Finite Group Databasebeta

← #119209

Subgroups — #119209

Order 512.

576 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21804C211yesmaximal, characteristic
2#17742C211yesmaximal, characteristic
3#17742C211yesmaximal, characteristic
4#8109C211yesmaximal, characteristic
5#7965C211yesmaximal, characteristic
6#7961C211yesmaximal, characteristic
7#7886C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#431C2211yescharacteristic
14#2285C2211yescharacteristic
15#1981—22no—
16#1956—22no—
17#1787—22no—
18#1556—22no—
19#1556—22no—
20#1549—22no—
21#1549—22no—
22#1546—22no—
23#1546C414yescomplemented
24#1305—22no—
25#1287—22no—
26#1286—22no—
27#1278—22no—
28C23×Q8—21no—
29C2×C4⋊Q8—21no—
30C2×C4⋊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⋊Q8C2×C412yes—
39C2×C4⋊Q8D412yes—
40C2×C4⋊Q8D412yes—
41C2×#117—21no—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C42C2311yescharacteristic
45#432—21no—
46#432—21no—
47#432—21no—
48#432—21no—
49#432—21no—
50#432—21no—
51#432—21no—
52#432D412yes—
53#431—21no—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431—21no—
63#431—21no—
64#431C2×C412yes—
65#423—21no—
66#423—21no—
67#423C2×C412yes—
68#423D412yes—
69C2×C4⋊C8—21no—
70C2×#103—28no—
71#407—21no—
72#407—21no—
73#407—21no—
74#407—21no—
75C2×C22⋊C8—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C2×C4×C8D412yes—
81#384—21no—
82#382—21no—
83#380—21no—
84#380D412yes—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87#332—28no—
88#332—28no—
89C22×Q8—22no—
90C22×Q8—22no—
91C22×Q8—22no—
92C22×Q8—41no—
93C22×Q8—41no—
94C22×Q8—41no—
95C22×Q8—84no—
96C23×C4—41no—
97C23×C4C2×D411yescharacteristic
98C23×C4#4111yescharacteristic
99C23×C4#4111yescharacteristic
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—41no—
103C2×M16—41no—
104C22×C8—22no—
105C4⋊Q8—22no—
106C4⋊Q8—22no—
107C4⋊Q8—22no—
108C4⋊Q8—22no—
109C4⋊Q8—22no—
110C4⋊Q8—22no—
111C4⋊Q8—22no—
112C4⋊Q8—22no—
113C4⋊Q8—22no—
114C4⋊Q8—41no—
115C4⋊Q8—41no—
116C4⋊Q8—41no—
117C4⋊Q8—41no—
118C4⋊Q8—41no—
119C4⋊Q8—41no—
120C4⋊Q8C22⋊C414yes—
121#117—82no—
122C2×C4⋊C4—22no—
123C2×C4⋊C4—22no—
124C2×C4⋊C4—22no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C4⋊C4—22no—
129C2×C4⋊C4—41no—
130C2×C4⋊C4—41no—
131C2×C4⋊C4—41no—
132C2×C4⋊C4—41no—
133C2×C4⋊C4—41no—
134C2×C4⋊C4—41no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
137C2×C22⋊C4—22no—
138C2×C42—41no—
139C2×C42C22×C411yescharacteristic
140C2×C42C2×D411yescharacteristic
141C2×C42C2×D411yescharacteristic
142C2×C42C2×D411yescharacteristic
143C2×C42C2×D411yescharacteristic
144C2×C42C2×D411yescharacteristic
145C2×C42#4111yescharacteristic
146C2×C42#4111yescharacteristic
147C2×C42#4111yescharacteristic
148C2×C42#4111yescharacteristic
149C2×C42#4111yescharacteristic
150C2×C42#4111yescharacteristic
151C4⋊C8—22no—
152C4⋊C8—22no—
153C4⋊C8—22no—
154C4⋊C8—22no—
155C4⋊C8—22no—
156C4⋊C8—22no—
157C4⋊C8—41no—
158C4⋊C8—41no—
159C4⋊C8—41no—
160C4⋊C8—41no—
161#103—88no—
162C22⋊C8—22no—
163C22⋊C8—22no—
164C22⋊C8—41no—
165C22⋊C8—41no—
166C22⋊C8—41no—
167C22⋊C8—41no—
168C22⋊C8—41no—
169C22⋊C8—41no—
170C22⋊C8—41no—
171C22⋊C8—41no—
172#97—22no—
173#97—41no—
174#97—41no—
175#97—41no—
176C4×C8—22no—
177C4×C8—22no—
178C4×C8—22no—
179C4×C8—41no—
180C4×C8—41no—
181C4×C8—41no—
182#95—22no—
183#95—41no—
184C24#12311yescharacteristic
185C2×Q8—24no—
186C2×Q8—48no—
187C2×Q8—42no—
188C2×Q8—42no—
189C2×Q8—42no—
190C2×Q8—84no—
191C2×Q8—84no—
192C2×Q8—84no—
193C2×Q8—84no—
194C2×Q8—84no—
195C22×C4—22no—
196C22×C4—22no—
197C22×C4—42no—
198C22×C4—42no—
199C22×C4—42no—
200C22×C4—42no—
201C22×C4—42no—
202C22×C4—42no—
203C22×C4—84no—
204C22×C4#11711yescharacteristic
205C22×C4C4×D411yescharacteristic
206C22×C4C4×D411yescharacteristic
207C22×C4#12011yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12311yescharacteristic
214C22×C4#12311yescharacteristic
215C22×C4#12311yescharacteristic
216C22×C4#12411yescharacteristic
217C22×C4#12411yescharacteristic
218C22×C4#12411yescharacteristic
219C22×C4#12611yescharacteristic
220C22×C4#12611yescharacteristic
221C22×C4#12611yescharacteristic
222M16—42no—
223M16—42no—
224M16—82no—
225M16—82no—
226M16—82no—
227C2×C8—44no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C4⋊C4—24no—
234C4⋊C4—24no—
235C4⋊C4—44no—
236C4⋊C4—42no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—82no—
243C4⋊C4—82no—
244C22⋊C4—82no—
245C42—82no—
246C42C2×C22⋊C411yescharacteristic
247C42#11711yescharacteristic
248C42C4×D411yescharacteristic
249C42C4×D411yescharacteristic
250C42#12011yescharacteristic
251C42#12111yescharacteristic
252C42#12111yescharacteristic
253C42#12111yescharacteristic
254C42#12311yescharacteristic
255C42#12311yescharacteristic
256C42#12311yescharacteristic
257C42#12311yescharacteristic
258C42#12311yescharacteristic
259C42#12311yescharacteristic
260C42#12411yescharacteristic
261C42#12611yescharacteristic
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23#38811yescharacteristic
267C23#39411yescharacteristic
268C23#39711yescharacteristic
269Q8—48no—
270Q8—84no—
271Q8—84no—
272Q8—84no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—84no—
293C2×C4—84no—
294C2×C4—84no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4#38111yescharacteristic
299C2×C4#38611yescharacteristic
300C2×C4#38611yescharacteristic
301C2×C4#38811yescharacteristic
302C2×C4#38811yescharacteristic
303C2×C4#38811yescharacteristic
304C2×C4#39411yescharacteristic
305C2×C4#39411yescharacteristic
306C2×C4#39411yescharacteristic
307C2×C4#39711yescharacteristic
308C2×C4#39711yescharacteristic
309C2×C4#39711yescharacteristic
310C2×C4#47112yes—
311C2×C4#47312yes—
312C8—82no—
313C8—82no—
314C8—82no—
315C8—82no—
316C8—82no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—42no—
326C22—42no—
327C22#144011yescharacteristic
328C22#204611yescharacteristic
329C22#204711yescharacteristic
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—84no—
339C4—162no—
340C4#190412yes—
341C4#193012yes—
342C4#205212yes—
343C4#208212yes—
344C2—21no—
345C2—21no—
346C2—41no—
347C2—41no—
348C2#1116411yescharacteristic
349C2#1118211yescharacteristic
350C2#1118311yescharacteristic
subgroupG/H
center, socleC22#1440
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#123
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
23. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#218042
#177422
#81092
#79652
#79612
#78862