Finite Group Databasebeta

← #118601

Subgroups — #118601

Order 512.

500 conjugacy classes of subgroups.

The subgroup lattice has 349 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17821C211yesmaximal, characteristic
2#17667C211yesmaximal, characteristic
3C2×#1534C211yesmaximal, characteristic
4#17394C211yesmaximal, characteristic
5#8086C211yesmaximal, characteristic
6#8010C211yesmaximal, characteristic
7#7998C211yesmaximal, characteristic
8C22×#125C2211yescharacteristic
9#2920C2211yescharacteristic
10#2920C2211yescharacteristic
11#2920C2211yescharacteristic
12#2920C2211yescharacteristic
13C2×#432C2211yescharacteristic
14C2×#432C2211yescharacteristic
15#1573—22no—
16#1572—22no—
17#1569—22no—
18#1534—22no—
19#1534—22no—
20#1534—22no—
21#1528—22no—
22#1528—22no—
23C2×#330—22no—
24#1470—22no—
25#1470C414yes—
26#1463—22no—
27#1302—22no—
28#1293—22no—
29#1291—22no—
30C2×#125—21no—
31C2×#125—21no—
32C2×#125—21no—
33C2×#125—21no—
34C2×#125—21no—
35C2×#125—21no—
36C2×#125—21no—
37C2×#125—21no—
38C2×#125—21no—
39C2×#125—21no—
40C2×#125C2×C412yes—
41C2×#125D412yes—
42C22×C4⋊C4—21no—
43C22×C4⋊C4—21no—
44C22×C4⋊C4Q811yescharacteristic
45C22×C4⋊C4Q811yescharacteristic
46C22×C42C2311yescharacteristic
47#432—21no—
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#432—21no—
57#432—21no—
58#432—21no—
59#432C2×C412yes—
60#432D412yes—
61#432D412yes—
62#432D412yes—
63#431—21no—
64#431—21no—
65#431—21no—
66#431—21no—
67#431—21no—
68#431—21no—
69#431C2×C412yes—
70#431D412yes—
71#423—21no—
72#423—21no—
73#423—21no—
74#423—21no—
75C2×C4⋊C8—21no—
76C2×C4⋊C8—21no—
77#407—21no—
78#407—21no—
79#407—21no—
80#407D412yes—
81C2×C22⋊C8—21no—
82C2×C22⋊C8—21no—
83C4×M16—21no—
84C4×M16—21no—
85C4×M16—21no—
86C4×M16—21no—
87C2×#97—21no—
88C2×#97D412yes—
89#335—24no—
90#334—24no—
91#330—28no—
92#330—28no—
93#324—28no—
94C23×C4—21no—
95C23×C4—21no—
96C23×C4C2×D411yescharacteristic
97C23×C4C2×Q811yescharacteristic
98C23×C4#4111yescharacteristic
99C2×M16—22no—
100C2×M16—41no—
101C2×M16—41no—
102C2×M16—41no—
103C22×C8—22no—
104C22×C8—41no—
105#125—22no—
106#125—22no—
107#125—22no—
108#125—22no—
109#125—22no—
110#125—22no—
111#125—22no—
112#125—22no—
113#125—41no—
114#125—41no—
115#125—41no—
116#125—41no—
117#125—41no—
118#125—41no—
119#125—41no—
120#125—41no—
121C2×C4⋊C4—21no—
122C2×C4⋊C4—21no—
123C2×C4⋊C4—21no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—21no—
126C2×C4⋊C4—21no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—21no—
129C2×C4⋊C4—22no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—22no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—21no—
135C2×C4⋊C4—41no—
136C2×C4⋊C4—41no—
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⋊C4C4⋊C412yes—
144C2×C4⋊C4C4⋊C412yes—
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
157C4⋊C8—22no—
158C4⋊C8—22no—
159C4⋊C8—22no—
160C4⋊C8—22no—
161C4⋊C8—22no—
162C4⋊C8—22no—
163C4⋊C8—41no—
164C4⋊C8—41no—
165C4⋊C8—41no—
166C4⋊C8—41no—
167C4⋊C8—41no—
168C4⋊C8—41no—
169C22⋊C8—22no—
170C22⋊C8—22no—
171C22⋊C8—22no—
172C22⋊C8—22no—
173C22⋊C8—22no—
174C22⋊C8—41no—
175C22⋊C8—41no—
176C22⋊C8—41no—
177C22⋊C8—41no—
178C22⋊C8—41no—
179C22⋊C8—41no—
180C22⋊C8C22⋊C414yes—
181#97—22no—
182#97—22no—
183#97—22no—
184#97—22no—
185#97—22no—
186#97—22no—
187#97—41no—
188#97—41no—
189C4×C8—22no—
190C4×C8—22no—
191C4×C8—41no—
192C4×C8—41no—
193C24C22⋊Q811yescharacteristic
194C22×C4—22no—
195C22×C4—22no—
196C22×C4—42no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—41no—
200C22×C4—41no—
201C22×C4—81no—
202C22×C4—81no—
203C22×C4—82no—
204C22×C4—82no—
205C22×C4C2×C22⋊C411yescharacteristic
206C22×C4C2×C4⋊C411yescharacteristic
207C22×C4#11711yescharacteristic
208C22×C4#12111yescharacteristic
209C22×C4#12111yescharacteristic
210C22×C4#12111yescharacteristic
211C22×C4#12111yescharacteristic
212C22×C4#12111yescharacteristic
213C22×C4#12111yescharacteristic
214C22×C4#12111yescharacteristic
215C22×C4#12111yescharacteristic
216C22×C4C22⋊Q811yescharacteristic
217C22×C4C22⋊Q811yescharacteristic
218C22×C4C22⋊Q811yescharacteristic
219C22×C4C22⋊Q811yescharacteristic
220C22×C4C22⋊Q811yescharacteristic
221C22×C4#12611yescharacteristic
222C22×C4#12611yescharacteristic
223M16—82no—
224M16—82no—
225M16—82no—
226M16—82no—
227C2×C8—24no—
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C2×C8—42no—
232C2×C8—42no—
233C2×C8—82no—
234C2×C8—82no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—22no—
238C4⋊C4—22no—
239C4⋊C4—42no—
240C4⋊C4—42no—
241C4⋊C4—42no—
242C4⋊C4—42no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C4⋊C4—41no—
246C4⋊C4—42no—
247C4⋊C4—42no—
248C4⋊C4—42no—
249C4⋊C4—41no—
250C4⋊C4—41no—
251C42C4×D411yescharacteristic
252C42C4×D411yescharacteristic
253C42C4×D411yescharacteristic
254C42C4×D411yescharacteristic
255C42#12011yescharacteristic
256C42#12011yescharacteristic
257C42#12011yescharacteristic
258C42#12311yescharacteristic
259C42#12311yescharacteristic
260C42#12311yescharacteristic
261C42#12311yescharacteristic
262C42#12311yescharacteristic
263C42#12411yescharacteristic
264C42#12411yescharacteristic
265C42#12411yescharacteristic
266C42#12711yescharacteristic
267C23—22no—
268C23—41no—
269C23—41no—
270C23#38011yescharacteristic
271C23#39611yescharacteristic
272C23#39611yescharacteristic
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—44no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—82no—
299C2×C4#38511yescharacteristic
300C2×C4#38511yescharacteristic
301C2×C4#38611yescharacteristic
302C2×C4#38811yescharacteristic
303C2×C4#38811yescharacteristic
304C2×C4#39011yescharacteristic
305C2×C4#39211yescharacteristic
306C2×C4#39311yescharacteristic
307C2×C4#39411yescharacteristic
308C2×C4#39411yescharacteristic
309C2×C4#39711yescharacteristic
310C2×C4#39911yescharacteristic
311C8—44no—
312C8—82no—
313C8—82no—
314C8—82no—
315C8—82no—
316C8—82no—
317C22—21no—
318C22—21no—
319C22—21no—
320C22—21no—
321C22—21no—
322C22—21no—
323C22—21no—
324C22—21no—
325C22—44no—
326C22#143911yescharacteristic
327C22#207012yes—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—21no—
332C4—21no—
333C4—21no—
334C4—21no—
335C4—21no—
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—82no—
341C4—82no—
342C2—22no—
343C2—41no—
344C2—41no—
345C2#1109411yescharacteristic
346C2#1110911yescharacteristic
347C2#1114011yescharacteristic
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
#178212
#176672
C2×#15342
#173942
#80862
#80102
#79982