Finite Group Databasebeta

← #116171

Subgroups — #116171

Order 512.

563 conjugacy classes of subgroups.

The subgroup lattice has 359 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21353C211yesmaximal, characteristic
2C2×#1531C211yesmaximal, characteristic
3#17584C211yesmaximal, characteristic
4#17479C211yesmaximal, characteristic
5#8088C211yesmaximal, characteristic
6#7978C211yesmaximal, characteristic
7#7864C211yesmaximal, characteristic
8C2×#533C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11C2×#432C2211yescharacteristic
12C2×#431C2211yescharacteristic
13C22×#97C2211yescharacteristic
14#2283C2211yescharacteristic
15#1940—22no—
16#1850—22no—
17#1768—22no—
18#1531—22no—
19#1531—22no—
20C2×#330—22no—
21#1518C414yes—
22#1515—22no—
23#1508—22no—
24#1497—22no—
25#1493—22no—
26#1479—22no—
27#1302—22no—
28#1288—22no—
29#1275—22no—
30C23×C8—21no—
31#533—21no—
32#533—21no—
33#533—21no—
34#533—21no—
35#533—21no—
36#533—21no—
37#533C2×C412yes—
38#533D412yes—
39C2×C4⋊Q8Q811yescharacteristic
40C2×C4⋊Q8Q811yescharacteristic
41C2×#125—21no—
42C2×C22⋊Q8—21no—
43C2×C22⋊Q8—21no—
44C2×C4×Q8—21no—
45C2×C4×Q8—21no—
46C2×#117—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#431—21no—
57#431—21no—
58#431—21no—
59#431—21no—
60#431—21no—
61#431—21no—
62#431C2×C412yes—
63#431D412yes—
64#423—21no—
65#423—21no—
66C2×C4⋊C8—24no—
67C2×C4⋊C8—21no—
68C2×C4⋊C8—21no—
69#407—21no—
70#407—21no—
71C2×C22⋊C8—21no—
72C2×C22⋊C8—21no—
73C4×M16—21no—
74C4×M16—21no—
75C2×#97—21no—
76C2×#97—21no—
77C2×#97—21no—
78C2×#97—21no—
79C2×#97—21no—
80C2×#97D412yes—
81C2×#97D412yes—
82C2×C4×C8D412yes—
83#383D412yes—
84#382C2×C412yes—
85#381—21no—
86#380—21no—
87C4×C4⋊C4—21no—
88C4×C22⋊C4D412yes—
89#376—21no—
90#336—24no—
91#330—28no—
92#328—28no—
93Q8⋊C8—28no—
94C22×Q8—21no—
95C22×Q8—21no—
96C23×C4C2×D411yescharacteristic
97C23×C4#4111yescharacteristic
98C23×C4#4111yescharacteristic
99C2×M16—41no—
100C2×M16—41no—
101C22×C8—22no—
102C22×C8—22no—
103C22×C8—22no—
104C22×C8—44no—
105C22×C8—41no—
106C22×C8—41no—
107C4⋊Q8—21no—
108C4⋊Q8—21no—
109C4⋊Q8—21no—
110C4⋊Q8—21no—
111C4⋊Q8—21no—
112C4⋊Q8—21no—
113C4⋊Q8C4⋊C412yes—
114C4⋊Q8C4⋊C412yes—
115#125—22no—
116#125—22no—
117#125—41no—
118#125—41no—
119C22⋊Q8—22no—
120C22⋊Q8—22no—
121C22⋊Q8—22no—
122C22⋊Q8—22no—
123C22⋊Q8—41no—
124C22⋊Q8—41no—
125C22⋊Q8—41no—
126C22⋊Q8—41no—
127C4×Q8—22no—
128C4×Q8—22no—
129C4×Q8—22no—
130C4×Q8—22no—
131C4×Q8—41no—
132C4×Q8—41no—
133C4×Q8—41no—
134C4×Q8—41no—
135#117—22no—
136#117—22no—
137#117—41no—
138#117—41no—
139C2×C4⋊C4—21no—
140C2×C4⋊C4—21no—
141C2×C4⋊C4—21no—
142C2×C4⋊C4—21no—
143C2×C4⋊C4—22no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C22⋊C4—22no—
147C2×C22⋊C4—41no—
148C2×C42—22no—
149C2×C42—41no—
150C2×C42C22×C411yescharacteristic
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
161C2×C42#4111yescharacteristic
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—48no—
165C4⋊C8—41no—
166C4⋊C8—41no—
167C4⋊C8—41no—
168C4⋊C8—41no—
169C4⋊C8—41no—
170C4⋊C8—41no—
171C22⋊C8—22no—
172C22⋊C8—22no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176C22⋊C8—41no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179#97—22no—
180#97—22no—
181#97—22no—
182#97—22no—
183#97—41no—
184#97—41no—
185#97—41no—
186#97—41no—
187C4×C8—22no—
188C4×C8—22no—
189C4×C8—22no—
190C4×C8—22no—
191#95—22no—
192#95—22no—
193#95—22no—
194#95C22⋊C414yes—
195C24#12311yescharacteristic
196C2×Q8—22no—
197C2×Q8—22no—
198C2×Q8—44no—
199C2×Q8—41no—
200C2×Q8—41no—
201C2×Q8—82no—
202C22×C4—24no—
203C22×C4—42no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—41no—
207C22×C4—41no—
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C4×D411yescharacteristic
210C22×C4C4×D411yescharacteristic
211C22×C4#12011yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4C22⋊Q811yescharacteristic
216C22×C4C22⋊Q811yescharacteristic
217C22×C4C22⋊Q811yescharacteristic
218C22×C4#12311yescharacteristic
219C22×C4#12311yescharacteristic
220C22×C4#12311yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12411yescharacteristic
223C22×C4#12611yescharacteristic
224C22×C4#12611yescharacteristic
225C22×C4#12711yescharacteristic
226M16—82no—
227M16—82no—
228C2×C8—24no—
229C2×C8—42no—
230C2×C8—44no—
231C2×C8—44no—
232C2×C8—42no—
233C2×C8—44no—
234C2×C8—44no—
235C2×C8—42no—
236C2×C8—42no—
237C2×C8—42no—
238C2×C8—42no—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—41no—
247C4⋊C4—42no—
248C4⋊C4—41no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C4⋊C4—82no—
252C22⋊C4—44no—
253C22⋊C4—42no—
254C22⋊C4—42no—
255C42—44no—
256C42—42no—
257C42—42no—
258C42C2×C4⋊C411yescharacteristic
259C42#11711yescharacteristic
260C42C4×D411yescharacteristic
261C42C4×D411yescharacteristic
262C42#12011yescharacteristic
263C42#12011yescharacteristic
264C42#12111yescharacteristic
265C42#12111yescharacteristic
266C42#12111yescharacteristic
267C42#12111yescharacteristic
268C42#12111yescharacteristic
269C42C22⋊Q811yescharacteristic
270C42C22⋊Q811yescharacteristic
271C42C22⋊Q811yescharacteristic
272C42#12411yescharacteristic
273C42#12411yescharacteristic
274C23—22no—
275C23—22no—
276C23—41no—
277C23#38611yescharacteristic
278C23#39711yescharacteristic
279C23#39911yescharacteristic
280Q8—44no—
281Q8—82no—
282Q8—82no—
283Q8—82no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
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—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—44no—
303C2×C4—44no—
304C2×C4—44no—
305C2×C4—42no—
306C2×C4—42no—
307C2×C4—82no—
308C2×C4—82no—
309C2×C4#38011yescharacteristic
310C2×C4#38511yescharacteristic
311C2×C4#38511yescharacteristic
312C2×C4#38811yescharacteristic
313C2×C4#38811yescharacteristic
314C2×C4#39011yescharacteristic
315C2×C4#39211yescharacteristic
316C2×C4#39311yescharacteristic
317C2×C4#39411yescharacteristic
318C2×C4#39411yescharacteristic
319C2×C4#39611yescharacteristic
320C2×C4#39611yescharacteristic
321C8—44no—
322C8—82no—
323C8—82no—
324C8—82no—
325C8—82no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22—21no—
333C22—44no—
334C22#143911yescharacteristic
335C22#190212yes—
336C22#209812yes—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—84no—
349C4—82no—
350C4—82no—
351C4#203112yes—
352C2—22no—
353C2—22no—
354C2—41no—
355C2#1105911yescharacteristic
356C2#1108411yescharacteristic
357C2#1109111yescharacteristic
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

Maximal subgroups

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

subgroupindex
#213532
C2×#15312
#175842
#174792
#80882
#79782
#78642