Finite Group Databasebeta

← #118641

Subgroups — #118641

Order 512.

542 conjugacy classes of subgroups.

The subgroup lattice has 371 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17818C211yesmaximal, characteristic
2#17676C211yesmaximal, characteristic
3#17596C211yesmaximal, characteristic
4C2×#1518C211yesmaximal, characteristic
5#8030C211yesmaximal, characteristic
6#8017C211yesmaximal, characteristic
7#7926C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9C2×#432C2211yescharacteristic
10C2×#432C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13#2283C2211yescharacteristic
14C2×#382C2211yescharacteristic
15#1573—22no—
16#1573—22no—
17#1568—22no—
18#1534—22no—
19#1531—22no—
20#1531—22no—
21#1520—22no—
22#1518—22no—
23#1518C414yescomplemented
24#1515—22no—
25#1506—22no—
26C2×#328—22no—
27#1295—22no—
28#1294—22no—
29#1282—22no—
30#533—21no—
31#533—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533C2×C412yes—
37#533D412yes—
38C2×C4⋊Q8Q811yescharacteristic
39C2×C4⋊Q8Q811yescharacteristic
40C2×#125—21no—
41C2×C22⋊Q8—21no—
42C2×C22⋊Q8—21no—
43C2×C4×Q8—21no—
44C2×C4×Q8—21no—
45C2×#117—21no—
46C22×C4⋊C4—21no—
47C22×C42C2311yescharacteristic
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#432D412yes—
60#431—21no—
61#431—21no—
62#431C2×C412yes—
63#431D412yes—
64C2×C4⋊C8—21no—
65C2×C4⋊C8—21no—
66C2×C4⋊C8—21no—
67C2×C4⋊C8—21no—
68C2×C22⋊C8—21no—
69C2×C22⋊C8—21no—
70C2×C22⋊C8—21no—
71C2×C22⋊C8D412yes—
72C2×#97—21no—
73C2×#97—21no—
74C2×#97D412yes—
75C2×C4×C8—21no—
76#383—21no—
77#382—21no—
78#382—21no—
79#382—21no—
80#382—21no—
81#382C2×C412yes—
82#381—21no—
83#380D412yes—
84C4×C4⋊C4—21no—
85C4×C22⋊C4—21no—
86#376—21no—
87C2×#95—21no—
88C2×#95—21no—
89#335—24no—
90#334—24no—
91#330—28no—
92#328—28no—
93#328—28no—
94C22×Q8—21no—
95C22×Q8—21no—
96C23×C4—41no—
97C23×C4C2×D411yescharacteristic
98C23×C4C2×D411yescharacteristic
99C23×C4#4111yescharacteristic
100C22×C8—22no—
101C22×C8—22no—
102C22×C8—41no—
103C22×C8—41no—
104C4⋊Q8—21no—
105C4⋊Q8—21no—
106C4⋊Q8—21no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8C4⋊C412yes—
111C4⋊Q8C4⋊C412yes—
112#125—22no—
113#125—22no—
114#125—41no—
115#125—41no—
116C22⋊Q8—22no—
117C22⋊Q8—22no—
118C22⋊Q8—22no—
119C22⋊Q8—22no—
120C22⋊Q8—41no—
121C22⋊Q8—41no—
122C22⋊Q8—41no—
123C22⋊Q8—41no—
124C4×Q8—22no—
125C4×Q8—22no—
126C4×Q8—22no—
127C4×Q8—22no—
128C4×Q8—41no—
129C4×Q8—41no—
130C4×Q8—41no—
131C4×Q8—41no—
132#117—22no—
133#117—22no—
134#117—41no—
135#117—41no—
136C2×C4⋊C4—21no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—21no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—82no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—41no—
148C2×C42—41no—
149C2×C42—41no—
150C2×C42C22×C411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×Q811yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C2×C42#4111yescharacteristic
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—41no—
167C4⋊C8—41no—
168C4⋊C8—41no—
169C4⋊C8—41no—
170C22⋊C8—22no—
171C22⋊C8—22no—
172C22⋊C8—22no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176C22⋊C8—41no—
177C22⋊C8C22⋊C414yes—
178#97—22no—
179#97—22no—
180#97—22no—
181#97—22no—
182#97—41no—
183#97—41no—
184C4×C8—22no—
185C4×C8—22no—
186#95—22no—
187#95—22no—
188#95—22no—
189#95—41no—
190#95—41no—
191#95—41no—
192#95—41no—
193#95—41no—
194C24#12111yescharacteristic
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—41no—
198C2×Q8—41no—
199C2×Q8—82no—
200C2×Q8—82no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—41no—
204C22×C4—41no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—81no—
208C22×C4—81no—
209C22×C4—82no—
210C22×C4—82no—
211C22×C4—82no—
212C22×C4C2×C22⋊C411yescharacteristic
213C22×C4C4×D411yescharacteristic
214C22×C4C4×D411yescharacteristic
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#12411yescharacteristic
227C22×C4#12411yescharacteristic
228C22×C4#12611yescharacteristic
229C22×C4#12711yescharacteristic
230C2×C8—24no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—42no—
234C2×C8—82no—
235C2×C8—82no—
236C2×C8—82no—
237C2×C8—82no—
238C4⋊C4—22no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—82no—
251C4⋊C4—82no—
252C4⋊C4—82no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C22⋊C4—42no—
256C22⋊C4—42no—
257C42—42no—
258C42—42no—
259C42—82no—
260C42C2×C4⋊C411yescharacteristic
261C42#11711yescharacteristic
262C42C4×D411yescharacteristic
263C42C4×D411yescharacteristic
264C42#12011yescharacteristic
265C42#12011yescharacteristic
266C42#12111yescharacteristic
267C42#12111yescharacteristic
268C42C22⋊Q811yescharacteristic
269C42C22⋊Q811yescharacteristic
270C42C22⋊Q811yescharacteristic
271C42#12311yescharacteristic
272C42#12311yescharacteristic
273C42#12311yescharacteristic
274C42#12411yescharacteristic
275C42#12611yescharacteristic
276C23—41no—
277C23—41no—
278C23#39011yescharacteristic
279C23#39411yescharacteristic
280C23#39611yescharacteristic
281C23#47112yes—
282C23#47312yes—
283Q8—82no—
284Q8—82no—
285Q8—82no—
286Q8—82no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—42no—
308C2×C4—42no—
309C2×C4—82no—
310C2×C4—82no—
311C2×C4—82no—
312C2×C4—82no—
313C2×C4—82no—
314C2×C4—82no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—164no—
318C2×C4#38011yescharacteristic
319C2×C4#38511yescharacteristic
320C2×C4#38511yescharacteristic
321C2×C4#38611yescharacteristic
322C2×C4#38811yescharacteristic
323C2×C4#38811yescharacteristic
324C2×C4#39211yescharacteristic
325C2×C4#39311yescharacteristic
326C2×C4#39411yescharacteristic
327C2×C4#39611yescharacteristic
328C2×C4#39711yescharacteristic
329C2×C4#39911yescharacteristic
330C8—44no—
331C8—82no—
332C8—82no—
333C8—82no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—42no—
341C22—42no—
342C22#143911yescharacteristic
343C22#197711yescharacteristic
344C22#197911yescharacteristic
345C22#205212yes—
346C22#206412yes—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—82no—
360C4—82no—
361C4—162no—
362C4—162no—
363C2—21no—
364C2—22no—
365C2—22no—
366C2—21no—
367C2#1109411yescharacteristic
368C2#1111911yescharacteristic
369C2#1112811yescharacteristic
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
23. #1518C4semidirect

Maximal subgroups

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

subgroupindex
#178182
#176762
#175962
C2×#15182
#80302
#80172
#79262