Finite Group Databasebeta

← #224453

Subgroups — #224453

Order 512.

835 conjugacy classes of subgroups.

The subgroup lattice has 343 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36562C211yesmaximal, characteristic
2#21345C212yesmaximal, complemented
3#9894C214yesmaximal, complemented
4#9890C212yesmaximal, complemented
5#8579C212yesmaximal
6#8579C211yesmaximal, characteristic
7#8224C212yesmaximal
8C4×#329C211yesmaximal, characteristic
9C2×#557C411yescharacteristic
10C2×#554C411yescharacteristic
11#2862C2211yescharacteristic
12#2859C2212yes—
13#2567C411yescharacteristic
14#2566C412yes—
15#2565C412yes—
16#2556—24no—
17#2542C411yescharacteristic
18#2429—22no—
19#2360—22no—
20C4×#124C2211yescharacteristic
21C4×#124C2212yes—
22C42×Q8C2211yescharacteristic
23#1984—22no—
24#1939—22no—
25#1917—24no—
26#1767C2212yes—
27#1767C2214yes—
28#1745C2214yes—
29C4×#97C2212yes—
30C2×#329C2211yescharacteristic
31C2×#329C2212yes—
32C2×#329C2211yescharacteristic
33#1295C2212yes—
34#1295C2214yes—
35#1295C2214yes—
36#1295C2212yes—
37#557—44no—
38#554—44no—
39C2×C4⋊Q8—21no—
40C2×C4⋊Q8C2×C411yescharacteristic
41C2×C4⋊Q8C2×C411yescharacteristic
42C2×#125—22no—
43C2×#125C2×C412yes—
44C2×#125C2×C412yes—
45C2×#124C2311yescharacteristic
46C2×C22⋊Q8—21no—
47C2×C22⋊Q8—22no—
48C2×C22⋊Q8—22no—
49C2×C4×Q8—21no—
50C2×C4×Q8C2×C412yes—
51C2×C4×Q8C2×C411yescharacteristic
52C2×C4×Q8C2×C411yescharacteristic
53C2×C4×Q8D412yes—
54C2×C4×D4C2×C412yes—
55C2×C4×D4C2×C411yescharacteristic
56C22×C4⋊C4—21no—
57C2×C4⋊C8—24no—
58C2×C4⋊C8—22no—
59C2×#97—22no—
60C2×#97C2312yes—
61C2×#97C2314yes—
62C2×#97C2312yes—
63C2×C4×C8—22no—
64#399—22no—
65#398—24no—
66#396—24no—
67#395—22no—
68#391—22no—
69#391—22no—
70#391C2×C411yescharacteristic
71#391C2×C412yes—
72#390C2×C411yescharacteristic
73#390C2×C412yes—
74#389—22no—
75#389—24no—
76#389—24no—
77#388—24no—
78#388—22no—
79#387—24no—
80#385—24no—
81#384C2×C412yes—
82#384C2×C412yes—
83#382C2×C412yes—
84#382C2311yescharacteristic
85#382C2312yes—
86C4×C4⋊C4C2×C412yes—
87C4×C4⋊C4C2×C411yescharacteristic
88C4×C4⋊C4D412yes—
89C4×C4⋊C4D414yes—
90C4×C22⋊C4—22no—
91C4×C22⋊C4—24no—
92C43C2312yes—
93C43C2311yescharacteristic
94#336—22no—
95#336—24no—
96#336—24no—
97#336—24no—
98#329—22no—
99#329—24no—
100#329Q814yescomplemented
101C22×Q8—21no—
102C22×Q8—21no—
103C22×Q8C22⋊C411yescharacteristic
104C22×Q8C22×C411yescharacteristic
105C22×D4C22×C411yescharacteristic
106C23×C4—21no—
107C23×C4—22no—
108C22×C8—22no—
109C22×C8—22no—
110C22×C8—24no—
111C4⋊Q8—42no—
112C4⋊Q8—42no—
113C4⋊Q8—44no—
114#125—44no—
115#125—44no—
116#125—48no—
117#124—22no—
118#124C2×Q812yes—
119#124#4112yes—
120C22⋊Q8—48no—
121C22⋊Q8—48no—
122C22⋊Q8—44no—
123C4×Q8—24no—
124C4×Q8—44no—
125C4×Q8—44no—
126C4×Q8—42no—
127C4×Q8—44no—
128C4×D4—44no—
129C4×D4—42no—
130C2×C4⋊C4—22no—
131C2×C4⋊C4—21no—
132C2×C4⋊C4—22no—
133C2×C4⋊C4—21no—
134C2×C4⋊C4—22no—
135C2×C4⋊C4—22no—
136C2×C4⋊C4—24no—
137C2×C4⋊C4—21no—
138C2×C4⋊C4—22no—
139C2×C4⋊C4—24no—
140C2×C4⋊C4—44no—
141C2×C4⋊C4C22⋊C412yes—
142C2×C4⋊C4C22⋊C412yes—
143C2×C4⋊C4C22⋊C414yes—
144C2×C4⋊C4C22⋊C411yescharacteristic
145C2×C4⋊C4C22×C411yescharacteristic
146C2×C4⋊C4C22×C412yes—
147C2×C4⋊C4C22×C411yescharacteristic
148C2×C4⋊C4C22×C412yes—
149C2×C22⋊C4—22no—
150C2×C22⋊C4—22no—
151C2×C22⋊C4—21no—
152C2×C22⋊C4—24no—
153C2×C22⋊C4—22no—
154C2×C42—22no—
155C2×C42—21no—
156C2×C42C22×C412yes—
157C2×C42C22×C411yescharacteristic
158C2×C42C22×C411yescharacteristic
159C2×C42C22×C412yes—
160C2×C42C2×D412yes—
161C2×C42C2×D414yes—
162C2×C42C2×D414yes—
163C2×C42C2×D412yes—
164C2×C42C2411yescharacteristic
165C4⋊C8—44no—
166C4⋊C8—48no—
167#97—22no—
168#97—24no—
169#97—22no—
170#97—22no—
171#97—24no—
172#97—44no—
173#97C2×Q814yes—
174C4×C8—44no—
175#95—22no—
176#95—24no—
177#95—24no—
178#95—22no—
179#95—24no—
180#95C22⋊C414yes—
181#95C22⋊C412yes—
182C24—21no—
183C2×Q8—21no—
184C2×Q8—22no—
185C2×Q8—24no—
186C2×Q8—42no—
187C2×Q8—41no—
188C2×Q8—41no—
189C2×Q8—42no—
190C2×Q8—44no—
191C2×Q8—82no—
192C2×Q8C4×Q812yes—
193C2×D4—21no—
194C2×D4—41no—
195C2×D4—41no—
196C2×D4C4×Q812yes—
197C22×C4—21no—
198C22×C4—22no—
199C22×C4—22no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—42no—
203C22×C4—44no—
204C22×C4—44no—
205C22×C4—84no—
206C22×C4C2×C22⋊C412yes—
207C22×C4C2×C22⋊C414yes—
208C22×C4C2×C22⋊C411yescharacteristic
209C22×C4C2×C22⋊C412yes—
210C22×C4C2×C22⋊C412yes—
211C22×C4C2×C22⋊C411yescharacteristic
212C22×C4C23×C411yescharacteristic
213C22×C4C22×D412yes—
214C2×C8—24no—
215C2×C8—24no—
216C2×C8—42no—
217C2×C8—42no—
218C2×C8—42no—
219C2×C8—42no—
220C2×C8—44no—
221C2×C8—44no—
222C2×C8—44no—
223C4⋊C4—24no—
224C4⋊C4—24no—
225C4⋊C4—44no—
226C4⋊C4—41no—
227C4⋊C4—42no—
228C4⋊C4—41no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—44no—
232C4⋊C4—48no—
233C4⋊C4—44no—
234C4⋊C4—42no—
235C4⋊C4—44no—
236C4⋊C4—44no—
237C4⋊C4—42no—
238C4⋊C4—44no—
239C4⋊C4—84no—
240C4⋊C4#10014yes—
241C22⋊C4—24no—
242C22⋊C4—24no—
243C22⋊C4—42no—
244C22⋊C4—44no—
245C22⋊C4—48no—
246C22⋊C4—82no—
247C42—24no—
248C42—22no—
249C42—24no—
250C42—42no—
251C42—44no—
252C42—41no—
253C42—42no—
254C42—44no—
255C42—44no—
256C42—42no—
257C42C22×Q811yescharacteristic
258C42C2×#4111yescharacteristic
259C422−1+412yes—
260C23—22no—
261C23—41no—
262C23—41no—
263C23—42no—
264C23—42no—
265C23C22×C22⋊C411yescharacteristic
266Q8—42no—
267Q8—42no—
268Q8—44no—
269Q8—84no—
270Q8—162no—
271D4—42no—
272D4—42no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—24no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—22no—
287C2×C4—24no—
288C2×C4—22no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—42no—
293C2×C4—41no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—42no—
297C2×C4—42no—
298C2×C4—44no—
299C2×C4—84no—
300C2×C4—164no—
301C2×C4C2×#10014yes—
302C2×C4C2×#10012yes—
303C2×C4#42114yes—
304C2×C4C2×C4×Q811yescharacteristic
305C2×C4#51911yescharacteristic
306C2×C4#54812yes—
307C2×C4D4×Q812yes—
308C8—44no—
309C8—44no—
310C8—84no—
311C22—44no—
312C22—42no—
313C22—82no—
314C22—81no—
315C22—81no—
316C22Q8×C22⋊C411yescharacteristic
317C22C22×#10011yescharacteristic
318C22#288911yescharacteristic
319C22C2×#42112yes—
320C22#290112yes—
321C4—22no—
322C4—22no—
323C4—24no—
324C4—21no—
325C4—21no—
326C4—42no—
327C4—44no—
328C4—41no—
329C4—42no—
330C4—42no—
331C4—81no—
332C4—82no—
333C4—81no—
334C4—82no—
335C4#153614yes—
336C2—82no—
337C2C2×#153611yescharacteristic
338C2#1768212yes—
339C2#1768511yescharacteristic
340C2Q8×#10011yescharacteristic
341C2#2252912yes—
subgroupG/H
center, socleC23C22×C22⋊C4
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC22×D4C22×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23 ◁ C42×Q8 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
2. #21345C2semidirect
3. #9894C2semidirect
4. #9890C2semidirect
100. #329Q8semidirect

Maximal subgroups

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

subgroupindex
#365622
#213452
#98942
#98902
#85792
#82242
C4×#3292