Finite Group Databasebeta

← #99980

Subgroups — #99980

Order 512.

406 conjugacy classes of subgroups.

The subgroup lattice has 347 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22018C211yesmaximal, characteristic
2#21615C211yesmaximal, characteristic
3#9864C211yesmaximal, characteristic
4#8566C211yesmaximal, characteristic
5#7590C211yesmaximal, characteristic
6#7027C211yesmaximal, characteristic
7#7022C211yesmaximal, characteristic
8C2×#475C411yescharacteristic
9C4×C4⋊Q8C2211yescharacteristic
10C42×Q8C2211yescharacteristic
11#1988C411yescharacteristic
12#1984—21no—
13#1947C411yescharacteristic
14#1939—21no—
15#1920—21no—
16#1916—22no—
17#1863—21no—
18#1839—21no—
19#1831—21no—
20#1831C2211yescharacteristic
21#1803C411yescharacteristic
22#1776—21no—
23#1767C2211yescharacteristic
24C4×C4⋊C8C2211yescharacteristic
25C4×C4⋊C8C2211yescharacteristic
26#1763—21no—
27#1744C2211yescharacteristic
28C2×Q8⋊C8—21no—
29#1279—22no—
30#1279—21no—
31#1272—21no—
32#1272—21no—
33C2×C4⋊Q8C2×C411yescharacteristic
34C2×C4×Q8—21no—
35C2×C4×Q8C2×C411yescharacteristic
36C2×C4×Q8D411yescharacteristic
37C2×C4×Q8D411yescharacteristic
38#475—22no—
39#475—22no—
40C2×#107—21no—
41C2×#106—21no—
42C2×C4⋊C8—21no—
43C2×C4⋊C8—21no—
44C2×C4⋊C8—21no—
45C2×C4⋊C8—22no—
46C2×C4⋊C8C2×C411yescharacteristic
47C2×C4⋊C8C2×C411yescharacteristic
48C2×#103—21no—
49C2×#103—21no—
50C2×#97—21no—
51C2×C4×C8—21no—
52C2×C4×C8—21no—
53C2×C4×C8D412yes—
54#391—21no—
55#389—21no—
56#389—21no—
57#384C2×C411yescharacteristic
58C4×C4⋊C4—21no—
59C4×C4⋊C4—21no—
60C4×C4⋊C4—21no—
61C4×C4⋊C4—21no—
62C4×C4⋊C4—21no—
63C4×C4⋊C4C2×C411yescharacteristic
64C4×C4⋊C4D411yescharacteristic
65C4×C4⋊C4D411yescharacteristic
66C43—21no—
67C43C2311yescharacteristic
68#340—21no—
69#340—21no—
70#340—21no—
71#340—21no—
72#336—22no—
73#336—22no—
74#336—22no—
75#336—21no—
76#336—21no—
77#336—21no—
78#336—21no—
79#336—21no—
80#336—21no—
81#336D411yescharacteristic
82#336D411yescharacteristic
83Q8⋊C8—42no—
84C22×Q8—21no—
85C22×Q8C22⋊C411yescharacteristic
86C22×C8—21no—
87C22×C8—21no—
88C22×C8—21no—
89C22×C8—22no—
90C22×C8—21no—
91C22×C8—41no—
92C22×C8—41no—
93C4⋊Q8—21no—
94C4⋊Q8—21no—
95C4⋊Q8—21no—
96C4⋊Q8M1612yes—
97C4×Q8—21no—
98C4×Q8—21no—
99C4×Q8—21no—
100C4×Q8—21no—
101C4×Q8—21no—
102C4×Q8—21no—
103C4×Q8—41no—
104C4×Q8—41no—
105C4×Q8—82no—
106C4×Q8SD1612yes—
107C4×Q8Q1612yes—
108C2×C4⋊C4—21no—
109C2×C4⋊C4—21no—
110C2×C4⋊C4—21no—
111C2×C4⋊C4—21no—
112C2×C4⋊C4—21no—
113C2×C4⋊C4—21no—
114C2×C4⋊C4—21no—
115C2×C4⋊C4—21no—
116C2×C4⋊C4—41no—
117C2×C4⋊C4C22⋊C411yescharacteristic
118C2×C4⋊C4C22⋊C411yescharacteristic
119C2×C4⋊C4C22⋊C411yescharacteristic
120C2×C42—21no—
121C2×C42—21no—
122C2×C42—21no—
123C2×C42—21no—
124C2×C42—21no—
125C2×C42—21no—
126C2×C42—41no—
127C2×C42C22×C411yescharacteristic
128C2×C42C2×D411yescharacteristic
129C2×C42C2×D411yescharacteristic
130C2×C42C2×D411yescharacteristic
131C2×C42C2×D411yescharacteristic
132C2×C42#4111yescharacteristic
133C2×C42#4111yescharacteristic
134#107—42no—
135#106—24no—
136C4⋊C8—22no—
137C4⋊C8—21no—
138C4⋊C8—21no—
139C4⋊C8—21no—
140C4⋊C8—41no—
141C4⋊C8—41no—
142C4⋊C8—41no—
143C4⋊C8—41no—
144C4⋊C8—41no—
145C4⋊C8—82no—
146C4⋊C8M1612yes—
147#103—24no—
148#103—42no—
149#97—41no—
150#97—41no—
151C4×C8—42no—
152C4×C8—41no—
153C4×C8—41no—
154C4×C8—41no—
155C4×C8—41no—
156#95—21no—
157#95—21no—
158#95—21no—
159#95—41no—
160#95—41no—
161C2×Q8—21no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—22no—
167C2×Q8—41no—
168C2×Q8—81no—
169C2×Q8—81no—
170C2×Q8#10312yes—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—41no—
176C22×C4—41no—
177C22×C4C2×C22⋊C411yescharacteristic
178C22×C4C4×D411yescharacteristic
179C22×C4C4×D411yescharacteristic
180C22×C4#12011yescharacteristic
181C22×C4#12111yescharacteristic
182C22×C4#12111yescharacteristic
183C22×C4#12311yescharacteristic
184C2×C8—22no—
185C2×C8—41no—
186C2×C8—41no—
187C2×C8—41no—
188C2×C8—41no—
189C2×C8—42no—
190C2×C8—42no—
191C2×C8—41no—
192C2×C8—42no—
193C2×C8—41no—
194C2×C8—41no—
195C2×C8—41no—
196C2×C8—41no—
197C2×C8—42no—
198C2×C8—41no—
199C2×C8—41no—
200C2×C8—81no—
201C2×C8—81no—
202C2×C8—81no—
203C2×C8—81no—
204C2×C8—81no—
205C4⋊C4—21no—
206C4⋊C4—22no—
207C4⋊C4—22no—
208C4⋊C4—22no—
209C4⋊C4—21no—
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—21no—
213C4⋊C4—22no—
214C4⋊C4—42no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—41no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—41no—
222C4⋊C4—81no—
223C4⋊C4—82no—
224C4⋊C4#10312yes—
225C42—21no—
226C42—21no—
227C42—21no—
228C42—21no—
229C42—22no—
230C42—22no—
231C42—41no—
232C42—41no—
233C42—41no—
234C42—42no—
235C42—41no—
236C42—41no—
237C42—41no—
238C42—41no—
239C42—81no—
240C42—81no—
241C42—82no—
242C42C2×M1611yescharacteristic
243C42C2×M1611yescharacteristic
244C42#13111yescharacteristic
245C42#13111yescharacteristic
246C42C2×SD1611yescharacteristic
247C42C2×Q1611yescharacteristic
248C42#13511yescharacteristic
249C42#13511yescharacteristic
250C23#38611yescharacteristic
251Q8—22no—
252Q8—22no—
253Q8—41no—
254Q8—41no—
255Q8—82no—
256Q8—82no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4#40711yescharacteristic
289C2×C4#40811yescharacteristic
290C2×C4C2×#10311yescharacteristic
291C2×C4#41611yescharacteristic
292C2×C4#42111yescharacteristic
293C2×C4#42111yescharacteristic
294C2×C4#43511yescharacteristic
295C2×C4#43511yescharacteristic
296C2×C4#43611yescharacteristic
297C2×C4#43611yescharacteristic
298C2×C4#45511yescharacteristic
299C2×C4#45611yescharacteristic
300C2×C4#46511yescharacteristic
301C2×C4#46511yescharacteristic
302C2×C4#46711yescharacteristic
303C2×C4#46711yescharacteristic
304C8—42no—
305C8—81no—
306C8—82no—
307C8—81no—
308C8—82no—
309C8—81no—
310C8—161no—
311C22#187011yescharacteristic
312C22#188111yescharacteristic
313C22#188211yescharacteristic
314C22#189211yescharacteristic
315C22#189711yescharacteristic
316C22#190311yescharacteristic
317C22#190511yescharacteristic
318C4—21no—
319C4—21no—
320C4—21no—
321C4—21no—
322C4—21no—
323C4—21no—
324C4—21no—
325C4—21no—
326C4—21no—
327C4—21no—
328C4—41no—
329C4—42no—
330C4—42no—
331C4—81no—
332C4—82no—
333C4—81no—
334C4—161no—
335C4#148111yescharacteristic
336C4#148211yescharacteristic
337C4#148711yescharacteristic
338C4#148811yescharacteristic
339C2#895511yescharacteristic
340C2#907511yescharacteristic
341C2#909411yescharacteristic
342C2#930411yescharacteristic
343C2#931311yescharacteristic
344C2#931511yescharacteristic
345C2#932811yescharacteristic
subgroupG/H
center, socle, prime radicalC23#386
commutatorC2×C42C22×C4
Frattini, cosocleC43C23
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C43 ◁ G
LECSG ▷ C43 ▷ C23 ▷ C1

Maximal subgroups

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

subgroupindex
#220182
#216152
#98642
#85662
#75902
#70272
#70222