Finite Group Databasebeta

← #118464

Subgroups — #118464

Order 512.

621 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20741C211yesmaximal, characteristic
2#20735C211yesmaximal, characteristic
3#17810C211yesmaximal, characteristic
4#17365C211yesmaximal, characteristic
5#8096C211yesmaximal, characteristic
6#7954C211yesmaximal, characteristic
7#7945C211yesmaximal, characteristic
8C22×C4×Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2283C2211yescharacteristic
14#2283C2211yescharacteristic
15#1824—21no—
16#1823—21no—
17#1805—22no—
18#1805—22no—
19#1787—21no—
20#1787—21no—
21#1763—22no—
22C4×#103—22no—
23#1573—22no—
24#1572—22no—
25#1565—22no—
26#1467C414yescomplemented
27#1463—22no—
28#1461—22no—
29#1303—22no—
30#1285—22no—
31#1284—22no—
32C23×Q8D411yescharacteristic
33C2×C4×Q8—21no—
34C2×C4×Q8—21no—
35C2×C4×Q8—21no—
36C2×C4×Q8—21no—
37C2×C4×Q8—21no—
38C2×C4×Q8—21no—
39C2×C4×Q8—21no—
40C2×C4×Q8—21no—
41C2×C4×Q8—21no—
42C2×C4×Q8—21no—
43C2×C4×Q8C2×C412yes—
44C2×C4×Q8Q812yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4D411yescharacteristic
47C22×C42—21no—
48C22×C42C2311yescharacteristic
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432C2×C412yes—
56#432C2×C412yes—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431D412yes—
65#423—21no—
66#423—21no—
67C2×C4⋊C8D412yes—
68C2×C4⋊C8D412yes—
69C2×#103—24no—
70C2×#103—24no—
71#407—21no—
72#407D412yes—
73C2×C22⋊C8—21no—
74C2×C22⋊C8—21no—
75C4×M16—21no—
76C4×M16—21no—
77C2×#97—21no—
78C2×C4×C8—21no—
79#383—21no—
80#383—21no—
81#382—21no—
82#382—21no—
83#381—21no—
84#381—21no—
85#380—21no—
86#380—21no—
87C4×C4⋊C4D412yes—
88C4×C4⋊C4D412yes—
89C4×C22⋊C4—21no—
90C4×C22⋊C4—21no—
91#376—21no—
92#376—21no—
93#340—24no—
94#340—24no—
95#335—24no—
96#334—24no—
97#324—28no—
98C22×Q8—21no—
99C22×Q8—21no—
100C22×Q8—21no—
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8—44no—
104C22×Q8C4⋊C412yes—
105C23×C4—21no—
106C23×C4—21no—
107C23×C4C2×D411yescharacteristic
108C23×C4C2×D411yescharacteristic
109C23×C4#4111yescharacteristic
110C2×M16—22no—
111C2×M16—41no—
112C22×C8—22no—
113C22×C8—22no—
114C4×Q8—22no—
115C4×Q8—22no—
116C4×Q8—22no—
117C4×Q8—22no—
118C4×Q8—22no—
119C4×Q8—22no—
120C4×Q8—22no—
121C4×Q8—22no—
122C4×Q8—41no—
123C4×Q8—41no—
124C4×Q8—41no—
125C4×Q8—41no—
126C4×Q8—41no—
127C4×Q8—41no—
128C4×Q8—41no—
129C4×Q8—41no—
130C2×C4⋊C4—21no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—21no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—21no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4C4⋊C412yes—
144C2×C22⋊C4—41no—
145C2×C22⋊C4—41no—
146C2×C42—22no—
147C2×C42—22no—
148C2×C42—22no—
149C2×C42—22no—
150C2×C42—41no—
151C2×C42—41no—
152C2×C42—41no—
153C2×C42—41no—
154C2×C42C22×C411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×Q811yescharacteristic
161C2×C42#4111yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—41no—
174#103—48no—
175#103—48no—
176C22⋊C8—22no—
177C22⋊C8—22no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8C22⋊C414yes—
184#97—22no—
185#97—22no—
186#97—41no—
187#97—41no—
188C4×C8—22no—
189C4×C8—41no—
190C4×C8—41no—
191C4×C8—41no—
192#95—22no—
193#95—22no—
194#95—41no—
195#95—41no—
196#95—41no—
197#95—41no—
198#95—41no—
199#95—41no—
200C24#12111yescharacteristic
201C2×Q8—24no—
202C2×Q8—24no—
203C2×Q8—24no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—44no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—42no—
210C2×Q8—44no—
211C2×Q8—44no—
212C2×Q8—41no—
213C2×Q8—41no—
214C22×C4—22no—
215C22×C4—22no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—44no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—82no—
227C22×C4C2×C22⋊C411yescharacteristic
228C22×C4C2×C4⋊C411yescharacteristic
229C22×C4#11711yescharacteristic
230C22×C4#12011yescharacteristic
231C22×C4#12011yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4#12111yescharacteristic
234C22×C4#12111yescharacteristic
235C22×C4#12111yescharacteristic
236C22×C4#12111yescharacteristic
237C22×C4C22⋊Q811yescharacteristic
238C22×C4C22⋊Q811yescharacteristic
239C22×C4#12311yescharacteristic
240C22×C4#12311yescharacteristic
241C22×C4#12311yescharacteristic
242C22×C4#12311yescharacteristic
243C22×C4#12411yescharacteristic
244C22×C4#12411yescharacteristic
245M16—44no—
246M16—82no—
247C2×C8—24no—
248C2×C8—44no—
249C2×C8—44no—
250C2×C8—42no—
251C2×C8—42no—
252C2×C8—42no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—44no—
256C4⋊C4—44no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C4⋊C4—41no—
263C22⋊C4—82no—
264C22⋊C4—82no—
265C42—42no—
266C42—42no—
267C42—42no—
268C42—42no—
269C42—82no—
270C42—82no—
271C42C4×D411yescharacteristic
272C42C4×D411yescharacteristic
273C42C4×D411yescharacteristic
274C42C4×D411yescharacteristic
275C42#12011yescharacteristic
276C42#12111yescharacteristic
277C42#12111yescharacteristic
278C42C22⋊Q811yescharacteristic
279C42C22⋊Q811yescharacteristic
280C42C22⋊Q811yescharacteristic
281C42C22⋊Q811yescharacteristic
282C42#12311yescharacteristic
283C42#12411yescharacteristic
284C42#12611yescharacteristic
285C42#12611yescharacteristic
286C42#12711yescharacteristic
287C23—22no—
288C23—22no—
289C23—41no—
290C23#38011yescharacteristic
291C23#39411yescharacteristic
292C23#39411yescharacteristic
293Q8—24no—
294Q8—24no—
295Q8—42no—
296Q8—42no—
297Q8—44no—
298Q8—44no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—24no—
307C2×C4—41no—
308C2×C4—41no—
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—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—82no—
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#38511yescharacteristic
333C2×C4#38511yescharacteristic
334C2×C4#38611yescharacteristic
335C2×C4#38811yescharacteristic
336C2×C4#38811yescharacteristic
337C2×C4#39011yescharacteristic
338C2×C4#39211yescharacteristic
339C2×C4#39311yescharacteristic
340C2×C4#39611yescharacteristic
341C2×C4#39611yescharacteristic
342C2×C4#39711yescharacteristic
343C2×C4#39911yescharacteristic
344C8—44no—
345C8—82no—
346C8—82no—
347C8—82no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—44no—
356C22#143911yescharacteristic
357C22#204912yes—
358C22#205312yes—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—21no—
370C4—44no—
371C4—82no—
372C4—162no—
373C4—162no—
374C4#201012yes—
375C2—22no—
376C2—22no—
377C2—41no—
378C2#1109211yescharacteristic
379C2#1110011yescharacteristic
380C2#1114311yescharacteristic
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
26. #1467C4semidirect

Maximal subgroups

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

subgroupindex
#207412
#207352
#178102
#173652
#80962
#79542
#79452