Finite Group Databasebeta

← #118709

Subgroups — #118709

Order 512.

551 conjugacy classes of subgroups.

The subgroup lattice has 379 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21425C211yesmaximal, characteristic
2C2×#1550C211yesmaximal, characteristic
3#17590C211yesmaximal, characteristic
4#8063C211yesmaximal, characteristic
5#7988C211yesmaximal, characteristic
6#7962C211yesmaximal, characteristic
7#7895C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#423C2211yescharacteristic
12C22×C4⋊C8C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2283C2211yescharacteristic
15#1997—21no—
16#1966—21no—
17#1901—22no—
18C4×#101C414yescomplemented
19#1550—22no—
20#1550—22no—
21C2×#333—21no—
22C2×#332—21no—
23#1518—22no—
24#1511—22no—
25#1506—22no—
26#1299—22no—
27#1290—22no—
28#1286—22no—
29#1279—22no—
30C22×M16—21no—
31C22×M16—21no—
32C23×C8—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533—21no—
41C2×C4⋊Q8D411yescharacteristic
42C2×C4⋊Q8D411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8D412yes—
46C2×C4×Q8—21no—
47C2×C4×Q8C2×C412yes—
48C2×#117—21no—
49C22×C42C2311yescharacteristic
50#432—21no—
51#432—21no—
52#432—21no—
53#432D412yes—
54#431—21no—
55#431—21no—
56#431—21no—
57#431—21no—
58#423—21no—
59#423—21no—
60#423—21no—
61#423—21no—
62#423—21no—
63#423D412yes—
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⋊C8D412yes—
72C2×#101—24no—
73#407—21no—
74#407—21no—
75C4×M16—21no—
76C4×M16—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16C2×C412yes—
80C4×M16C2×C412yes—
81C2×#97—21no—
82C2×C4×C8—21no—
83#383—21no—
84#382D412yes—
85#381—21no—
86#380—21no—
87C4×C4⋊C4—21no—
88C4×C22⋊C4—21no—
89#376—21no—
90#343—24no—
91#333—24no—
92#332—24no—
93#328—28no—
94C22×Q8—21no—
95C22×Q8C22⋊C412yes—
96C23×C4C2×D411yescharacteristic
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C2×M16—22no—
100C2×M16—22no—
101C2×M16—22no—
102C2×M16—44no—
103C2×M16—41no—
104C2×M16—41no—
105C2×M16—41no—
106C2×M16—82no—
107C22×C8—22no—
108C22×C8—41no—
109C22×C8—41no—
110C22×C8—41no—
111C22×C8—41no—
112C22×C8—82no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121#125—41no—
122#125—41no—
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—41no—
142#117—41no—
143#117—41no—
144#117—41no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4C22⋊C412yes—
152C2×C22⋊C4—22no—
153C2×C22⋊C4—41no—
154C2×C42—22no—
155C2×C42—41no—
156C2×C42C22×C411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180#101—48no—
181C22⋊C8—22no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185#97—22no—
186#97—22no—
187#97—22no—
188#97—41no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—41no—
192C4×C8—41no—
193#95—22no—
194#95—22no—
195#95—41no—
196#95—41no—
197C24#12311yescharacteristic
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—44no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—81no—
204C2×Q8—81no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4C2×C22⋊C411yescharacteristic
214C22×C4#11711yescharacteristic
215C22×C4#11711yescharacteristic
216C22×C4#12011yescharacteristic
217C22×C4#12111yescharacteristic
218C22×C4#12111yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12311yescharacteristic
224C22×C4#12311yescharacteristic
225C22×C4#12311yescharacteristic
226C22×C4#12311yescharacteristic
227C22×C4#12411yescharacteristic
228C22×C4#12411yescharacteristic
229C22×C4#12611yescharacteristic
230C22×C4#12611yescharacteristic
231M16—24no—
232M16—42no—
233M16—44no—
234M16—44no—
235M16—82no—
236M16—82no—
237M16—82no—
238C2×C8—44no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—82no—
245C2×C8—82no—
246C2×C8—82no—
247C2×C8—82no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—41no—
257C4⋊C4—41no—
258C4⋊C4—42no—
259C4⋊C4—41no—
260C4⋊C4—42no—
261C4⋊C4—42no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—82no—
265C42—42no—
266C42—42no—
267C42—81no—
268C42—81no—
269C42C4×D411yescharacteristic
270C42C4×D411yescharacteristic
271C42C4×D411yescharacteristic
272C42C4×D411yescharacteristic
273C42#12011yescharacteristic
274C42#12111yescharacteristic
275C42#12111yescharacteristic
276C42#12111yescharacteristic
277C42#12111yescharacteristic
278C42#12311yescharacteristic
279C42#12311yescharacteristic
280C42#12311yescharacteristic
281C42#12411yescharacteristic
282C42#12411yescharacteristic
283C42#12611yescharacteristic
284C42#12611yescharacteristic
285C23—22no—
286C23—22no—
287C23—41no—
288C23#38111yescharacteristic
289C23#39411yescharacteristic
290C23#39711yescharacteristic
291Q8—82no—
292Q8—82no—
293Q8—82no—
294Q8—81no—
295Q8—81no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—44no—
319C2×C4—42no—
320C2×C4—42no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4#38611yescharacteristic
326C2×C4#38611yescharacteristic
327C2×C4#38811yescharacteristic
328C2×C4#38811yescharacteristic
329C2×C4#38811yescharacteristic
330C2×C4#38811yescharacteristic
331C2×C4#39411yescharacteristic
332C2×C4#39411yescharacteristic
333C2×C4#39411yescharacteristic
334C2×C4#39711yescharacteristic
335C2×C4#39711yescharacteristic
336C2×C4#39711yescharacteristic
337C2×C4#47112yes—
338C2×C4#47312yes—
339C8—44no—
340C8—82no—
341C8—82no—
342C8—82no—
343C8—82no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—44no—
352C22#144011yescharacteristic
353C22#182712yes—
354C22#205212yes—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—82no—
365C4—82no—
366C4—162no—
367C4#193012yes—
368C4#204611yescharacteristic
369C4#204711yescharacteristic
370C4#204811yescharacteristic
371C4#204911yescharacteristic
372C2—22no—
373C2—22no—
374C2—41no—
375C2#1114811yescharacteristic
376C2#1115211yescharacteristic
377C2#1116411yescharacteristic
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
18. C4×#101C4semidirect

Maximal subgroups

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

subgroupindex
#214252
C2×#15502
#175902
#80632
#79882
#79622
#78952