Finite Group Databasebeta

← #1494813

Subgroups — #1494813

Order 512.

3056 conjugacy classes of subgroups.

The subgroup lattice has 407 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#2732C213yesmaximal, complemented
2C2×#2423C213yesmaximal, complemented
3#15765C216yesmaximal, complemented
4#15505C216yesmaximal, complemented
5#15155C213yesmaximal, complemented
6#15145C213yesmaximal, complemented
7#14230C213yesmaximal, complemented
8#14228C213yesmaximal
9#13826C211yesmaximal, characteristic, complemented
10C22×C22⋊Q8C2213yes—
11C22×C4×Q8C2211yescharacteristic
12#2732—224no—
13#2423—224no—
14C2×#399C2213yes—
15C2×#399C2216yes—
16C2×#399C2216yes—
17C2×#399C2216yes—
18C2×#398C2216yes—
19C2×#398C2216yes—
20C2×#396C2216yes—
21C2×#392C2211yescharacteristic
22C2×#391C2213yes—
23C2×#390C2213yes—
24C2×#390C2216yes—
25C2×#389C2213yes—
26C2×#389C2213yes—
27C2×#388C2216yes—
28C2×#387C2216yes—
29C2×#386C2216yes—
30C2×#385C2216yes—
31C2×#384C2213yes—
32C2×#384C2216yes—
33C2×#380C2213yes—
34C2×C4×C22⋊C4C2213yes—
35#1445C2213yes—
36#1444C2213yes—
37#1444C2216yes—
38#1444C2212yes—
39#1444C2211yescharacteristic
40#1439C2216yes—
41#1439C2213yes—
42#1439C2213yes—
43#1438C2213yes—
44#1437C2216yes—
45#1436C2216yes—
46C4⋊#95C2216yes—
47C4×#95C2213yes—
48C4×#95C2213yes—
49C23×Q8C2311yescharacteristic
50C23×D4C2313yes—
51C24×C4C2313yes—
52C2×C22⋊Q8—212no—
53C2×C22⋊Q8—224no—
54C2×C4×Q8—212no—
55C22×C4⋊C4C2316yes—
56C22×C4⋊C4C2313yes—
57C22×C4⋊C4C2313yes—
58C22×C4⋊C4C2316yes—
59C22×C4⋊C4C2313yes—
60C22×C4⋊C4C2316yes—
61C22×C4⋊C4C2313yes—
62C22×C4⋊C4C2316yes—
63C22×C4⋊C4C2316yes—
64C22×C4⋊C4C2313yes—
65C22×C4⋊C4C2313yes—
66C22×C22⋊C4C2316yes—
67C22×C22⋊C4C2316yes—
68C22×C22⋊C4C2313yes—
69C22×C22⋊C4C2313yes—
70C22×C22⋊C4C2316yes—
71C22×C22⋊C4C2316yes—
72C22×C22⋊C4C2316yes—
73C22×C42C2313yes—
74C22×C42C2316yes—
75C22×C42C2313yes—
76#399—224no—
77#399—224no—
78#399—212no—
79#399—412no—
80#398—224no—
81#398—224no—
82#396—224no—
83#392Q818yescomplemented
84#391D4124yescomplemented
85#390—212no—
86#390—412no—
87#389—212no—
88#389—212no—
89#388—224no—
90#387—224no—
91#386—224no—
92#385—224no—
93#384—212no—
94#384—412no—
95#380—212no—
96C4×C22⋊C4—212no—
97C2×#95C2316yes—
98C2×#95C2313yes—
99C2×#95C2316yes—
100C2×#95C2316yes—
101C2×#95C2313yes—
102C2×#95C2313yes—
103C2×#95C2313yes—
104C2×#95C2316yes—
105C2×#95C2312yes—
106C2×#95C2316yes—
107C2×#95C2316yes—
108C2×#95C2311yescharacteristic
109C2×#95C2311yescharacteristic
110C25C2413yes—
111C22×Q8—43no—
112C22×Q8—41no—
113C22×Q8C2×D4112yes—
114C22×D4—26no—
115C22×D4—26no—
116C22×D4—46no—
117C22×D4—46no—
118C22×D4C2×Q8112yes—
119C23×C4—26no—
120C23×C4—212no—
121C23×C4—46no—
122C23×C4—43no—
123C23×C4—43no—
124C23×C4C2416yes—
125C23×C4C2413yes—
126C23×C4C2413yes—
127C23×C4C2416yes—
128C23×C4C2411yescharacteristic
129C23×C4C2413yes—
130C23×C4C2413yes—
131C23×C4C2413yes—
132C22⋊Q8—448no—
133C4×Q8—416no—
134C2×C4⋊C4—26no—
135C2×C4⋊C4—212no—
136C2×C4⋊C4—26no—
137C2×C4⋊C4—212no—
138C2×C4⋊C4—212no—
139C2×C4⋊C4—212no—
140C2×C4⋊C4—212no—
141C2×C4⋊C4—212no—
142C2×C4⋊C4—26no—
143C2×C4⋊C4—212no—
144C2×C4⋊C4—26no—
145C2×C4⋊C4—26no—
146C2×C4⋊C4—212no—
147C2×C4⋊C4—212no—
148C2×C4⋊C4—26no—
149C2×C4⋊C4—212no—
150C2×C4⋊C4—26no—
151C2×C4⋊C4—212no—
152C2×C4⋊C4—43no—
153C2×C4⋊C4—43no—
154C2×C4⋊C4—46no—
155C2×C4⋊C4—46no—
156C2×C4⋊C4—46no—
157C2×C4⋊C4—46no—
158C2×C4⋊C4—46no—
159C2×C4⋊C4—46no—
160C2×C4⋊C4—46no—
161C2×C4⋊C4—46no—
162C2×C4⋊C4C2×D4112yes—
163C2×C22⋊C4—26no—
164C2×C22⋊C4—26no—
165C2×C22⋊C4—212no—
166C2×C22⋊C4—212no—
167C2×C22⋊C4—212no—
168C2×C22⋊C4—26no—
169C2×C22⋊C4—212no—
170C2×C22⋊C4—212no—
171C2×C22⋊C4—26no—
172C2×C22⋊C4—26no—
173C2×C22⋊C4—212no—
174C2×C22⋊C4—212no—
175C2×C22⋊C4—212no—
176C2×C22⋊C4—212no—
177C2×C22⋊C4—46no—
178C2×C22⋊C4—46no—
179C2×C22⋊C4—46no—
180C2×C22⋊C4—46no—
181C2×C22⋊C4—46no—
182C2×C22⋊C4—46no—
183C2×C22⋊C4C2×Q8112yes—
184C2×C42—26no—
185C2×C42—212no—
186C2×C42—212no—
187C2×C42—212no—
188C2×C42—46no—
189C2×C42—46no—
190C2×C42C2×D4112yes—
191#95—26no—
192#95—26no—
193#95—212no—
194#95—212no—
195#95—212no—
196#95—212no—
197#95—26no—
198#95—26no—
199#95—42no—
200#95C2×D4124yes—
201#95C2×D4124yes—
202#95C2×Q814yes—
203#95#4114yes—
204C24—23no—
205C24—26no—
206C24—23no—
207C24—26no—
208C24—23no—
209C24—23no—
210C24—26no—
211C24—23no—
212C24—26no—
213C24—23no—
214C24C22×Q816yes—
215C24C2511yescharacteristic
216C2×Q8—412no—
217C2×Q8—412no—
218C2×Q8#127116yes—
219C2×D4—224no—
220C2×D4—424no—
221C2×D4—86no—
222C2×D4—86no—
223C2×D4—812no—
224C22×C4—26no—
225C22×C4—26no—
226C22×C4—26no—
227C22×C4—26no—
228C22×C4—26no—
229C22×C4—23no—
230C22×C4—23no—
231C22×C4—23no—
232C22×C4—26no—
233C22×C4—26no—
234C22×C4—26no—
235C22×C4—26no—
236C22×C4—23no—
237C22×C4—26no—
238C22×C4—26no—
239C22×C4—23no—
240C22×C4—26no—
241C22×C4—23no—
242C22×C4—26no—
243C22×C4—26no—
244C22×C4—26no—
245C22×C4—26no—
246C22×C4—26no—
247C22×C4—23no—
248C22×C4—23no—
249C22×C4—26no—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—26no—
253C22×C4—23no—
254C22×C4—26no—
255C22×C4—23no—
256C22×C4—412no—
257C22×C4—412no—
258C22×C4—412no—
259C22×C4—812no—
260C22×C4—812no—
261C22×C4C22×D4112yes—
262C22×C4C22×D4112yes—
263C22×C4C22×D416yes—
264C22×C4C22×D416yes—
265C22×C4C22×D416yes—
266C22×C4C22×Q812yes—
267C22×C4C22×Q816yes—
268C22×C4C2×#4116yes—
269C22×C4H(5,𝔽2)16yes—
270C22×C42−1+416yes—
271C4⋊C4—412no—
272C4⋊C4—412no—
273C4⋊C4—412no—
274C4⋊C4—424no—
275C4⋊C4—424no—
276C4⋊C4—412no—
277C4⋊C4—86no—
278C4⋊C4—812no—
279C4⋊C4—86no—
280C4⋊C4—812no—
281C4⋊C4—812no—
282C22⋊C4—224no—
283C22⋊C4—424no—
284C22⋊C4—412no—
285C22⋊C4—424no—
286C22⋊C4—812no—
287C22⋊C4—812no—
288C22⋊C4—812no—
289C42—412no—
290C42—424no—
291C42—412no—
292C23—26no—
293C23—212no—
294C23—26no—
295C23—26no—
296C23—212no—
297C23—43no—
298C23—43no—
299C23—46no—
300C23—46no—
301C23—43no—
302C23—43no—
303C23—46no—
304C23—43no—
305C23—46no—
306C23—43no—
307C23—46no—
308C23—43no—
309C23—46no—
310C23—46no—
311C23—46no—
312C23—46no—
313C23—46no—
314C23—43no—
315C23C23×D413yes—
316C23C23×Q811yescharacteristic
317C23C22×#4111yescharacteristic
318C23C2×H(5,𝔽2)13yes—
319C23C2×2−1+413yes—
320C23#58511yescharacteristic
321C23#58513yes—
322Q8—416no—
323D4—824no—
324C2×C4—26no—
325C2×C4—26no—
326C2×C4—26no—
327C2×C4—26no—
328C2×C4—212no—
329C2×C4—212no—
330C2×C4—26no—
331C2×C4—26no—
332C2×C4—212no—
333C2×C4—212no—
334C2×C4—212no—
335C2×C4—212no—
336C2×C4—26no—
337C2×C4—212no—
338C2×C4—212no—
339C2×C4—212no—
340C2×C4—41no—
341C2×C4—43no—
342C2×C4—46no—
343C2×C4—46no—
344C2×C4—46no—
345C2×C4—46no—
346C2×C4—43no—
347C2×C4—46no—
348C2×C4—43no—
349C2×C4—43no—
350C2×C4—46no—
351C2×C4—46no—
352C2×C4—46no—
353C2×C4—46no—
354C2×C4—46no—
355C2×C4—46no—
356C2×C4—43no—
357C2×C4—43no—
358C2×C4—46no—
359C2×C4—46no—
360C2×C4—43no—
361C2×C4—46no—
362C2×C4—46no—
363C2×C4—824no—
364C2×C4C2×#127112yes—
365C2×C4D4×Q8112yes—
366C2×C4#550112yes—
367C22—46no—
368C22—412no—
369C22—46no—
370C22—412no—
371C22—46no—
372C22—86no—
373C22—83no—
374C22—83no—
375C22—83no—
376C22—86no—
377C22—83no—
378C22C22×#12711yescharacteristic
379C22C2×D4×Q813yes—
380C22C2×#55013yes—
381C22Q8×#4111yescharacteristic
382C22#348513yes—
383C22#348513yes—
384C22#349813yes—
385C22#349913yes—
386C22#350113yes—
387C22#351513yes—
388C22#353516yes—
389C22#354713yes—
390C4—212no—
391C4—212no—
392C4—212no—
393C4—412no—
394C4—42no—
395C4—83no—
396C4—83no—
397C4—86no—
398C2—86no—
399C2#3546311yescharacteristic
400C2Q8×#12711yescharacteristic
401C2#3659211yescharacteristic
402C2#4349413yes—
403C2#4349713yes—
404C2#5154613yes—
405C2#5905213yes—
subgroupG/H
center, commutator, Frattini, socle, cosocleC24C25
prime radicalC2×#392C22
LCSG ▷ C24 ▷ C1
UCSC1 ◁ C24 ◁ G
LECSG ▷ C24 ▷ C1

Complements

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

NH ≅ G/Ntype
1. C2×#2732C2semidirect
2. C2×#2423C2semidirect
3. #15765C2semidirect
4. #15505C2semidirect
5. #15155C2semidirect
6. #15145C2semidirect
7. #14230C2semidirect
9. #13826C2semidirect
83. #392Q8semidirect
84. #391D4semidirect

Maximal subgroups

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

subgroupindex
C2×#27322
C2×#24232
#157652
#155052
#151552
#151452
#142302
#142282
#138262