Finite Group Databasebeta

← #117346

Subgroups — #117346

Order 512.

554 conjugacy classes of subgroups.

The subgroup lattice has 381 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17803C211yesmaximal, characteristic
2#17718C211yesmaximal, characteristic
3#17648C211yesmaximal, characteristic
4#17575C211yesmaximal, characteristic
5#8105C211yesmaximal, characteristic
6#7896C211yesmaximal, characteristic
7#7888C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#431C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C2×C4×M16C2211yescharacteristic
14#2285C2211yescharacteristic
15#1572—22no—
16#1571—21no—
17#1570—21no—
18#1564—22no—
19#1554—21no—
20#1554—21no—
21#1548C412yescomplemented
22#1546C412yescomplemented
23#1542—22no—
24#1531—22no—
25#1528—22no—
26#1526—22no—
27#1520—22no—
28#1518—22no—
29#1503—22no—
30#1305—22no—
31#1279—22no—
32#1278—22no—
33C22×M16—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40#533D412yes—
41#533Q812yes—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
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×C42C2311yescharacteristic
52#432—21no—
53#432—21no—
54#432—21no—
55#432D412yes—
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#431—21no—
65#431C2×C412yes—
66#431D412yes—
67#431D411yescharacteristic
68#431D411yescharacteristic
69#423—21no—
70#423C2×C412yes—
71C2×C4⋊C8—21no—
72C2×C4⋊C8D412yes—
73#407—21no—
74#407—21no—
75C2×C22⋊C8—21no—
76C2×C22⋊C8—21no—
77C4×M16—21no—
78C4×M16—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16D412yes—
83C2×#97—21no—
84C2×C4×C8—21no—
85C2×C4×C8—21no—
86C2×C4×C8—21no—
87#384—21no—
88#382—21no—
89#380—21no—
90#380D412yes—
91C4×C4⋊C4—21no—
92C4×C4⋊C4—21no—
93#335—24no—
94#334—24no—
95#333—22no—
96#333—22no—
97#332—22no—
98#332—22no—
99#330—28no—
100#328—28no—
101C22×Q8—21no—
102C22×Q8—21no—
103C23×C4C2×D411yescharacteristic
104C23×C4#4111yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—44no—
109C2×M16—41no—
110C2×M16—41no—
111C22×C8—22no—
112C22×C8—41no—
113C22×C8—41no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8C4⋊C412yes—
121C4⋊Q8C4⋊C412yes—
122#125—22no—
123#125—22no—
124#125—41no—
125#125—41no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—41no—
131C22⋊Q8—41no—
132C22⋊Q8—41no—
133C22⋊Q8—41no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—41no—
139C4×Q8—41no—
140C4×Q8—41no—
141C4×Q8—41no—
142#117—22no—
143#117—22no—
144#117—41no—
145#117—41no—
146#117—82no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—41no—
153C2×C4⋊C4—41no—
154C2×C4⋊C4—41no—
155C2×C4⋊C4—41no—
156C2×C22⋊C4—22no—
157C2×C22⋊C4—41no—
158C2×C42—41no—
159C2×C42—41no—
160C2×C42C22×C411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×D411yescharacteristic
165C2×C42C2×D411yescharacteristic
166C2×C42C2×D411yescharacteristic
167C2×C42C2×Q811yescharacteristic
168C2×C42#4111yescharacteristic
169C2×C42#4111yescharacteristic
170C2×C42#4111yescharacteristic
171C2×C42#4111yescharacteristic
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—22no—
176C4⋊C8—22no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8C22⋊C412yes—
180C4⋊C8C22⋊C412yes—
181C22⋊C8—22no—
182C22⋊C8—21no—
183C22⋊C8—21no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190#97—22no—
191#97—22no—
192#97—41no—
193#97—41no—
194C4×C8—22no—
195C4×C8—21no—
196C4×C8—21no—
197C4×C8—22no—
198C4×C8—22no—
199C4×C8—41no—
200C4×C8—41no—
201C4×C8—41no—
202C4×C8—41no—
203#95—22no—
204#95—41no—
205C24#12311yescharacteristic
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—82no—
211C2×Q8—82no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—41no—
217C22×C4—41no—
218C22×C4—42no—
219C22×C4#11711yescharacteristic
220C22×C4C4×D411yescharacteristic
221C22×C4C4×D411yescharacteristic
222C22×C4#12011yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4#12111yescharacteristic
226C22×C4#12111yescharacteristic
227C22×C4C22⋊Q811yescharacteristic
228C22×C4C22⋊Q811yescharacteristic
229C22×C4C22⋊Q811yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12311yescharacteristic
232C22×C4#12311yescharacteristic
233C22×C4#12411yescharacteristic
234C22×C4#12411yescharacteristic
235C22×C4#12611yescharacteristic
236C22×C4#12711yescharacteristic
237M16—24no—
238M16—42no—
239M16—44no—
240M16—82no—
241M16—82no—
242C2×C8—22no—
243C2×C8—22no—
244C2×C8—44no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—82no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—44no—
257C4⋊C4—41no—
258C4⋊C4—41no—
259C4⋊C4—41no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—42no—
264C4⋊C4—42no—
265C4⋊C4—82no—
266C4⋊C4—82no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—82no—
270C42—42no—
271C42—42no—
272C42—82no—
273C42C2×C22⋊C411yescharacteristic
274C42C2×C4⋊C411yescharacteristic
275C42C4×D411yescharacteristic
276C42C4×D411yescharacteristic
277C42#12011yescharacteristic
278C42#12011yescharacteristic
279C42#12111yescharacteristic
280C42#12111yescharacteristic
281C42#12111yescharacteristic
282C42#12111yescharacteristic
283C42C22⋊Q811yescharacteristic
284C42C22⋊Q811yescharacteristic
285C42C22⋊Q811yescharacteristic
286C42#12311yescharacteristic
287C42#12411yescharacteristic
288C42#12611yescharacteristic
289C23—21no—
290C23—21no—
291C23—41no—
292C23—41no—
293C23#38811yescharacteristic
294C23#39411yescharacteristic
295C23#39911yescharacteristic
296Q8—82no—
297Q8—82no—
298Q8—82no—
299Q8—82no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—22no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—22no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—41no—
318C2×C4—42no—
319C2×C4—42no—
320C2×C4—82no—
321C2×C4—82no—
322C2×C4—82no—
323C2×C4—82no—
324C2×C4—82no—
325C2×C4#38011yescharacteristic
326C2×C4#38511yescharacteristic
327C2×C4#38511yescharacteristic
328C2×C4#38611yescharacteristic
329C2×C4#38811yescharacteristic
330C2×C4#39011yescharacteristic
331C2×C4#39211yescharacteristic
332C2×C4#39311yescharacteristic
333C2×C4#39411yescharacteristic
334C2×C4#39611yescharacteristic
335C2×C4#39611yescharacteristic
336C2×C4#39711yescharacteristic
337C2×C4#47112yes—
338C2×C4#47312yes—
339C8—44no—
340C8—42no—
341C8—42no—
342C8—82no—
343C8—82no—
344C8—82no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—81no—
354C22—81no—
355C22#143911yescharacteristic
356C22#204611yescharacteristic
357C22#204711yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—21no—
365C4—21no—
366C4—82no—
367C4—82no—
368C4—162no—
369C4#190812yes—
370C4#193212yes—
371C4#197612yes—
372C4#206412yes—
373C2—21no—
374C2—21no—
375C2—41no—
376C2—41no—
377C2#1106211yescharacteristic
378C2#1112611yescharacteristic
379C2#1114611yescharacteristic
subgroupG/H
centerC2×C4#396
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
socleC22#1439
prime radicalC24#123
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
21. #1548C4semidirect
22. #1546C4semidirect

Maximal subgroups

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

subgroupindex
#178032
#177182
#176482
#175752
#81052
#78962
#78882