Finite Group Databasebeta

← #117670

Subgroups — #117670

Order 512.

616 conjugacy classes of subgroups.

The subgroup lattice has 361 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21834C211yesmaximal, characteristic
2#21811C211yesmaximal, characteristic
3#17818C211yesmaximal, characteristic
4C2×#1556C211yesmaximal, characteristic
5#7984C211yesmaximal, characteristic
6#7952C211yesmaximal, characteristic
7#7928C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2285C2211yescharacteristic
14#2283C2211yescharacteristic
15#1990—22no—
16#1987C414yescomplemented
17#1981—22no—
18#1960—22no—
19#1824—22no—
20#1787—22no—
21#1573—22no—
22#1573—22no—
23#1568—22no—
24#1556—22no—
25#1556—22no—
26C2×#332—22no—
27#1289—22no—
28#1285—22no—
29#1282—22no—
30C23×Q8—21no—
31C2×C4⋊Q8—21no—
32C2×C4⋊Q8—21no—
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⋊Q8C2×C412yes—
41C2×C4⋊Q8D412yes—
42C2×C4⋊Q8Q812yes—
43C2×#117—21no—
44C22×C4⋊C4—21no—
45C22×C4⋊C4—21no—
46C22×C42C2311yescharacteristic
47#432—21no—
48#432—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432D412yes—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8C2×C412yes—
65C2×C4⋊C8D412yes—
66C2×C4⋊C8D412yes—
67C2×#103—28no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8D412yes—
72C2×#97—21no—
73C2×#97—21no—
74C2×#97—21no—
75C2×C4×C8D412yes—
76#384—21no—
77#383—21no—
78#382—21no—
79#382—21no—
80#381—21no—
81#380—21no—
82#380—21no—
83#380D412yes—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86C4×C4⋊C4C2×C412yes—
87C4×C22⋊C4—21no—
88#376—21no—
89#340—28no—
90#335—24no—
91#334—24no—
92#332—28no—
93C22×Q8—22no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—41no—
97C22×Q8—41no—
98C22×Q8—41no—
99C22×Q8—84no—
100C23×C4—41no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×D411yescharacteristic
103C23×C4#4111yescharacteristic
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—41no—
107C22×C8C22⋊C414yes—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—22no—
115C4⋊Q8—22no—
116C4⋊Q8—22no—
117C4⋊Q8—41no—
118C4⋊Q8—41no—
119C4⋊Q8—41no—
120C4⋊Q8—41no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8C4⋊C414yes—
124#117—82no—
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×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C22⋊C4—22no—
142C2×C22⋊C4—41no—
143C2×C42—22no—
144C2×C42—41no—
145C2×C42C22×C411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×Q811yescharacteristic
152C2×C42#4111yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42#4111yescharacteristic
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—41no—
165#103—88no—
166C22⋊C8—22no—
167C22⋊C8—22no—
168C22⋊C8—22no—
169C22⋊C8—41no—
170C22⋊C8—41no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174#97—22no—
175#97—41no—
176#97—41no—
177#97—41no—
178#97—41no—
179#97—41no—
180C4×C8—22no—
181C4×C8—22no—
182#95—22no—
183#95—22no—
184#95—41no—
185#95—41no—
186#95—41no—
187#95—41no—
188C24#12111yescharacteristic
189C2×Q8—24no—
190C2×Q8—48no—
191C2×Q8—42no—
192C2×Q8—42no—
193C2×Q8—42no—
194C2×Q8—84no—
195C2×Q8—84no—
196C2×Q8—84no—
197C2×Q8—84no—
198C2×Q8—84no—
199C22×C4—22no—
200C22×C4—22no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—84no—
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4C4×D411yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4C22⋊Q811yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12411yescharacteristic
224C22×C4#12411yescharacteristic
225C22×C4#12611yescharacteristic
226C22×C4#12711yescharacteristic
227C2×C8—24no—
228C2×C8—44no—
229C2×C8—44no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—82no—
234C2×C8—82no—
235C4⋊C4—24no—
236C4⋊C4—24no—
237C4⋊C4—44no—
238C4⋊C4—44no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—82no—
246C4⋊C4—82no—
247C22⋊C4—42no—
248C22⋊C4—42no—
249C22⋊C4—82no—
250C42—82no—
251C42—82no—
252C42C2×C4⋊C411yescharacteristic
253C42#11711yescharacteristic
254C42C4×D411yescharacteristic
255C42C4×D411yescharacteristic
256C42#12011yescharacteristic
257C42#12011yescharacteristic
258C42#12111yescharacteristic
259C42#12111yescharacteristic
260C42C22⋊Q811yescharacteristic
261C42C22⋊Q811yescharacteristic
262C42C22⋊Q811yescharacteristic
263C42#12311yescharacteristic
264C42#12311yescharacteristic
265C42#12311yescharacteristic
266C42#12411yescharacteristic
267C42#12611yescharacteristic
268C23—41no—
269C23—41no—
270C23#39011yescharacteristic
271C23#39411yescharacteristic
272C23#39611yescharacteristic
273C23#47112yes—
274C23#47312yes—
275Q8—48no—
276Q8—84no—
277Q8—84no—
278Q8—84no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—84no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4—82no—
304C2×C4—82no—
305C2×C4—82no—
306C2×C4—82no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4#38011yescharacteristic
310C2×C4#38511yescharacteristic
311C2×C4#38511yescharacteristic
312C2×C4#38611yescharacteristic
313C2×C4#38811yescharacteristic
314C2×C4#38811yescharacteristic
315C2×C4#39211yescharacteristic
316C2×C4#39311yescharacteristic
317C2×C4#39411yescharacteristic
318C2×C4#39611yescharacteristic
319C2×C4#39711yescharacteristic
320C2×C4#39911yescharacteristic
321C8—44no—
322C8—82no—
323C8—82no—
324C8—82no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—42no—
332C22—42no—
333C22#143911yescharacteristic
334C22#197711yescharacteristic
335C22#197911yescharacteristic
336C22#205212yes—
337C22#206412yes—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—84no—
349C4—162no—
350C4—162no—
351C4#190912yes—
352C4#200812yes—
353C2—21no—
354C2—22no—
355C2—22no—
356C2—21no—
357C2#1107411yescharacteristic
358C2#1112811yescharacteristic
359C2#1112911yescharacteristic
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. #1987C4semidirect

Maximal subgroups

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

subgroupindex
#218342
#218112
#178182
C2×#15562
#79842
#79522
#79282