Finite Group Databasebeta

← #116692

Subgroups — #116692

Order 512.

508 conjugacy classes of subgroups.

The subgroup lattice has 353 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17821C211yesmaximal, characteristic
2#17748C211yesmaximal, characteristic
3#17739C211yesmaximal, characteristic
4#17479C211yesmaximal, characteristic
5#7996C211yesmaximal, characteristic
6#7936C211yesmaximal, characteristic
7#7863C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#431C2211yescharacteristic
15#1573—22no—
16#1572—22no—
17#1569—22no—
18#1561—22no—
19#1558—21no—
20#1556—21no—
21#1556—21no—
22#1556—21no—
23#1554—22no—
24#1550—22no—
25#1544—22no—
26#1497—22no—
27#1493C414yes—
28#1479—22no—
29#1291—22no—
30#1283—22no—
31#1275—22no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533C2×C412yes—
39#533Q812yes—
40C2×C4⋊Q8D411yescharacteristic
41C2×C4⋊Q8D411yescharacteristic
42C2×#125—21no—
43C2×C22⋊Q8—21no—
44C2×C22⋊Q8—21no—
45C2×C4×Q8—21no—
46C2×C4×Q8—21no—
47C2×#117—21no—
48C22×C42C2311yescharacteristic
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#432—21no—
59#432—21no—
60#432D412yes—
61#431—21no—
62#431—21no—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431—21no—
70#431D412yes—
71#431D412yes—
72#431D412yes—
73#423—21no—
74#423—21no—
75#423C2×C412yes—
76#423D412yes—
77C2×C4⋊C8—21no—
78C2×C4⋊C8D412yes—
79#407—21no—
80#407—21no—
81#407—21no—
82#407—21no—
83C2×C22⋊C8—21no—
84C2×C22⋊C8—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16—21no—
88C4×M16—21no—
89C2×#97—21no—
90C2×C4×C8C2×C412yes—
91#335—24no—
92#334—24no—
93#333—24no—
94#332—24no—
95#332—24no—
96#332—24no—
97Q8⋊C8—28no—
98C22×Q8—21no—
99C22×Q8—21no—
100C23×C4#4111yescharacteristic
101C23×C4#4111yescharacteristic
102C23×C4#4111yescharacteristic
103C2×M16—22no—
104C2×M16—22no—
105C2×M16—41no—
106C2×M16—41no—
107C22×C8—22no—
108C22×C8—22no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8—21no—
114C4⋊Q8—21no—
115C4⋊Q8C4⋊C412yes—
116C4⋊Q8C4⋊C412yes—
117#125—22no—
118#125—22no—
119#125—41no—
120#125—41no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C22⋊Q8—22no—
124C22⋊Q8—22no—
125C22⋊Q8—41no—
126C22⋊Q8—41no—
127C22⋊Q8—41no—
128C22⋊Q8—41no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—41no—
134C4×Q8—41no—
135C4×Q8—41no—
136C4×Q8—41no—
137#117—22no—
138#117—22no—
139#117—41no—
140#117—41no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—21no—
144C2×C4⋊C4—21no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4—41no—
147C2×C22⋊C4—41no—
148C2×C42—41no—
149C2×C42C22×C411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×Q811yescharacteristic
158C2×C42#4111yescharacteristic
159C2×C42#4111yescharacteristic
160C2×C42#4111yescharacteristic
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—41no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C4⋊C8—41no—
172C4⋊C8C22⋊C414yes—
173C22⋊C8—22no—
174C22⋊C8—22no—
175C22⋊C8—22no—
176C22⋊C8—22no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181C22⋊C8—41no—
182C22⋊C8—41no—
183C22⋊C8—41no—
184C22⋊C8—41no—
185#97—22no—
186#97—41no—
187#97—41no—
188#97—41no—
189#97—41no—
190#97—41no—
191C4×C8—22no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—41no—
197C24#12611yescharacteristic
198C2×Q8—22no—
199C2×Q8—22no—
200C2×Q8—44no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—82no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4—81no—
209C22×C4#11711yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4C4×D411yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4#12111yescharacteristic
217C22×C4C22⋊Q811yescharacteristic
218C22×C4C22⋊Q811yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4#12311yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12311yescharacteristic
224C22×C4#12411yescharacteristic
225C22×C4#12411yescharacteristic
226C22×C4#12611yescharacteristic
227M16—44no—
228M16—82no—
229M16—82no—
230M16—82no—
231C2×C8—24no—
232C2×C8—44no—
233C2×C8—42no—
234C2×C8—42no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—82no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C22⋊C4—42no—
252C22⋊C4—42no—
253C42—42no—
254C42—42no—
255C42C2×C22⋊C411yescharacteristic
256C42C2×C4⋊C411yescharacteristic
257C42C4×D411yescharacteristic
258C42C4×D411yescharacteristic
259C42#12011yescharacteristic
260C42#12011yescharacteristic
261C42#12011yescharacteristic
262C42#12111yescharacteristic
263C42#12111yescharacteristic
264C42#12111yescharacteristic
265C42C22⋊Q811yescharacteristic
266C42C22⋊Q811yescharacteristic
267C42C22⋊Q811yescharacteristic
268C42#12311yescharacteristic
269C42#12411yescharacteristic
270C42#12711yescharacteristic
271C23—41no—
272C23—41no—
273C23—41no—
274C23#38811yescharacteristic
275C23#39611yescharacteristic
276C23#39711yescharacteristic
277Q8—44no—
278Q8—82no—
279Q8—82no—
280Q8—82no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—42no—
300C2×C4—42no—
301C2×C4—82no—
302C2×C4—82no—
303C2×C4#38011yescharacteristic
304C2×C4#38511yescharacteristic
305C2×C4#38511yescharacteristic
306C2×C4#38611yescharacteristic
307C2×C4#38811yescharacteristic
308C2×C4#39011yescharacteristic
309C2×C4#39211yescharacteristic
310C2×C4#39311yescharacteristic
311C2×C4#39411yescharacteristic
312C2×C4#39411yescharacteristic
313C2×C4#39611yescharacteristic
314C2×C4#39911yescharacteristic
315C8—44no—
316C8—82no—
317C8—82no—
318C8—82no—
319C8—82no—
320C8—82no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—82no—
331C22#143911yescharacteristic
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—82no—
343C4—82no—
344C4#190212yes—
345C4#201712yes—
346C2—41no—
347C2—41no—
348C2—41no—
349C2#1105911yescharacteristic
350C2#1111111yescharacteristic
351C2#1111611yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24#126
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C22 ◁ C22×C42 ◁ G
LECSG ▷ C22×C42 ▷ C22 ▷ C1

Maximal subgroups

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

subgroupindex
#178212
#177482
#177392
#174792
#79962
#79362
#78632