Finite Group Databasebeta

← #117164

Subgroups — #117164

Order 512.

606 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21337C211yesmaximal, characteristic
2#20765C211yesmaximal, characteristic
3#17575C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5#8095C211yesmaximal, characteristic
6C2×#1286C211yesmaximal, characteristic
7#7882C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#431C2211yescharacteristic
10C2×#431C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C2×C4×M16C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1998—22no—
16#1997C412yescomplemented
17#1997C412yescomplemented
18#1836—22no—
19#1835—22no—
20#1742—22no—
21#1736—22no—
22#1520—22no—
23#1518—22no—
24#1518—22no—
25#1518—22no—
26#1503—22no—
27C2×#328—22no—
28#1303—22no—
29#1286—22no—
30#1278—22no—
31C22×M16—21no—
32C22×M16D412yes—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8D412yes—
46C2×C4×Q8—21no—
47C2×C4×Q8Q812yes—
48C2×#117—21no—
49C2×#117—21no—
50C22×C4⋊C4—21no—
51C22×C42C2311yescharacteristic
52#431—21no—
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431D412yes—
59#431D412yes—
60#423—21no—
61#423—21no—
62#423—21no—
63#423C2×C412yes—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C4×M16—21no—
71C4×M16—21no—
72C4×M16C2×C412yes—
73C4×M16D412yes—
74C2×#97—21no—
75C2×C4×C8—21no—
76C2×C4×C8—21no—
77C2×C4×C8—21no—
78#384—21no—
79#382—21no—
80#382—21no—
81#382—21no—
82#382D412yes—
83#382D412yes—
84#380—21no—
85#380—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C2×#95—21no—
89C2×#95—21no—
90#343—24no—
91#343—24no—
92#340—28no—
93#328—28no—
94#328—28no—
95C22×Q8—21no—
96C22×Q8C4⋊C412yes—
97C23×C4—41no—
98C23×C4C2×D411yescharacteristic
99C23×C4C2×D411yescharacteristic
100C23×C4#4111yescharacteristic
101C2×M16—22no—
102C2×M16—22no—
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—22no—
106C2×M16—22no—
107C2×M16—41no—
108C2×M16—42no—
109C2×M16—42no—
110C2×M16C22⋊C414yes—
111C22×C8—22no—
112C22×C8—41no—
113C22×C8—41no—
114C22×C8—41no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122C4⋊Q8—21no—
123#125—41no—
124#125—41no—
125#125—41no—
126#125—41no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C22⋊Q8—41no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—41no—
143#117—41no—
144#117—41no—
145#117—41no—
146#117—41no—
147#117—42no—
148#117—42no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—21no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—22no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—42no—
157C2×C4⋊C4—42no—
158C2×C4⋊C4—41no—
159C2×C4⋊C4—41no—
160C2×C4⋊C4—41no—
161C2×C4⋊C4C4⋊C412yes—
162C2×C22⋊C4—22no—
163C2×C22⋊C4—41no—
164C2×C42—22no—
165C2×C42—41no—
166C2×C42C22×C411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×D411yescharacteristic
170C2×C42C2×D411yescharacteristic
171C2×C42C2×D411yescharacteristic
172C2×C42C2×Q811yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C2×C42#4111yescharacteristic
176C2×C42#4111yescharacteristic
177C2×C42#4111yescharacteristic
178C4⋊C8—22no—
179C4⋊C8—22no—
180C4⋊C8—22no—
181C4⋊C8—22no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190#97—22no—
191#97—22no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—41no—
197C4×C8—41no—
198#95—22no—
199#95—22no—
200#95—22no—
201#95—22no—
202#95—41no—
203#95—41no—
204C24#12111yescharacteristic
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—82no—
210C2×Q8—82no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—81no—
216C22×C4—81no—
217C22×C4—82no—
218C22×C4—82no—
219C22×C4—82no—
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×C4⋊C411yescharacteristic
222C22×C4#11711yescharacteristic
223C22×C4#12011yescharacteristic
224C22×C4#12011yescharacteristic
225C22×C4#12011yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4#12111yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4C22⋊Q811yescharacteristic
230C22×C4C22⋊Q811yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12311yescharacteristic
234C22×C4#12411yescharacteristic
235C22×C4#12611yescharacteristic
236C22×C4#12611yescharacteristic
237C22×C4#12711yescharacteristic
238C22×C4#13512yes—
239C22×C4#13512yes—
240C22×C4#13712yes—
241C22×C4#13712yes—
242M16—24no—
243M16—24no—
244M16—42no—
245M16—42no—
246M16—44no—
247M16—81no—
248M16—81no—
249C2×C8—24no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C2×C8—42no—
254C2×C8—42no—
255C2×C8—82no—
256C2×C8—82no—
257C2×C8—82no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—42no—
267C4⋊C4—42no—
268C4⋊C4—42no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—44no—
272C4⋊C4—82no—
273C4⋊C4—82no—
274C4⋊C4—81no—
275C4⋊C4—81no—
276C22⋊C4—42no—
277C22⋊C4—42no—
278C22⋊C4—82no—
279C42—42no—
280C42—42no—
281C42—81no—
282C42—81no—
283C42C4×D411yescharacteristic
284C42C4×D411yescharacteristic
285C42C4×D411yescharacteristic
286C42C4×D411yescharacteristic
287C42#12111yescharacteristic
288C42#12111yescharacteristic
289C42#12111yescharacteristic
290C42#12111yescharacteristic
291C42C22⋊Q811yescharacteristic
292C42C22⋊Q811yescharacteristic
293C42C22⋊Q811yescharacteristic
294C42C22⋊Q811yescharacteristic
295C42#12311yescharacteristic
296C42#12311yescharacteristic
297C42#12411yescharacteristic
298C42#12411yescharacteristic
299C23—41no—
300C23—41no—
301C23#38011yescharacteristic
302C23#39211yescharacteristic
303C23#39711yescharacteristic
304C23#46412yes—
305C23#46412yes—
306Q8—82no—
307Q8—82no—
308Q8—82no—
309Q8—82no—
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—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—164no—
333C2×C4#38511yescharacteristic
334C2×C4#38511yescharacteristic
335C2×C4#38611yescharacteristic
336C2×C4#38811yescharacteristic
337C2×C4#38811yescharacteristic
338C2×C4#39011yescharacteristic
339C2×C4#39311yescharacteristic
340C2×C4#39411yescharacteristic
341C2×C4#39411yescharacteristic
342C2×C4#39611yescharacteristic
343C2×C4#39611yescharacteristic
344C2×C4#39911yescharacteristic
345C2×C4#41612yes—
346C2×C4#41912yes—
347C2×C4#45212yes—
348C2×C4#45212yes—
349C2×C4#47912yes—
350C2×C4#47912yes—
351C2×C4#48512yes—
352C2×C4#48512yes—
353C2×C4#48812yes—
354C2×C4#49112yes—
355C2×C4#49511yescharacteristic
356C2×C4#49511yescharacteristic
357C2×C4#49711yescharacteristic
358C2×C4#49711yescharacteristic
359C8—44no—
360C8—82no—
361C8—82no—
362C8—82no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—21no—
368C22—21no—
369C22—42no—
370C22—42no—
371C22#143911yescharacteristic
372C22#180512yes—
373C22#201511yescharacteristic
374C22#201511yescharacteristic
375C22#207412yes—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—82no—
385C4—82no—
386C4—162no—
387C4—162no—
388C4#187312yes—
389C4#195811yescharacteristic
390C4#196011yescharacteristic
391C4#202612yes—
392C4#209711yescharacteristic
393C4#209711yescharacteristic
394C2—21no—
395C2—21no—
396C2—22no—
397C2—22no—
398C2#1105211yescharacteristic
399C2#1113811yescharacteristic
400C2#1114111yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
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
16. #1997C4semidirect
17. #1997C4semidirect

Maximal subgroups

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

subgroupindex
#213372
#207652
#175752
C2×#15182
#80952
C2×#12862
#78822