Finite Group Databasebeta

← #116198

Subgroups — #116198

Order 512.

579 conjugacy classes of subgroups.

The subgroup lattice has 375 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21353C211yesmaximal, characteristic
2C2×#1531C211yesmaximal, characteristic
3#17596C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic
5#8089C211yesmaximal, characteristic
6#8034C211yesmaximal, characteristic
7#7984C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C22×#97C2211yescharacteristic
13#2285C2211yescharacteristic
14#2283C2211yescharacteristic
15#1940—22no—
16#1850—22no—
17#1768—22no—
18#1531—22no—
19#1531—22no—
20C2×#330—22no—
21#1520—22no—
22#1520—22no—
23#1518—22no—
24#1515C414yescomplemented
25#1506—22no—
26#1503—22no—
27#1302—22no—
28#1295—22no—
29#1289—22no—
30C23×C8—21no—
31#533—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533C2×C412yes—
38#533D412yes—
39C2×C4⋊Q8Q811yescharacteristic
40C2×C4⋊Q8Q811yescharacteristic
41C2×#125—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×#117—21no—
47C2×#117—21no—
48C22×C42C2311yescharacteristic
49#432—21no—
50#432—21no—
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432C2×C412yes—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—24no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C22⋊C8—21no—
66C2×C22⋊C8—21no—
67C2×C22⋊C8D412yes—
68C2×#97—21no—
69C2×#97—21no—
70C2×#97—21no—
71C2×#97—21no—
72C2×#97—21no—
73C2×#97—21no—
74C2×#97D412yes—
75C2×#97D412yes—
76C2×C4×C8—21no—
77#384—21no—
78#383C2×C412yes—
79#382—21no—
80#382D412yes—
81#381—21no—
82#380—21no—
83#380—21no—
84#380D412yes—
85C4×C4⋊C4—21no—
86C4×C4⋊C4—21no—
87C4×C4⋊C4—21no—
88C4×C22⋊C4—21no—
89#376D412yes—
90#336—24no—
91#330—28no—
92#328—28no—
93#328—28no—
94C22×Q8—21no—
95C22×Q8—21no—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×D411yescharacteristic
98C23×C4#4111yescharacteristic
99C22×C8—22no—
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—44no—
103C22×C8—41no—
104C22×C8—41no—
105C22×C8—41no—
106C4⋊Q8—21no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8C4⋊C412yes—
113C4⋊Q8C4⋊C412yes—
114#125—22no—
115#125—22no—
116#125—41no—
117#125—41no—
118C22⋊Q8—22no—
119C22⋊Q8—22no—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C22⋊Q8—41no—
123C22⋊Q8—41no—
124C22⋊Q8—41no—
125C22⋊Q8—41no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—41no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134#117—22no—
135#117—22no—
136#117—41no—
137#117—41no—
138#117—82no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C4⋊C4—41no—
148C2×C4⋊C4—41no—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—41no—
151C2×C22⋊C4—41no—
152C2×C42—22no—
153C2×C42—41no—
154C2×C42—41no—
155C2×C42C22×C411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×Q811yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C4⋊C8—22no—
168C4⋊C8—48no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180#97—22no—
181#97—22no—
182#97—22no—
183#97—22no—
184#97—22no—
185#97—41no—
186#97—41no—
187#97—41no—
188C4×C8—41no—
189C4×C8—41no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95—22no—
195#95C22⋊C414yes—
196C24#12111yescharacteristic
197C2×Q8—22no—
198C2×Q8—22no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—82no—
202C2×Q8—82no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—24no—
206C22×C4—42no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—42no—
210C22×C4—81no—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C4×D411yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4#12011yescharacteristic
215C22×C4#12011yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4C22⋊Q811yescharacteristic
222C22×C4C22⋊Q811yescharacteristic
223C22×C4C22⋊Q811yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4#12311yescharacteristic
226C22×C4#12311yescharacteristic
227C22×C4#12411yescharacteristic
228C22×C4#12611yescharacteristic
229C2×C8—24no—
230C2×C8—42no—
231C2×C8—44no—
232C2×C8—44no—
233C2×C8—42no—
234C2×C8—44no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—82no—
239C2×C8—82no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—41no—
252C4⋊C4—82no—
253C4⋊C4—82no—
254C4⋊C4—82no—
255C4⋊C4—82no—
256C22⋊C4—42no—
257C22⋊C4—42no—
258C22⋊C4—42no—
259C22⋊C4—42no—
260C22⋊C4—82no—
261C42—44no—
262C42—42no—
263C42—42no—
264C42—82no—
265C42C2×C4⋊C411yescharacteristic
266C42#11711yescharacteristic
267C42C4×D411yescharacteristic
268C42C4×D411yescharacteristic
269C42#12011yescharacteristic
270C42#12111yescharacteristic
271C42#12111yescharacteristic
272C42C22⋊Q811yescharacteristic
273C42C22⋊Q811yescharacteristic
274C42C22⋊Q811yescharacteristic
275C42#12311yescharacteristic
276C42#12311yescharacteristic
277C42#12411yescharacteristic
278C42#12411yescharacteristic
279C42#12611yescharacteristic
280C42#12711yescharacteristic
281C23—41no—
282C23—41no—
283C23#38611yescharacteristic
284C23#39411yescharacteristic
285C23#39611yescharacteristic
286C23#47112yes—
287C23#47312yes—
288Q8—82no—
289Q8—82no—
290Q8—82no—
291Q8—82no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—22no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—44no—
313C2×C4—44no—
314C2×C4—44no—
315C2×C4—42no—
316C2×C4—42no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4—82no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4#38011yescharacteristic
323C2×C4#38511yescharacteristic
324C2×C4#38511yescharacteristic
325C2×C4#38811yescharacteristic
326C2×C4#38811yescharacteristic
327C2×C4#39011yescharacteristic
328C2×C4#39211yescharacteristic
329C2×C4#39311yescharacteristic
330C2×C4#39411yescharacteristic
331C2×C4#39611yescharacteristic
332C2×C4#39711yescharacteristic
333C2×C4#39911yescharacteristic
334C8—44no—
335C8—82no—
336C8—82no—
337C8—82no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—42no—
345C22—42no—
346C22#143911yescharacteristic
347C22#190812yes—
348C22#204611yescharacteristic
349C22#204711yescharacteristic
350C22#206412yes—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—84no—
363C4—82no—
364C4—82no—
365C4—162no—
366C4#203112yes—
367C2—22no—
368C2—21no—
369C2—22no—
370C2—21no—
371C2#1106211yescharacteristic
372C2#1108511yescharacteristic
373C2#1109411yescharacteristic
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
24. #1515C4semidirect

Maximal subgroups

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

subgroupindex
#213532
C2×#15312
#175962
#175752
#80892
#80342
#79842