Finite Group Databasebeta

← #117254

Subgroups — #117254

Order 512.

538 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17816C211yesmaximal, characteristic
2C2×#1554C211yesmaximal, characteristic
3#17672C211yesmaximal, characteristic
4#17545C211yesmaximal, characteristic
5#7997C211yesmaximal, characteristic
6#7930C211yesmaximal, characteristic
7#7870C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#432C2211yescharacteristic
13C2×#431C2211yescharacteristic
14C4×#117C2211yescharacteristic
15#1572—22no—
16#1571—21no—
17#1570—21no—
18#1568—22no—
19#1554—21no—
20#1554—21no—
21#1554—22no—
22C2×#333C412yescomplemented
23C2×#332C412yescomplemented
24#1537—22no—
25#1531—22no—
26#1528—22no—
27#1508—22no—
28#1506—22no—
29#1503—22no—
30#1291—22no—
31#1282—22no—
32#1276—22no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533D412yes—
40#533Q812yes—
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×#125—21no—
44C2×C22⋊Q8—21no—
45C2×C22⋊Q8—21no—
46C2×C4×Q8—21no—
47C2×C4×Q8—21no—
48C2×#117—21no—
49C2×#117—21no—
50C22×C42C2311yescharacteristic
51#432—21no—
52#432—21no—
53#432—21no—
54#432—21no—
55#432—21no—
56#432—21no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432D412yes—
62#432D412yes—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431D412yes—
69#431D411yescharacteristic
70#431D411yescharacteristic
71#431D412yes—
72#423—21no—
73#423—21no—
74C2×C4⋊C8—21no—
75C2×C4⋊C8C2×C412yes—
76C2×C4⋊C8C2×C412yes—
77#407—21no—
78#407—21no—
79C2×C22⋊C8—21no—
80C2×C22⋊C8—21no—
81C2×C22⋊C8—21no—
82C4×M16—21no—
83C4×M16—21no—
84C2×#97—21no—
85C2×#97—21no—
86C2×C4×C8—21no—
87C4×C4⋊C4—21no—
88C4×C4⋊C4—21no—
89C4×C22⋊C4—21no—
90C4×C22⋊C4D412yes—
91#376—21no—
92C43—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—41no—
107C2×M16—41no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—41no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8C4⋊C412yes—
118C4⋊Q8C4⋊C412yes—
119#125—22no—
120#125—22no—
121#125—41no—
122#125—41no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—22no—
126C22⋊Q8—22no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C22⋊Q8—41no—
130C22⋊Q8—41no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—41no—
136C4×Q8—41no—
137C4×Q8—41no—
138C4×Q8—41no—
139#117—22no—
140#117—22no—
141#117—44no—
142#117—41no—
143#117—41no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—21no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—21no—
148C2×C4⋊C4—41no—
149C2×C4⋊C4—41no—
150C2×C4⋊C4—41no—
151C2×C22⋊C4—22no—
152C2×C22⋊C4—41no—
153C2×C42—22no—
154C2×C42—41no—
155C2×C42—41no—
156C2×C42—41no—
157C2×C42C22×C411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×D411yescharacteristic
162C2×C42C2×D411yescharacteristic
163C2×C42C2×D411yescharacteristic
164C2×C42C2×Q811yescharacteristic
165C2×C42#4111yescharacteristic
166C2×C42#4111yescharacteristic
167C2×C42#4111yescharacteristic
168C2×C42#4111yescharacteristic
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—22no—
173C4⋊C8—22no—
174C4⋊C8—22no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8C22⋊C412yes—
179C4⋊C8C22⋊C412yes—
180C22⋊C8—22no—
181C22⋊C8—21no—
182C22⋊C8—21no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—41no—
187C22⋊C8—41no—
188C22⋊C8—41no—
189C22⋊C8—41no—
190C22⋊C8—41no—
191#97—22no—
192#97—22no—
193#97—41no—
194#97—41no—
195#97—41no—
196#97—41no—
197C4×C8—21no—
198C4×C8—21no—
199C4×C8—22no—
200C4×C8—22no—
201C4×C8—41no—
202#95—22no—
203#95—41no—
204C24#12311yescharacteristic
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—82no—
210C2×Q8—82no—
211C22×C4—42no—
212C22×C4—42no—
213C22×C4—42no—
214C22×C4—41no—
215C22×C4—41no—
216C22×C4—42no—
217C22×C4#11711yescharacteristic
218C22×C4C4×D411yescharacteristic
219C22×C4C4×D411yescharacteristic
220C22×C4#12011yescharacteristic
221C22×C4#12111yescharacteristic
222C22×C4#12111yescharacteristic
223C22×C4#12111yescharacteristic
224C22×C4#12111yescharacteristic
225C22×C4C22⋊Q811yescharacteristic
226C22×C4C22⋊Q811yescharacteristic
227C22×C4C22⋊Q811yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12311yescharacteristic
230C22×C4#12311yescharacteristic
231C22×C4#12411yescharacteristic
232C22×C4#12411yescharacteristic
233C22×C4#12611yescharacteristic
234C22×C4#12711yescharacteristic
235M16—82no—
236M16—82no—
237C2×C8—22no—
238C2×C8—22no—
239C2×C8—42no—
240C2×C8—42no—
241C2×C8—42no—
242C2×C8—42no—
243C2×C8—82no—
244C2×C8—82no—
245C2×C8—82no—
246C4⋊C4—22no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—44no—
251C4⋊C4—41no—
252C4⋊C4—41no—
253C4⋊C4—41no—
254C4⋊C4—41no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C22⋊C4—24no—
260C22⋊C4—42no—
261C22⋊C4—42no—
262C22⋊C4—42no—
263C42—44no—
264C42—44no—
265C42—42no—
266C42—42no—
267C42—82no—
268C42C2×C22⋊C411yescharacteristic
269C42C2×C4⋊C411yescharacteristic
270C42C4×D411yescharacteristic
271C42C4×D411yescharacteristic
272C42#12011yescharacteristic
273C42#12011yescharacteristic
274C42#12111yescharacteristic
275C42#12111yescharacteristic
276C42#12111yescharacteristic
277C42#12111yescharacteristic
278C42C22⋊Q811yescharacteristic
279C42C22⋊Q811yescharacteristic
280C42C22⋊Q811yescharacteristic
281C42#12311yescharacteristic
282C42#12411yescharacteristic
283C42#12611yescharacteristic
284C23—22no—
285C23—41no—
286C23—41no—
287C23#38811yescharacteristic
288C23#39411yescharacteristic
289C23#39911yescharacteristic
290Q8—82no—
291Q8—82no—
292Q8—82no—
293Q8—82no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—41no—
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—42no—
314C2×C4—42no—
315C2×C4—82no—
316C2×C4—82no—
317C2×C4—82no—
318C2×C4—82no—
319C2×C4#38011yescharacteristic
320C2×C4#38511yescharacteristic
321C2×C4#38511yescharacteristic
322C2×C4#38611yescharacteristic
323C2×C4#38811yescharacteristic
324C2×C4#39011yescharacteristic
325C2×C4#39211yescharacteristic
326C2×C4#39311yescharacteristic
327C2×C4#39411yescharacteristic
328C2×C4#39611yescharacteristic
329C2×C4#39611yescharacteristic
330C2×C4#39711yescharacteristic
331C8—42no—
332C8—42no—
333C8—82no—
334C8—82no—
335C8—82no—
336C8—82no—
337C22—21no—
338C22—21no—
339C22—21no—
340C22—21no—
341C22—21no—
342C22—21no—
343C22—21no—
344C22—21no—
345C22—82no—
346C22#143911yescharacteristic
347C22#193212yes—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—21no—
352C4—21no—
353C4—21no—
354C4—21no—
355C4—21no—
356C4—21no—
357C4—21no—
358C4—84no—
359C4—82no—
360C4—82no—
361C4#190212yes—
362C4#197612yes—
363C2—22no—
364C2—41no—
365C2—41no—
366C2#1105911yescharacteristic
367C2#1112611yescharacteristic
368C2#1114611yescharacteristic
subgroupG/H
center, socleC22#1439
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
22. C2×#333C4semidirect
23. C2×#332C4semidirect

Maximal subgroups

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

subgroupindex
#178162
C2×#15542
#176722
#175452
#79972
#79302
#78702