Finite Group Databasebeta

← #107209

Subgroups — #107209

Order 512.

678 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#60249C211yesmaximal, characteristic
2#17404C211yesmaximal, characteristic
3#17385C211yesmaximal, characteristic
4#8124C211yesmaximal, characteristic
5#8122C211yesmaximal, characteristic
6#7936C211yesmaximal, characteristic
7#7921C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2987—22no—
10#2986—22no—
11#2985—22no—
12#2985—22no—
13Q8×M16—22no—
14Q8×M16—22no—
15#2921C414yes—
16#2920C2211yescharacteristic
17#2920C2211yescharacteristic
18#2920C2211yescharacteristic
19#2920C2211yescharacteristic
20C2×#432C2211yescharacteristic
21C2×#431C2211yescharacteristic
22#1472C414yes—
23#1472C414yes—
24#1470—22no—
25#1470—22no—
26#1467—22no—
27#1461—22no—
28#1308—22no—
29#1307—22no—
30#1283—22no—
31#1281—22no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533—21no—
38#533—21no—
39#533—21no—
40C2×C4⋊Q8Q811yescharacteristic
41C2×C4⋊Q8Q811yescharacteristic
42C2×#125D412yes—
43C2×C22⋊Q8C2×C412yes—
44C2×C22⋊Q8C2×C412yes—
45C2×C4×Q8D412yes—
46C2×C4×Q8D412yes—
47C2×#117C2×C412yes—
48C22×C42C2311yescharacteristic
49#446—24no—
50#446—24no—
51#446—28no—
52#446—28no—
53C8×Q8—24no—
54C8×Q8—24no—
55#433—28no—
56#433—28no—
57#432—21no—
58#432—21no—
59#432—21no—
60#432—21no—
61#432—21no—
62#432—21no—
63#432—21no—
64#432—21no—
65#432—21no—
66#432—21no—
67#432D412yes—
68#432D412yes—
69#431—21no—
70#431—21no—
71#431—21no—
72#431—21no—
73#431—21no—
74#431—21no—
75#431—21no—
76#431—21no—
77#431—21no—
78#431—21no—
79#431D412yes—
80#431D412yes—
81#423—24no—
82#423—21no—
83#423—21no—
84#423—21no—
85#423—21no—
86C2×C4⋊C8—21no—
87C2×C4⋊C8—21no—
88#407C2×C412yes—
89#407C2×C412yes—
90#407C2×C412yes—
91#407C2×C412yes—
92C2×C22⋊C8C2×C412yes—
93C2×C22⋊C8C2×C412yes—
94C4×M16—24no—
95C4×M16—21no—
96C4×M16—21no—
97C4×M16—21no—
98C4×M16—21no—
99C2×#97—21no—
100C2×C4×C8—21no—
101#324—28no—
102#324—28no—
103C22×Q8C4⋊C412yes—
104C22×Q8C4⋊C412yes—
105C23×C4C22×C411yescharacteristic
106C23×C4C22×C411yescharacteristic
107C23×C4C22×C411yescharacteristic
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—22no—
112C22×C8—22no—
113C22×C8—22no—
114C4⋊Q8—21no—
115C4⋊Q8—21no—
116C4⋊Q8—21no—
117C4⋊Q8—21no—
118C4⋊Q8—21no—
119C4⋊Q8—21no—
120C4⋊Q8—21no—
121C4⋊Q8—21no—
122#125—22no—
123#125—22no—
124#125—22no—
125#125—22no—
126C22⋊Q8—22no—
127C22⋊Q8—22no—
128C22⋊Q8—22no—
129C22⋊Q8—22no—
130C22⋊Q8—22no—
131C22⋊Q8—22no—
132C22⋊Q8—22no—
133C22⋊Q8—22no—
134C4×Q8—22no—
135C4×Q8—22no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142#117—22no—
143#117—22no—
144#117—22no—
145#117—22no—
146C2×C4⋊C4C22⋊C414yes—
147C2×C4⋊C4C22⋊C414yes—
148C2×C4⋊C4C4⋊C412yes—
149C2×C4⋊C4C4⋊C412yes—
150C2×C4⋊C4C4⋊C412yes—
151C2×C4⋊C4C4⋊C412yes—
152C2×C22⋊C4C4214yes—
153C2×C42C22⋊C414yes—
154C2×C42C2×D411yescharacteristic
155C2×C42C2×D411yescharacteristic
156C2×C42C2×D411yescharacteristic
157C2×C42C2×D411yescharacteristic
158C2×C42C2×D411yescharacteristic
159C2×C42C2×D411yescharacteristic
160C2×C42C2×D411yescharacteristic
161C2×C42C2×Q811yescharacteristic
162C2×C42#4111yescharacteristic
163C2×C42#4111yescharacteristic
164C2×C42#4111yescharacteristic
165C2×C42#4111yescharacteristic
166C4⋊C8—28no—
167C4⋊C8—28no—
168C4⋊C8—22no—
169C4⋊C8—22no—
170C4⋊C8—22no—
171C4⋊C8—22no—
172C4⋊C8—41no—
173C4⋊C8—41no—
174C4⋊C8—41no—
175C4⋊C8—41no—
176C4⋊C8—41no—
177C4⋊C8—41no—
178C4⋊C8—41no—
179C4⋊C8—41no—
180C22⋊C8—22no—
181C22⋊C8—22no—
182C22⋊C8—22no—
183C22⋊C8—22no—
184C22⋊C8—22no—
185C22⋊C8—22no—
186C22⋊C8—22no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8C22⋊C414yes—
191C22⋊C8C22⋊C414yes—
192#97—28no—
193#97—22no—
194#97—22no—
195#97—41no—
196#97—41no—
197#97—41no—
198#97—41no—
199C4×C8—28no—
200C4×C8—22no—
201C4×C8—22no—
202C4×C8—41no—
203C4×C8—41no—
204C4×C8—41no—
205C4×C8—41no—
206C24C2×C4211yescharacteristic
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—44no—
212C2×Q8—82no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4—81no—
216C22×C4#9514yes—
217C22×C4#9514yes—
218C22×C4C2×C22⋊C411yescharacteristic
219C22×C4C2×C22⋊C411yescharacteristic
220C22×C4C2×C22⋊C411yescharacteristic
221C22×C4C2×C22⋊C411yescharacteristic
222C22×C4C2×C22⋊C411yescharacteristic
223C22×C4C2×C4⋊C411yescharacteristic
224C22×C4C2×C4⋊C411yescharacteristic
225C22×C4C2×C4⋊C411yescharacteristic
226C22×C4#11711yescharacteristic
227C22×C4#11711yescharacteristic
228C22×C4C4×D411yescharacteristic
229C22×C4C4×D411yescharacteristic
230C22×C4C4×D411yescharacteristic
231C22×C4C4×D411yescharacteristic
232C22×C4C4×D411yescharacteristic
233C22×C4C4×D411yescharacteristic
234C22×C4C4×D411yescharacteristic
235C22×C4C4×D411yescharacteristic
236M16—44no—
237M16—44no—
238M16—44no—
239M16—82no—
240C2×C8—24no—
241C2×C8—24no—
242C2×C8—44no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—82no—
248C4⋊C4—22no—
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—22no—
252C4⋊C4—22no—
253C4⋊C4—22no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C22⋊C4—24no—
261C22⋊C4—24no—
262C42—24no—
263C42—24no—
264C42#12011yescharacteristic
265C42#12011yescharacteristic
266C42#12011yescharacteristic
267C42#12011yescharacteristic
268C42#12111yescharacteristic
269C42#12111yescharacteristic
270C42#12111yescharacteristic
271C42#12111yescharacteristic
272C42C22⋊Q811yescharacteristic
273C42C22⋊Q811yescharacteristic
274C42C22⋊Q811yescharacteristic
275C42C22⋊Q811yescharacteristic
276C42#12311yescharacteristic
277C42#12311yescharacteristic
278C42#12311yescharacteristic
279C42#12311yescharacteristic
280C23—41no—
281C23—41no—
282C23—41no—
283C23C2×#9511yescharacteristic
284C23C4×C22⋊C411yescharacteristic
285C23C4×C22⋊C411yescharacteristic
286Q8—44no—
287Q8—44no—
288Q8—82no—
289Q8—82no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—22no—
295C2×C4—22no—
296C2×C4—24no—
297C2×C4—24no—
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—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—44no—
311C2×C4—44no—
312C2×C4#37911yescharacteristic
313C2×C4#38011yescharacteristic
314C2×C4#38011yescharacteristic
315C2×C4#38111yescharacteristic
316C2×C4#38511yescharacteristic
317C2×C4#38511yescharacteristic
318C2×C4#38511yescharacteristic
319C2×C4#38511yescharacteristic
320C2×C4#38611yescharacteristic
321C2×C4#38611yescharacteristic
322C2×C4#38611yescharacteristic
323C2×C4#38611yescharacteristic
324C8—44no—
325C8—44no—
326C8—82no—
327C8—82no—
328C8—82no—
329C8—82no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—21no—
338C22—21no—
339C22—82no—
340C22C22⋊#9511yescharacteristic
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—21no—
350C4—21no—
351C4—44no—
352C4—44no—
353C4#183012yes—
354C4#186812yes—
355C2—41no—
356C2—41no—
357C2—41no—
358C2#1074811yescharacteristic
359C2#1075711yescharacteristic
360C2#1077011yescharacteristic
subgroupG/H
center, socleC22C22⋊#95
commutator, prime radicalC24C2×C42
Frattini, cosocleC22×C42C23
LCSG ▷ C24 ▷ 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
#602492
#174042
#173852
#81242
#81222
#79362
#79212