Finite Group Databasebeta

← #117218

Subgroups — #117218

Order 512.

602 conjugacy classes of subgroups.

The subgroup lattice has 355 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21970C211yesmaximal, characteristic
2C2×#1997C211yesmaximal, characteristic
3#17746C211yesmaximal, characteristic
4#17746C211yesmaximal, characteristic
5#8064C211yesmaximal, characteristic
6#8061C211yesmaximal, characteristic
7#7992C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C2×C4×M16C2211yescharacteristic
15#1997—22no—
16#1997C414yes—
17#1950—22no—
18#1950—22no—
19#1811—22no—
20C2×#343—22no—
21#1558—22no—
22#1558—22no—
23#1548—22no—
24#1548—22no—
25#1548—22no—
26#1548—22no—
27#1299—22no—
28#1299—22no—
29#1291—22no—
30C23×Q8Q812yes—
31C22×M16—21no—
32C22×M16D412yes—
33C2×C4⋊Q8—21no—
34C2×C4⋊Q8—21no—
35C2×C4⋊Q8—21no—
36C2×C4⋊Q8—21no—
37C2×C4⋊Q8—21no—
38C2×C4⋊Q8—21no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8—21no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8—21no—
43C2×C4⋊Q8C2×C412yes—
44C2×C4⋊Q8D412yes—
45C22×C4⋊C4—21no—
46C22×C4⋊C4—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#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—
64C2×#108—24no—
65#423—21no—
66#423—21no—
67#423—21no—
68#423C2×C412yes—
69#423D412yes—
70#423D412yes—
71#423D412yes—
72#423D412yes—
73C2×C4⋊C8—21no—
74C2×C4⋊C8—21no—
75#407—21no—
76#407—21no—
77#407—21no—
78#407—21no—
79C4×M16—21no—
80C4×M16—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16C2×C412yes—
87C2×#97—21no—
88C2×C4×C8—21no—
89#343—28no—
90#341—24no—
91#333—28no—
92#333—28no—
93C22×Q8—22no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—22no—
97C22×Q8—22no—
98C22×Q8—84no—
99C22×Q8C4⋊C414yes—
100C23×C4—22no—
101C23×C4C2×D411yescharacteristic
102C23×C4C2×Q811yescharacteristic
103C23×C4#4111yescharacteristic
104C2×M16—22no—
105C2×M16—24no—
106C2×M16—22no—
107C2×M16—22no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—42no—
111C2×M16—41no—
112C2×M16—41no—
113C2×M16—44no—
114C2×M16C22⋊C414yes—
115C22×C8—22no—
116C22×C8—41no—
117C4⋊Q8—22no—
118C4⋊Q8—22no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—41no—
126C4⋊Q8—41no—
127C4⋊Q8—41no—
128C4⋊Q8—41no—
129C4⋊Q8—41no—
130C4⋊Q8—41no—
131C4⋊Q8—41no—
132C4⋊Q8—41no—
133C2×C4⋊C4—22no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—41no—
138C2×C4⋊C4—41no—
139C2×C4⋊C4—41no—
140C2×C4⋊C4—41no—
141C2×C4⋊C4—41no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C42C22×C411yescharacteristic
146C2×C42C2×D411yescharacteristic
147C2×C42C2×D411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42#4111yescharacteristic
153C2×C42#4111yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42#4111yescharacteristic
157#108—84no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—22no—
164C4⋊C8—22no—
165C4⋊C8—22no—
166C4⋊C8—22no—
167C4⋊C8—22no—
168C4⋊C8—22no—
169C4⋊C8—41no—
170C22⋊C8—22no—
171C22⋊C8—41no—
172C22⋊C8—41no—
173C22⋊C8—41no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176C22⋊C8—41no—
177C22⋊C8—41no—
178#97—22no—
179#97—22no—
180#97—41no—
181#97—41no—
182#97—41no—
183#97—41no—
184C4×C8—22no—
185C4×C8—41no—
186C4×C8—41no—
187C4×C8—41no—
188C4×C8—41no—
189C4×C8—41no—
190C24C22⋊Q811yescharacteristic
191C2×Q8—24no—
192C2×Q8—24no—
193C2×Q8—42no—
194C2×Q8—42no—
195C2×Q8—84no—
196C2×Q8—84no—
197C2×Q8—84no—
198C2×Q8—84no—
199C2×Q8—84no—
200C2×Q8—84no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—24no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—42no—
207C22×C4—42no—
208C22×C4—84no—
209C22×C4C2×C22⋊C411yescharacteristic
210C22×C4C2×C4⋊C411yescharacteristic
211C22×C4#11711yescharacteristic
212C22×C4#12011yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4C22⋊Q811yescharacteristic
216C22×C4C22⋊Q811yescharacteristic
217C22×C4C22⋊Q811yescharacteristic
218C22×C4C22⋊Q811yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4#12311yescharacteristic
221C22×C4#12311yescharacteristic
222C22×C4#12311yescharacteristic
223C22×C4#12411yescharacteristic
224C22×C4#12411yescharacteristic
225C22×C4#12411yescharacteristic
226C22×C4#12711yescharacteristic
227M16—24no—
228M16—42no—
229M16—44no—
230M16—44no—
231M16—42no—
232M16—42no—
233M16—42no—
234M16—42no—
235M16—82no—
236M16—82no—
237M16—82no—
238C2×C8—24no—
239C2×C8—44no—
240C2×C8—42no—
241C2×C8—44no—
242C2×C8—42no—
243C2×C8—42no—
244C2×C8—42no—
245C2×C8—42no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—42no—
252C4⋊C4—42no—
253C4⋊C4—42no—
254C42C4×D411yescharacteristic
255C42C4×D411yescharacteristic
256C42C4×D411yescharacteristic
257C42C4×D411yescharacteristic
258C42#12011yescharacteristic
259C42#12011yescharacteristic
260C42#12111yescharacteristic
261C42#12111yescharacteristic
262C42#12111yescharacteristic
263C42#12111yescharacteristic
264C42#12111yescharacteristic
265C42#12111yescharacteristic
266C42#12311yescharacteristic
267C42#12311yescharacteristic
268C42#12611yescharacteristic
269C42#12611yescharacteristic
270C23—22no—
271C23—41no—
272C23—41no—
273C23#38011yescharacteristic
274C23#39311yescharacteristic
275C23#39911yescharacteristic
276Q8—84no—
277Q8—84no—
278Q8—84no—
279Q8—84no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—22no—
286C2×C4—22no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—44no—
300C2×C4—84no—
301C2×C4—84no—
302C2×C4—84no—
303C2×C4#38511yescharacteristic
304C2×C4#38511yescharacteristic
305C2×C4#38611yescharacteristic
306C2×C4#38811yescharacteristic
307C2×C4#38811yescharacteristic
308C2×C4#39011yescharacteristic
309C2×C4#39211yescharacteristic
310C2×C4#39411yescharacteristic
311C2×C4#39411yescharacteristic
312C2×C4#39611yescharacteristic
313C2×C4#39611yescharacteristic
314C2×C4#39711yescharacteristic
315C2×C4#45512yes—
316C2×C4#45612yes—
317C8—44no—
318C8—44no—
319C8—82no—
320C8—82no—
321C8—82no—
322C8—82no—
323C22—21no—
324C22—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—21no—
329C22—21no—
330C22—21no—
331C22—44no—
332C22#143911yescharacteristic
333C22#202812yes—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—84no—
343C4#188511yescharacteristic
344C4#188611yescharacteristic
345C4#200712yes—
346C4#204912yes—
347C4#206912yes—
348C2—22no—
349C2—41no—
350C2—41no—
351C2#1105511yescharacteristic
352C2#1113911yescharacteristic
353C2#1114311yescharacteristic
subgroupG/H
center, socleC22#1439
commutatorC2×C42C22×C4
Frattini, cosocleC22×C42C23
prime radicalC24C22⋊Q8
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
#219702
C2×#19972
#177462
#80642
#80612
#79922