Finite Group Databasebeta

← #497604

Subgroups — #497604

Order 512.

776 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#39619C211yesmaximal, characteristic
2Q8×#101C212yesmaximal
3#13303C214yesmaximal
4#13301C212yesmaximal
5#12804C212yesmaximal
6#12802C211yesmaximal, characteristic
7#12631C212yesmaximal
8C4×#356C211yesmaximal, characteristic
9C2×Q82C2211yescharacteristic
10C2×#557C411yescharacteristic
11Q8×M16—22no—
12#2862C411yescharacteristic
13#2713—24no—
14#2660C412yes—
15#2568C412yes—
16#2568C2211yescharacteristic
17#2567C411yescharacteristic
18#2565C2212yes—
19#2451—22no—
20#2446—22no—
21#2327C2212yes—
22C4×C4⋊Q8C2211yescharacteristic
23#2295C411yescharacteristic
24C2×#356C2212yes—
25C2×#356C2211yescharacteristic
26C2×#356C2211yescharacteristic
27#1997—22no—
28#1997—24no—
29#1801C2212yes—
30#1801C2214yes—
31C4×#101C2212yes—
32C4×#101C2214yes—
33#1394C2212yes—
34#1394C2214yes—
35#1394C2212yes—
36#1394C2214yes—
37Q82—42no—
38Q82—42no—
39#557—42no—
40#557—42no—
41C2×C4⋊Q8—21no—
42C2×C4⋊Q8C2×C411yescharacteristic
43C2×C4⋊Q8C2×C411yescharacteristic
44C2×C4⋊Q8C2×C412yes—
45C2×C4⋊Q8D412yes—
46C2×C4⋊Q8D414yes—
47C2×C4⋊Q8C2311yescharacteristic
48C2×#125—22no—
49C2×#125C2×C412yes—
50C2×C4×Q8—21no—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C412yes—
53C2×C4×Q8C2×C411yescharacteristic
54C2×C4×Q8D412yes—
55C2×C4×Q8C2311yescharacteristic
56C2×C4×Q8C2312yes—
57#446—42no—
58#446—44no—
59C8×Q8—42no—
60#423—22no—
61#423—24no—
62C2×#101C2314yes—
63C2×#101C2312yes—
64C2×#101C2312yes—
65C4×M16—22no—
66C4×M16—24no—
67#400—24no—
68#400—22no—
69#398—24no—
70#398—24no—
71#395—22no—
72#395—24no—
73#391—24no—
74#391—22no—
75#391—21no—
76#391—22no—
77#391—22no—
78#391C2×C411yescharacteristic
79#391C2×C412yes—
80#391C2311yescharacteristic
81#391C2312yes—
82#389—22no—
83#389—21no—
84#389—22no—
85#387—22no—
86#387—24no—
87#387—24no—
88#387—24no—
89#384C2×C411yescharacteristic
90#384C2×C412yes—
91#382C2×C412yes—
92#382C2×C412yes—
93#382C2×C412yes—
94C4×C4⋊C4—22no—
95C4×C4⋊C4C2×C412yes—
96C4×C4⋊C4C2×C411yescharacteristic
97#376C2×C412yes—
98C43C2×C411yescharacteristic
99#356—24no—
100#356—22no—
101#356Q814yescomplemented
102#343—24no—
103#343—22no—
104#343—24no—
105#343—22no—
106C22×Q8—21no—
107C22×Q8C22⋊C412yes—
108C22×Q8C22⋊C414yes—
109C22×Q8C22⋊C411yescharacteristic
110C22×Q8C22×C411yescharacteristic
111C22×Q8C2411yescharacteristic
112C2×M16—22no—
113C2×M16—22no—
114C2×M16—24no—
115C4⋊Q8—22no—
116C4⋊Q8—44no—
117C4⋊Q8—41no—
118C4⋊Q8—41no—
119C4⋊Q8—42no—
120C4⋊Q8—44no—
121C4⋊Q8—42no—
122C4⋊Q8—44no—
123C4⋊Q8—41no—
124C4⋊Q8—41no—
125C4⋊Q8—42no—
126C4⋊Q8—42no—
127C4⋊Q8C2×Q812yes—
128C4⋊Q8#4112yes—
129#125—44no—
130#125—48no—
131C4×Q8—22no—
132C4×Q8—22no—
133C4×Q8—22no—
134C4×Q8—22no—
135C4×Q8—24no—
136C4×Q8—41no—
137C4×Q8—44no—
138C4×Q8—41no—
139C4×Q8—44no—
140C4×Q8—41no—
141C4×Q8—41no—
142C4×Q8—42no—
143C4×Q8—42no—
144C2×C4⋊C4—22no—
145C2×C4⋊C4—22no—
146C2×C4⋊C4—21no—
147C2×C4⋊C4—22no—
148C2×C4⋊C4—21no—
149C2×C4⋊C4—22no—
150C2×C4⋊C4—21no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4C22⋊C411yescharacteristic
155C2×C4⋊C4C22⋊C412yes—
156C2×C4⋊C4C22⋊C412yes—
157C2×C4⋊C4C22⋊C414yes—
158C2×C4⋊C4C22×C411yescharacteristic
159C2×C4⋊C4C22×C412yes—
160C2×C4⋊C4C22×C411yescharacteristic
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C2×D414yes—
163C2×C4⋊C4C2×D412yes—
164C2×C42—22no—
165C2×C42—21no—
166C2×C42—21no—
167C2×C42—21no—
168C2×C42C22×C412yes—
169C2×C42C22×C411yescharacteristic
170C2×C42C22×C411yescharacteristic
171C2×C42C22×C411yescharacteristic
172C2×C42C22×C412yes—
173C2×C42C2×D412yes—
174C2×C42C2×D414yes—
175C4⋊C8—44no—
176C4⋊C8—42no—
177C4⋊C8—42no—
178C4⋊C8—44no—
179#101—22no—
180#101—24no—
181#101—22no—
182#101—24no—
183#101—22no—
184#101C2×Q814yes—
185#97—42no—
186#97—44no—
187C4×C8—44no—
188C4×C8—42no—
189#95—22no—
190#95—22no—
191#95—22no—
192#95—22no—
193#95—21no—
194#95—24no—
195#95—24no—
196#95—24no—
197#95—24no—
198#95—22no—
199#95—24no—
200C2×Q8—21no—
201C2×Q8—41no—
202C2×Q8—41no—
203C2×Q8—44no—
204C2×Q8—42no—
205C2×Q8—42no—
206C2×Q8—44no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—82no—
212C2×Q8—84no—
213C2×Q8#9912yes—
214C2×Q8#9912yes—
215C2×Q8C4×Q812yes—
216C2×Q8C22×Q811yescharacteristic
217C2×Q8C2×#4111yescharacteristic
218C2×Q82−1+412yes—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—22no—
223C22×C4—22no—
224C22×C4—21no—
225C22×C4C2×C22⋊C411yescharacteristic
226C22×C4C2×C22⋊C412yes—
227C22×C4C2×C22⋊C414yes—
228C22×C4C2×C22⋊C412yes—
229C22×C4C2×C22⋊C411yescharacteristic
230C22×C4C2×C22⋊C412yes—
231C22×C4C23×C411yescharacteristic
232C22×C4C22×D412yes—
233M16—24no—
234M16—42no—
235M16—42no—
236M16—44no—
237M16—44no—
238M16—42no—
239C2×C8—42no—
240C2×C8—44no—
241C2×C8—42no—
242C4⋊C4—24no—
243C4⋊C4—24no—
244C4⋊C4—21no—
245C4⋊C4—21no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—24no—
249C4⋊C4—24no—
250C4⋊C4—22no—
251C4⋊C4—41no—
252C4⋊C4—42no—
253C4⋊C4—44no—
254C4⋊C4—42no—
255C4⋊C4—41no—
256C4⋊C4—42no—
257C4⋊C4—42no—
258C4⋊C4—44no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C4⋊C4—44no—
262C4⋊C4—44no—
263C4⋊C4—82no—
264C4⋊C4—82no—
265C4⋊C4—84no—
266C42—21no—
267C42—24no—
268C42—22no—
269C42—22no—
270C42—24no—
271C42—24no—
272C42—21no—
273C42—21no—
274C42—41no—
275C42—42no—
276C42—42no—
277C42—44no—
278C42—41no—
279C42—41no—
280C42—81no—
281C42C4×Q812yes—
282C23C22×C22⋊C411yescharacteristic
283Q8—24no—
284Q8—42no—
285Q8—41no—
286Q8—41no—
287Q8—42no—
288Q8—42no—
289Q8—42no—
290Q8—82no—
291Q8—82no—
292Q8—164no—
293Q8—162no—
294C2×C4—21no—
295C2×C4—22no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—22no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—24no—
304C2×C4—22no—
305C2×C4—42no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—42no—
312C2×C4—44no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—41no—
321C2×C4—42no—
322C2×C4—41no—
323C2×C4—42no—
324C2×C4C2×#9911yescharacteristic
325C2×C4C2×#9914yes—
326C2×C4C2×#9911yescharacteristic
327C2×C4C2×C4×Q811yescharacteristic
328C2×C4#51911yescharacteristic
329C2×C4#54812yes—
330C2×C4D4×Q812yes—
331C8—44no—
332C8—84no—
333C8—82no—
334C22—21no—
335C22—21no—
336C22Q8×C22⋊C411yescharacteristic
337C22C22×#9911yescharacteristic
338C22#288411yescharacteristic
339C4—21no—
340C4—24no—
341C4—22no—
342C4—22no—
343C4—21no—
344C4—21no—
345C4—42no—
346C4—42no—
347C4—41no—
348C4—44no—
349C4—42no—
350C4—42no—
351C4—81no—
352C4—82no—
353C4—82no—
354C4—81no—
355C4#213312yes—
356C2—21no—
357C2—21no—
358C2Q8×#9911yescharacteristic
359C2C2×#213311yescharacteristic
360C2#3337111yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC23C22×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C22×Q8 ◁ C2×Q82 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
101. #356Q8semidirect

Maximal subgroups

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

subgroupindex
#396192
Q8×#1012
#133032
#133012
#128042
#128022
#126312
C4×#3562