Finite Group Databasebeta

← #116227

Subgroups — #116227

Order 512.

626 conjugacy classes of subgroups.

The subgroup lattice has 395 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21885C211yesmaximal, characteristic
2C2×#1832C211yesmaximal, characteristic
3C2×#1552C211yesmaximal, characteristic
4C2×#1506C211yesmaximal, characteristic
5#8044C211yesmaximal, characteristic
6#8022C211yesmaximal, characteristic
7C2×#1290C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C22×C4⋊C8C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13#2285C2211yescharacteristic
14C2×#376C2211yescharacteristic
15#1987—21no—
16#1981—21no—
17#1892—22no—
18#1832—22no—
19#1832—22no—
20#1803—22no—
21C2×#336—22no—
22#1552—22no—
23#1552—22no—
24C2×#332—21no—
25C2×#332—21no—
26#1506—22no—
27#1506C414yescomplemented
28C2×#328—22no—
29#1296—22no—
30#1294—22no—
31#1290—22no—
32C23×C8—21no—
33C23×C8D412yes—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533C2×C412yes—
41#533Q812yes—
42C2×C4⋊Q8D411yescharacteristic
43C2×C4⋊Q8D411yescharacteristic
44C2×#125—21no—
45C2×C22⋊Q8—21no—
46C2×C22⋊Q8—21no—
47C2×C4×Q8—21no—
48C2×C4×Q8—21no—
49C2×#117—21no—
50C2×#117—21no—
51C22×C42—21no—
52C22×C42C2311yescharacteristic
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432C2×C412yes—
60#432D412yes—
61C2×C4⋊C8—21no—
62C2×C4⋊C8—21no—
63C2×C4⋊C8—21no—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69C2×C4⋊C8—21no—
70C2×C4⋊C8—21no—
71C2×C4⋊C8—21no—
72C2×C4⋊C8—21no—
73C2×C4⋊C8—21no—
74C2×C4⋊C8D412yes—
75C2×#103—24no—
76C2×C22⋊C8—21no—
77C2×C22⋊C8—21no—
78C2×#97—21no—
79C2×#97—21no—
80#384D412yes—
81#382—21no—
82#380—21no—
83#380—21no—
84C4×C4⋊C4—21no—
85C4×C4⋊C4—21no—
86#376—21no—
87#376—21no—
88#376C2×C412yes—
89#376D412yes—
90C2×#95—21no—
91C2×#95D412yes—
92#340—24no—
93#336—28no—
94#332—24no—
95#332—24no—
96#328—28no—
97C22×Q8—21no—
98C22×Q8—21no—
99C23×C4—22no—
100C23×C4C2×D411yescharacteristic
101C23×C4C2×D411yescharacteristic
102C23×C4#4111yescharacteristic
103C22×C8—22no—
104C22×C8—22no—
105C22×C8—22no—
106C22×C8—22no—
107C22×C8—44no—
108C22×C8—44no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—41no—
112C22×C8—41no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8C4⋊C412yes—
120C4⋊Q8C4⋊C412yes—
121#125—22no—
122#125—22no—
123#125—41no—
124#125—41no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—41no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141#117—22no—
142#117—22no—
143#117—41no—
144#117—41no—
145#117—82no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C22⋊C4—41no—
156C2×C22⋊C4—41no—
157C2×C42—22no—
158C2×C42—22no—
159C2×C42—42no—
160C2×C42—42no—
161C2×C42—41no—
162C2×C42—41no—
163C2×C42C22×C411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×D411yescharacteristic
168C2×C42C2×D411yescharacteristic
169C2×C42C2×Q811yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C2×C42#4111yescharacteristic
173C2×C42#4111yescharacteristic
174C2×C42#4111yescharacteristic
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—22no—
178C4⋊C8—22no—
179C4⋊C8—41no—
180C4⋊C8—41no—
181C4⋊C8—41no—
182C4⋊C8—41no—
183C4⋊C8—41no—
184C4⋊C8—41no—
185C4⋊C8—41no—
186C4⋊C8—41no—
187#103—84no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—41no—
191C22⋊C8—41no—
192#97—22no—
193#97—22no—
194#97—41no—
195#97—41no—
196#95—22no—
197#95—22no—
198#95—22no—
199#95—41no—
200#95—41no—
201#95C22⋊C414yes—
202C24#12111yescharacteristic
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—41no—
206C2×Q8—41no—
207C2×Q8—42no—
208C2×Q8—42no—
209C2×Q8—82no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—24no—
215C22×C4—42no—
216C22×C4—44no—
217C22×C4—44no—
218C22×C4—41no—
219C22×C4—41no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4C2×C22⋊C411yescharacteristic
224C22×C4C4×D411yescharacteristic
225C22×C4C4×D411yescharacteristic
226C22×C4#12011yescharacteristic
227C22×C4#12011yescharacteristic
228C22×C4#12111yescharacteristic
229C22×C4#12111yescharacteristic
230C22×C4#12111yescharacteristic
231C22×C4#12111yescharacteristic
232C22×C4#12111yescharacteristic
233C22×C4C22⋊Q811yescharacteristic
234C22×C4C22⋊Q811yescharacteristic
235C22×C4C22⋊Q811yescharacteristic
236C22×C4#12311yescharacteristic
237C22×C4#12311yescharacteristic
238C22×C4#12311yescharacteristic
239C22×C4#12411yescharacteristic
240C22×C4#12611yescharacteristic
241C2×C8—44no—
242C2×C8—44no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—44no—
248C2×C8—82no—
249C2×C8—82no—
250C2×C8—82no—
251C2×C8—82no—
252C2×C8—82no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—41no—
262C4⋊C4—42no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—42no—
267C4⋊C4—82no—
268C4⋊C4—82no—
269C22⋊C4—42no—
270C22⋊C4—42no—
271C22⋊C4—82no—
272C42—44no—
273C42—42no—
274C42—42no—
275C42—42no—
276C42—42no—
277C42—82no—
278C42C2×C4⋊C411yescharacteristic
279C42#11711yescharacteristic
280C42C4×D411yescharacteristic
281C42C4×D411yescharacteristic
282C42#12011yescharacteristic
283C42#12111yescharacteristic
284C42#12111yescharacteristic
285C42C22⋊Q811yescharacteristic
286C42C22⋊Q811yescharacteristic
287C42C22⋊Q811yescharacteristic
288C42#12311yescharacteristic
289C42#12311yescharacteristic
290C42#12411yescharacteristic
291C42#12411yescharacteristic
292C42#12611yescharacteristic
293C42#12711yescharacteristic
294C23—22no—
295C23—22no—
296C23#38611yescharacteristic
297C23#39411yescharacteristic
298C23#39611yescharacteristic
299C23#47112yes—
300C23#47312yes—
301Q8—44no—
302Q8—82no—
303Q8—82no—
304Q8—81no—
305Q8—81no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—22no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—41no—
326C2×C4—41no—
327C2×C4—44no—
328C2×C4—44no—
329C2×C4—44no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—84no—
333C2×C4—84no—
334C2×C4—82no—
335C2×C4—82no—
336C2×C4—82no—
337C2×C4—82no—
338C2×C4—82no—
339C2×C4#38011yescharacteristic
340C2×C4#38511yescharacteristic
341C2×C4#38511yescharacteristic
342C2×C4#38811yescharacteristic
343C2×C4#38811yescharacteristic
344C2×C4#39011yescharacteristic
345C2×C4#39211yescharacteristic
346C2×C4#39311yescharacteristic
347C2×C4#39411yescharacteristic
348C2×C4#39611yescharacteristic
349C2×C4#39711yescharacteristic
350C2×C4#39911yescharacteristic
351C2×C4#45412yes—
352C2×C4#45512yes—
353C8—82no—
354C8—82no—
355C8—82no—
356C8—82no—
357C22—24no—
358C22—24no—
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22#143911yescharacteristic
364C22#190812yes—
365C22#204012yes—
366C22#204611yescharacteristic
367C22#204711yescharacteristic
368C22#205312yes—
369C22#206412yes—
370C4—21no—
371C4—21no—
372C4—21no—
373C4—21no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—21no—
378C4—21no—
379C4—21no—
380C4—84no—
381C4—82no—
382C4—82no—
383C4—162no—
384C4#186012yes—
385C4#200311yescharacteristic
386C4#200411yescharacteristic
387C2—22no—
388C2—22no—
389C2#1051512yes—
390C2#1051712yes—
391C2#1106211yescharacteristic
392C2#1107911yescharacteristic
393C2#1109211yescharacteristic
subgroupG/H
center, socleC23#394
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#121
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
27. #1506C4semidirect

Maximal subgroups

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

subgroupindex
#218852
C2×#18322
C2×#15522
C2×#15062
#80442
#80222
C2×#12902