Finite Group Databasebeta

← #117216

Subgroups — #117216

Order 512.

646 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#21970C211yesmaximal, characteristic
2C2×#1966C211yesmaximal, characteristic
3#17742C211yesmaximal, characteristic
4#17742C211yesmaximal, characteristic
5#8064C211yesmaximal, characteristic
6#8061C211yesmaximal, characteristic
7#7992C211yesmaximal, characteristic
8C22×C4⋊Q8C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#423C2211yescharacteristic
14C22×#101—22no—
15C2×C4×M16C2211yescharacteristic
16#1966—22no—
17#1966C414yes—
18#1950—22no—
19#1950—22no—
20#1811—22no—
21#1556—22no—
22#1556—22no—
23#1549—22no—
24#1549—22no—
25#1546—22no—
26#1546—22no—
27#1299—22no—
28#1299—22no—
29#1291—22no—
30C23×Q8—21no—
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⋊C4Q812yes—
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—
75C2×#101—28no—
76C2×#101—48no—
77#407—21no—
78#407—21no—
79#407—21no—
80#407—21no—
81C4×M16—21no—
82C4×M16—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16—21no—
87C4×M16—21no—
88C4×M16C2×C412yes—
89C2×#97—21no—
90C2×C4×C8—21no—
91#341—24no—
92#332—28no—
93#332—28no—
94C22×Q8—22no—
95C22×Q8—22no—
96C22×Q8—44no—
97C22×Q8—44no—
98C22×Q8—41no—
99C22×Q8—41no—
100C22×Q8—41no—
101C22×Q8—41no—
102C23×C4—22no—
103C23×C4C2×D411yescharacteristic
104C23×C4C2×Q811yescharacteristic
105C23×C4#4111yescharacteristic
106C2×M16—22no—
107C2×M16—24no—
108C2×M16—22no—
109C2×M16—22no—
110C2×M16—22no—
111C2×M16—22no—
112C2×M16—42no—
113C2×M16—41no—
114C2×M16—41no—
115C2×M16—44no—
116C2×M16C22⋊C414yes—
117C22×C8—22no—
118C22×C8—41no—
119C4⋊Q8—22no—
120C4⋊Q8—22no—
121C4⋊Q8—22no—
122C4⋊Q8—22no—
123C4⋊Q8—22no—
124C4⋊Q8—22no—
125C4⋊Q8—22no—
126C4⋊Q8—22no—
127C4⋊Q8—41no—
128C4⋊Q8—41no—
129C4⋊Q8—41no—
130C4⋊Q8—41no—
131C4⋊Q8—41no—
132C4⋊Q8—41no—
133C4⋊Q8—41no—
134C4⋊Q8—41no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—22no—
137C2×C4⋊C4—22no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—22no—
140C2×C4⋊C4—22no—
141C2×C4⋊C4—22no—
142C2×C4⋊C4—41no—
143C2×C4⋊C4—41no—
144C2×C4⋊C4—41no—
145C2×C4⋊C4—41no—
146C2×C4⋊C4C4⋊C414yes—
147C2×C42C22×C411yescharacteristic
148C2×C42C2×D411yescharacteristic
149C2×C42C2×D411yescharacteristic
150C2×C42C2×D411yescharacteristic
151C2×C42C2×D411yescharacteristic
152C2×C42C2×D411yescharacteristic
153C2×C42C2×D411yescharacteristic
154C2×C42#4111yescharacteristic
155C2×C42#4111yescharacteristic
156C2×C42#4111yescharacteristic
157C2×C42#4111yescharacteristic
158C2×C42#4111yescharacteristic
159#108—84no—
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—22no—
170C4⋊C8—22no—
171C4⋊C8—41no—
172#101—416no—
173C22⋊C8—22no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176C22⋊C8—41no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C22⋊C8—41no—
181#97—22no—
182#97—22no—
183#97—41no—
184#97—41no—
185#97—41no—
186#97—41no—
187C4×C8—22no—
188C4×C8—41no—
189C4×C8—41no—
190C4×C8—41no—
191C4×C8—41no—
192C4×C8—41no—
193C24C22⋊Q811yescharacteristic
194C2×Q8—44no—
195C2×Q8—44no—
196C2×Q8—48no—
197C2×Q8—42no—
198C2×Q8—42no—
199C2×Q8—42no—
200C2×Q8—42no—
201C2×Q8—84no—
202C2×Q8—82no—
203C2×Q8—82no—
204C2×Q8—84no—
205C2×Q8—82no—
206C2×Q8—82no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—24no—
210C22×C4—42no—
211C22×C4—42no—
212C22×C4—48no—
213C22×C4—42no—
214C22×C4—42no—
215C22×C4C2×C22⋊C411yescharacteristic
216C22×C4C2×C4⋊C411yescharacteristic
217C22×C4#11711yescharacteristic
218C22×C4#12011yescharacteristic
219C22×C4#12111yescharacteristic
220C22×C4#12111yescharacteristic
221C22×C4C22⋊Q811yescharacteristic
222C22×C4C22⋊Q811yescharacteristic
223C22×C4C22⋊Q811yescharacteristic
224C22×C4C22⋊Q811yescharacteristic
225C22×C4C22⋊Q811yescharacteristic
226C22×C4#12311yescharacteristic
227C22×C4#12311yescharacteristic
228C22×C4#12311yescharacteristic
229C22×C4#12411yescharacteristic
230C22×C4#12411yescharacteristic
231C22×C4#12411yescharacteristic
232C22×C4#12711yescharacteristic
233M16—24no—
234M16—42no—
235M16—44no—
236M16—44no—
237M16—42no—
238M16—42no—
239M16—42no—
240M16—42no—
241M16—82no—
242M16—82no—
243M16—82no—
244C2×C8—24no—
245C2×C8—44no—
246C2×C8—42no—
247C2×C8—44no—
248C2×C8—42no—
249C2×C8—42no—
250C2×C8—42no—
251C2×C8—42no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—42no—
255C4⋊C4—42no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C42C4×D411yescharacteristic
261C42C4×D411yescharacteristic
262C42C4×D411yescharacteristic
263C42C4×D411yescharacteristic
264C42#12011yescharacteristic
265C42#12011yescharacteristic
266C42#12111yescharacteristic
267C42#12111yescharacteristic
268C42#12111yescharacteristic
269C42#12111yescharacteristic
270C42#12111yescharacteristic
271C42#12111yescharacteristic
272C42#12311yescharacteristic
273C42#12311yescharacteristic
274C42#12611yescharacteristic
275C42#12611yescharacteristic
276C23—22no—
277C23—41no—
278C23—41no—
279C23#38011yescharacteristic
280C23#39311yescharacteristic
281C23#39911yescharacteristic
282Q8—84no—
283Q8—84no—
284Q8—82no—
285Q8—82no—
286Q8—84no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—22no—
292C2×C4—22no—
293C2×C4—22no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—48no—
307C2×C4—44no—
308C2×C4—84no—
309C2×C4—84no—
310C2×C4#38511yescharacteristic
311C2×C4#38511yescharacteristic
312C2×C4#38611yescharacteristic
313C2×C4#38811yescharacteristic
314C2×C4#38811yescharacteristic
315C2×C4#39011yescharacteristic
316C2×C4#39211yescharacteristic
317C2×C4#39411yescharacteristic
318C2×C4#39411yescharacteristic
319C2×C4#39611yescharacteristic
320C2×C4#39611yescharacteristic
321C2×C4#39711yescharacteristic
322C2×C4#45312yes—
323C2×C4#45412yes—
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—44no—
339C22#143911yescharacteristic
340C22#202712yes—
341C4—21no—
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—21no—
347C4—21no—
348C4—21no—
349C4—84no—
350C4#188711yescharacteristic
351C4#188811yescharacteristic
352C4#200612yes—
353C4#204912yes—
354C4#207012yes—
355C2—22no—
356C2—41no—
357C2—41no—
358C2#1105411yescharacteristic
359C2#1114011yescharacteristic
360C2#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×#19662
#177422
#80642
#80612
#79922