C8—44no—
341C8—82no—
342C8—82no—
343C8—82no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—82no—
354C22C22⋊#9511yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—44no—
366C4—44no—
367C4—162no—
368C4#179612yes—
369C4#181412yes—
370C2—41no—
371C2—41no—
372C2—41no—
373C2#1075711yescharacteristic
374C2#1077311yescharacteristic
375C2#1077711yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutatorC22×C4C2×C42
Frattini, cosocleC22×C42C23
prime radicalC24C4×D4
LCSG ▷ C22×C4 ▷ 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
14. #2962C4semidirect
26. #1497C4semidirect

Maximal subgroups

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

subgroupindex
#602502
#175902
#174912
#81222
#81062
#80962
#80662