Finite Group Databasebeta

← #497601

Subgroups — #497601

Order 512.

817 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#36560C211yesmaximal, characteristic
2#22517C212yesmaximal
3#13303C214yesmaximal
4#13301C212yesmaximal
5#12802C211yesmaximal, characteristic
6#12802C212yesmaximal
7C4×#356C212yesmaximal
8C4×#356C211yesmaximal, characteristic
9C2×#557C411yescharacteristic
10C2×#557C2211yescharacteristic
11Q8×M16—22no—
12#2868C411yescharacteristic
13#2859C412yes—
14#2568C411yescharacteristic
15#2568C2211yescharacteristic
16#2565C412yes—
17#2565C2212yes—
18#2546—22no—
19#2506—24no—
20#2365—22no—
21C4×C4⋊Q8C2212yes—
22C4×C4⋊Q8C2211yescharacteristic
23C42×Q8C411yescharacteristic
24C2×#356C2212yes—
25C2×#356C2211yescharacteristic
26C2×#356C2211yescharacteristic
27#1998—24no—
28#1997—22no—
29#1801C2212yes—
30#1801C2214yes—
31C4×#101C2212yes—
32C4×#101C2214yes—
33#1394C2214yes—
34#1394C2212yes—
35#1394C2214yes—
36#1394C2212yes—
37#557—41no—
38#557—42no—
39#557—41no—
40#557—42no—
41#557—41no—
42#557—41no—
43C2×C4⋊Q8—22no—
44C2×C4⋊Q8—21no—
45C2×C4⋊Q8C2×C411yescharacteristic
46C2×C4⋊Q8C2×C411yescharacteristic
47C2×C4⋊Q8D412yes—
48C2×C4⋊Q8C2311yescharacteristic
49C2×#125—22no—
50C2×#125C2×C412yes—
51C2×#125C2×C412yes—
52C2×#125D414yes—
53C2×C4×Q8—21no—
54C2×C4×Q8—21no—
55C2×C4×Q8C2×C412yes—
56C2×C4×Q8C2×C411yescharacteristic
57C2×C4×Q8C2×C411yescharacteristic
58C2×C4×Q8D412yes—
59C2×C4×Q8C2312yes—
60C2×C4×Q8C2311yescharacteristic
61#446—44no—
62#446—42no—
63C8×Q8—42no—
64#423—22no—
65#423—24no—
66C2×#101C2314yes—
67C2×#101C2312yes—
68C2×#101C2312yes—
69C4×M16—22no—
70C4×M16—24no—
71#400—22no—
72#400—22no—
73#398—24no—
74#398—24no—
75#391—22no—
76#391—22no—
77#391C2×C411yescharacteristic
78#391C2×C412yes—
79#391C2311yescharacteristic
80#391C2312yes—
81#389—24no—
82#389—24no—
83#389—22no—
84#389—24no—
85#387—24no—
86#387—24no—
87#384—22no—
88#384C2×C411yescharacteristic
89#384C2×C412yes—
90#384C2×C412yes—
91#382C2×C412yes—
92#382C2×C412yes—
93C4×C4⋊C4—22no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—24no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2×C412yes—
98C4×C4⋊C4C2×C411yescharacteristic
99C43C2×C412yes—
100C43C2×C411yescharacteristic
101#356—24no—
102#356—22no—
103#356Q814yescomplemented
104#343—24no—
105#343—22no—
106#343—24no—
107#343—22no—
108C22×Q8—21no—
109C22×Q8C22⋊C411yescharacteristic
110C22×Q8C22⋊C412yes—
111C22×Q8C22×C411yescharacteristic
112C22×Q8C2411yescharacteristic
113C2×M16—22no—
114C2×M16—22no—
115C2×M16—24no—
116C4⋊Q8—22no—
117C4⋊Q8—41no—
118C4⋊Q8—41no—
119C4⋊Q8—42no—
120C4⋊Q8—42no—
121C4⋊Q8—41no—
122C4⋊Q8—41no—
123C4⋊Q8—48no—
124C4⋊Q8—42no—
125C4⋊Q8—42no—
126C4⋊Q8C2×Q812yes—
127C4⋊Q8#4112yes—
128#125—42no—
129#125—42no—
130#125—44no—
131#125—44no—
132#125—42no—
133#125—42no—
134#125—44no—
135#125—44no—
136C4×Q8—22no—
137C4×Q8—21no—
138C4×Q8—21no—
139C4×Q8—22no—
140C4×Q8—24no—
141C4×Q8—22no—
142C4×Q8—42no—
143C4×Q8—41no—
144C4×Q8—42no—
145C4×Q8—41no—
146C4×Q8—44no—
147C4×Q8—42no—
148C4×Q8—42no—
149C4×Q8—41no—
150C4×Q8—41no—
151C4×Q8—42no—
152C4×Q8—41no—
153C4×Q8—41no—
154C2×C4⋊C4—22no—
155C2×C4⋊C4—22no—
156C2×C4⋊C4—21no—
157C2×C4⋊C4—22no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—22no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—24no—
163C2×C4⋊C4—24no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C22⋊C412yes—
166C2×C4⋊C4C22⋊C414yes—
167C2×C4⋊C4C22⋊C414yes—
168C2×C4⋊C4C22⋊C412yes—
169C2×C4⋊C4C22⋊C411yescharacteristic
170C2×C4⋊C4C22×C411yescharacteristic
171C2×C4⋊C4C22×C412yes—
172C2×C4⋊C4C22×C411yescharacteristic
173C2×C4⋊C4C22×C412yes—
174C2×C4⋊C4C2×D412yes—
175C2×C4⋊C4C2×D414yes—
176C2×C42—21no—
177C2×C42—22no—
178C2×C42—21no—
179C2×C42—22no—
180C2×C42—22no—
181C2×C42—21no—
182C2×C42C22×C412yes—
183C2×C42C22×C411yescharacteristic
184C2×C42C22×C411yescharacteristic
185C2×C42C22×C412yes—
186C2×C42C22×C411yescharacteristic
187C2×C42C2×D414yes—
188C2×C42C2×D412yes—
189C4⋊C8—44no—
190C4⋊C8—42no—
191C4⋊C8—44no—
192C4⋊C8—42no—
193#101—22no—
194#101—24no—
195#101—22no—
196#101—24no—
197#101—22no—
198#101C2×Q814yes—
199#97—44no—
200#97—42no—
201C4×C8—44no—
202C4×C8—42no—
203#95—22no—
204#95—21no—
205#95—22no—
206#95—24no—
207#95—24no—
208#95—24no—
209#95—22no—
210#95—24no—
211C2×Q8—21no—
212C2×Q8—21no—
213C2×Q8—21no—
214C2×Q8—22no—
215C2×Q8—22no—
216C2×Q8—41no—
217C2×Q8—41no—
218C2×Q8—42no—
219C2×Q8—44no—
220C2×Q8—44no—
221C2×Q8—41no—
222C2×Q8—41no—
223C2×Q8—41no—
224C2×Q8—41no—
225C2×Q8C4×Q812yes—
226C2×Q8C22×Q811yescharacteristic
227C2×Q8C2×#4111yescharacteristic
228C2×Q82−1+412yes—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—22no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4C2×C22⋊C411yescharacteristic
236C22×C4C2×C22⋊C412yes—
237C22×C4C2×C22⋊C414yes—
238C22×C4C2×C22⋊C411yescharacteristic
239C22×C4C2×C22⋊C412yes—
240C22×C4C2×C22⋊C412yes—
241C22×C4C23×C411yescharacteristic
242C22×C4C22×D412yes—
243M16—24no—
244M16—42no—
245M16—42no—
246M16—44no—
247M16—42no—
248M16—44no—
249C2×C8—42no—
250C2×C8—44no—
251C2×C8—42no—
252C4⋊C4—24no—
253C4⋊C4—24no—
254C4⋊C4—24no—
255C4⋊C4—24no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—42no—
259C4⋊C4—44no—
260C4⋊C4—41no—
261C4⋊C4—42no—
262C4⋊C4—42no—
263C4⋊C4—44no—
264C4⋊C4—41no—
265C4⋊C4—42no—
266C4⋊C4—44no—
267C4⋊C4—42no—
268C4⋊C4—44no—
269C4⋊C4—42no—
270C4⋊C4—44no—
271C4⋊C4—42no—
272C4⋊C4—44no—
273C4⋊C4—48no—
274C4⋊C4—48no—
275C4⋊C4—42no—
276C4⋊C4—42no—
277C4⋊C4#9912yes—
278C4⋊C4#9911yescharacteristic
279C4⋊C4#9911yescharacteristic
280C42—24no—
281C42—21no—
282C42—24no—
283C42—22no—
284C42—24no—
285C42—22no—
286C42—22no—
287C42—22no—
288C42—21no—
289C42—21no—
290C42—41no—
291C42—42no—
292C42—44no—
293C42—42no—
294C42—44no—
295C42—42no—
296C42—42no—
297C42—41no—
298C42—41no—
299C42C4×Q812yes—
300C23C22×C22⋊C411yescharacteristic
301Q8—41no—
302Q8—41no—
303Q8—42no—
304Q8—42no—
305Q8—42no—
306Q8—42no—
307Q8—42no—
308Q8—44no—
309Q8—82no—
310Q8—82no—
311Q8—82no—
312C2×C4—22no—
313C2×C4—22no—
314C2×C4—22no—
315C2×C4—21no—
316C2×C4—22no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—24no—
326C2×C4—42no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—44no—
334C2×C4—42no—
335C2×C4—42no—
336C2×C4—42no—
337C2×C4—44no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4C2×#9912yes—
343C2×C4C2×#9912yes—
344C2×C4C2×#9911yescharacteristic
345C2×C4C2×#9911yescharacteristic
346C2×C4C2×C4×Q811yescharacteristic
347C2×C4#51911yescharacteristic
348C2×C4#54812yes—
349C2×C4D4×Q812yes—
350C8—44no—
351C8—84no—
352C8—82no—
353C22—21no—
354C22—21no—
355C22Q8×C22⋊C411yescharacteristic
356C22C22×#9911yescharacteristic
357C22#288411yescharacteristic
358C4—24no—
359C4—21no—
360C4—22no—
361C4—22no—
362C4—22no—
363C4—22no—
364C4—21no—
365C4—21no—
366C4—42no—
367C4—42no—
368C4—44no—
369C4—41no—
370C4—42no—
371C4—42no—
372C4—82no—
373C4—82no—
374C4—81no—
375C4#213212yes—
376C2—21no—
377C2—21no—
378C2Q8×#9911yescharacteristic
379C2C2×#213211yescharacteristic
380C2#3337211yescharacteristic
subgroupG/H
center, socleC22C22×#99
commutatorC22×C4C23×C4
Frattini, cosocleC22×Q8C24
prime radicalC23C22×C22⋊C4
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22 ◁ C2×C4⋊C4 ◁ C2×#557 ◁ G
LECSG ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1

Complements

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

NH ≅ G/Ntype
103. #356Q8semidirect

Maximal subgroups

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

subgroupindex
#365602
#225172
#133032
#133012
#128022
C4×#3562