Finite Group Databasebeta

← #213821

Subgroups — #213821

Order 512.

801 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#35319C211yesmaximal, characteristic
2#17391C212yesmaximal
3#9104C212yesmaximal, complemented
4#9100C212yesmaximal, complemented
5#8962C212yesmaximal, complemented
6#8958C212yesmaximal, complemented
7#8731C211yesmaximal, characteristic
8#8730C211yesmaximal, characteristic
9#8327C212yesmaximal
10C2×#541C2211yescharacteristic
11C2×#541C2211yescharacteristic
12#2983C2212yes—
13#2982C2212yes—
14#2868C411yescharacteristic
15#2859C411yescharacteristic
16#2857C412yes—
17#2642C411yescharacteristic
18#2642C411yescharacteristic
19#2621C412yes—
20#2506—24no—
21#2496—24no—
22#2327C2212yes—
23C4×C4⋊Q8C2212yes—
24C4×#117C2211yescharacteristic
25#1479C2212yes—
26#1479C2212yes—
27#1479C2212yes—
28#1479C2212yes—
29#1473—24no—
30#1470—24no—
31#1461C2212yes—
32#1461C2212yes—
33#1458C2212yes—
34#1458C2212yes—
35#1276C2211yescharacteristic
36#1276C2212yes—
37#1276C2211yescharacteristic
38#1274C2212yes—
39#1274C2211yescharacteristic
40#1274C2211yescharacteristic
41#541—41no—
42#541—42no—
43#541—41no—
44#541—41no—
45#541—41no—
46#541—42no—
47C2×C4⋊Q8C2×C412yes—
48C2×C4⋊Q8C2×C412yes—
49C2×C4⋊Q8C2×C411yescharacteristic
50C2×C4⋊Q8C2×C411yescharacteristic
51C2×#126—22no—
52C2×#126D414yes—
53C2×#125—22no—
54C2×#125C2×C411yescharacteristic
55C2×#125C2×C411yescharacteristic
56C2×#125D414yes—
57C2×C22⋊Q8C2×C412yes—
58C2×C22⋊Q8C2×C412yes—
59C2×C4×Q8C2312yes—
60C2×C4×Q8C2312yes—
61C2×#117C2×C411yescharacteristic
62C2×#117C2311yescharacteristic
63C22×C42C2×C411yescharacteristic
64#446—42no—
65#446—42no—
66C8×Q8—42no—
67C8×Q8—42no—
68#432—24no—
69#431—24no—
70#431C2312yes—
71#431C2312yes—
72#431C2312yes—
73#431C2312yes—
74#400—24no—
75#400—24no—
76#398—24no—
77#397—24no—
78#391C2×C412yes—
79#391C2×C412yes—
80#391C2×C412yes—
81#391C2×C412yes—
82#389—24no—
83#389—24no—
84#389—24no—
85#387—24no—
86#387—24no—
87#384—24no—
88#384C2×C412yes—
89#382C2×C412yes—
90#380—24no—
91C4×C4⋊C4—22no—
92C4×C4⋊C4—24no—
93C4×C22⋊C4—22no—
94#376C2×C411yescharacteristic
95#376C2311yescharacteristic
96C43C2×C411yescharacteristic
97C43C2311yescharacteristic
98Q8⋊C8—42no—
99Q8⋊C8—42no—
100Q8⋊C8—42no—
101Q8⋊C8—42no—
102#324—22no—
103#324—22no—
104#324—24no—
105#324—24no—
106#323—22no—
107#323—22no—
108C22×Q8C22×C412yes—
109C22×Q8C22×C412yes—
110C23×C4—21no—
111C23×C4C22×C411yescharacteristic
112C4⋊Q8—24no—
113C4⋊Q8—24no—
114C4⋊Q8—42no—
115C4⋊Q8—42no—
116C4⋊Q8—41no—
117C4⋊Q8—41no—
118C4⋊Q8—41no—
119C4⋊Q8—41no—
120#126—44no—
121#126—44no—
122#126—44no—
123#126—44no—
124#125—41no—
125#125—41no—
126#125—41no—
127#125—41no—
128#125—44no—
129#125—44no—
130#125—44no—
131#125—44no—
132C22⋊Q8—42no—
133C22⋊Q8—42no—
134C22⋊Q8—42no—
135C22⋊Q8—42no—
136C4×Q8—22no—
137C4×Q8—22no—
138C4×Q8—22no—
139C4×Q8—22no—
140C4×Q8—22no—
141C4×Q8—22no—
142C4×Q8—22no—
143C4×Q8—22no—
144#117—22no—
145#117—21no—
146#117—21no—
147#117—22no—
148#117—41no—
149C2×C4⋊C4—21no—
150C2×C4⋊C4—22no—
151C2×C4⋊C4—22no—
152C2×C4⋊C4—22no—
153C2×C4⋊C4—24no—
154C2×C4⋊C4—24no—
155C2×C4⋊C4—24no—
156C2×C4⋊C4—22no—
157C2×C4⋊C4C22⋊C414yes—
158C2×C4⋊C4C22⋊C414yes—
159C2×C4⋊C4C22⋊C414yes—
160C2×C4⋊C4C22×C412yes—
161C2×C4⋊C4C22×C412yes—
162C2×C4⋊C4C2×D412yes—
163C2×C4⋊C4C2×D414yes—
164C2×C22⋊C4—21no—
165C2×C22⋊C4—22no—
166C2×C22⋊C4C22⋊C414yes—
167C2×C22⋊C4C2×D412yes—
168C2×C42—22no—
169C2×C42—22no—
170C2×C42—21no—
171C2×C42—22no—
172C2×C42—22no—
173C2×C42—41no—
174C2×C42C22×C411yescharacteristic
175C2×C42C22×C411yescharacteristic
176C2×C42C22×C411yescharacteristic
177C2×C42C22×C411yescharacteristic
178C2×C42C22×C411yescharacteristic
179C2×C42C2×D414yes—
180C2×C42C2411yescharacteristic
181C4⋊C8—22no—
182C4⋊C8—22no—
183C4⋊C8—22no—
184C4⋊C8—22no—
185C4⋊C8—44no—
186C4⋊C8—44no—
187C22⋊C8—22no—
188C22⋊C8—22no—
189C22⋊C8—22no—
190C22⋊C8—22no—
191#97—44no—
192C4×C8—22no—
193C4×C8—22no—
194C4×C8—22no—
195C4×C8—22no—
196C4×C8—44no—
197#95—22no—
198#95—24no—
199#95—24no—
200#95—24no—
201#95—24no—
202#95—24no—
203#95—22no—
204C24C2×C22⋊C411yescharacteristic
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—22no—
208C2×Q8—22no—
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8—22no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—22no—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—44no—
231C22×C4C2×C22⋊C414yes—
232C22×C4C2×C22⋊C411yescharacteristic
233C22×C4C2×C22⋊C414yes—
234C22×C4C2×C22⋊C412yes—
235C22×C4C23×C411yescharacteristic
236C22×C4C22×D412yes—
237C2×C8—42no—
238C2×C8—42no—
239C2×C8—42no—
240C2×C8—42no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—24no—
246C4⋊C4—24no—
247C4⋊C4—22no—
248C4⋊C4—22no—
249C4⋊C4—44no—
250C4⋊C4—44no—
251C4⋊C4—44no—
252C4⋊C4—42no—
253C4⋊C4—48no—
254C4⋊C4—48no—
255C4⋊C4—48no—
256C4⋊C4—44no—
257C4⋊C4—44no—
258C4⋊C4—44no—
259C4⋊C4—42no—
260C4⋊C4—42no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—44no—
264C22⋊C4—42no—
265C22⋊C4—44no—
266C42—21no—
267C42—21no—
268C42—24no—
269C42—24no—
270C42—21no—
271C42—21no—
272C42—21no—
273C42—21no—
274C42—21no—
275C42—21no—
276C42—42no—
277C42—44no—
278C42—44no—
279C42—42no—
280C42—42no—
281C42#13112yes—
282C42#13112yes—
283C42H(5,𝔽2)11yescharacteristic
284C42H(5,𝔽2)11yescharacteristic
285C42H(5,𝔽2)12yes—
286C23—21no—
287C23—21no—
288C23—21no—
289C23—42no—
290C23C22×C22⋊C411yescharacteristic
291Q8—42no—
292Q8—42no—
293Q8—42no—
294Q8—42no—
295Q8—42no—
296Q8—42no—
297Q8—42no—
298Q8—42no—
299C2×C4—24no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—24no—
303C2×C4—24no—
304C2×C4—21no—
305C2×C4—22no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—24no—
309C2×C4—24no—
310C2×C4—42no—
311C2×C4—42no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—42no—
317C2×C4—44no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—42no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—84no—
328C2×C4#51811yescharacteristic
329C2×C4#51811yescharacteristic
330C2×C4#53412yes—
331C2×C4#53412yes—
332C2×C4C2×#13111yescharacteristic
333C2×C4C2×#13111yescharacteristic
334C2×C4#56811yescharacteristic
335C2×C4#56811yescharacteristic
336C8—82no—
337C8—82no—
338C8—82no—
339C8—82no—
340C22—22no—
341C22—42no—
342C22—41no—
343C22—81no—
344C22—81no—
345C22#230811yescharacteristic
346C22#288011yescharacteristic
347C22#288011yescharacteristic
348C4—24no—
349C4—22no—
350C4—22no—
351C4—42no—
352C4—42no—
353C4—44no—
354C4—41no—
355C4—44no—
356C4—81no—
357C4—81no—
358C4—82no—
359C4—82no—
360C4#297911yescharacteristic
361C4#298011yescharacteristic
362C4#298111yescharacteristic
363C4#298111yescharacteristic
364C2—21no—
365C2—21no—
366C2—81no—
367C2#2174511yescharacteristic
368C2#2178312yes—
subgroupG/H
center, socleC22#2308
commutatorC22×C4C23×C4
Frattini, cosocleC2×C42C24
prime radicalC24C2×C22⋊C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C42 ◁ G
LECSG ▷ C2×C42 ▷ C22 ▷ C1

Complements

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

NH ≅ G/Ntype
3. #9104C2semidirect
4. #9100C2semidirect
5. #8962C2semidirect
6. #8958C2semidirect

Maximal subgroups

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

subgroupindex
#353192
#173912
#91042
#91002
#89622
#89582
#87312
#87302
#83272