Finite Group Databasebeta

← #26375749

Subgroups — #26375749

Order 12032.

2446 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D47C211yesmaximal, characteristic
2#20246355C213yesmaximal, complemented
3#20246351C213yesmaximal, complemented
4#20246309C211yesmaximal, characteristic, complemented
5#20246256C216yesmaximal
6Q8×(C2×D47)⋊C4C212yesmaximal
7#20245859C216yesmaximal, complemented
8#20245781C213yesmaximal
9C4×Q8×Dic47C212yesmaximal, complemented
10C4×#13093982C213yesmaximal
11C47×#2351C211yesmaximal, characteristic, complemented
12C22×Q8×D47C2211yescharacteristic
13C2×Q8×Dic47C2212yes—
14C2×Q8×Dic47C2211yescharacteristic
15C2×C4×Dic94C2213yes—
16C2×#13093989C2213yes—
17C4×Q8×D47C4116yescomplemented
18C2×#13093982C2216yes—
19C2×C4⋊C4×D47C2213yes—
20C2×C42×D47C2213yes—
21#15689029C2216yes—
22#15689028C2216yes—
23#15689022C2216yes—
24#15689010C22112yes—
25#15689004C22112yes—
26#15688990C2216yes—
27#15688989C2213yes—
28#15688985C2216yes—
29#15688980C2216yes—
30C4⋊C4×Dic47C2216yes—
31C4⋊C4×Dic47C2216yes—
32C4⋊C4×Dic47C2216yes—
33C4×C4⋊Dic47C2213yes—
34C4×C4⋊Dic47C2216yes—
35C4×(C2×D47)⋊C4C2216yes—
36#15688970C2213yes—
37#15688970C2211yescharacteristic
38#15688968C2216yes—
39#15688968C2213yes—
40#15688968C2213yes—
41C42×Dic47C2216yes—
42C2×C4×Q8×C47C2212yes—
43C2×C4×Q8×C47C2211yescharacteristic
44C47×#389C2216yes—
45C47×#382C2213yes—
46C4×C4⋊C4×C47C2213yes—
47C22×Dic94C2313yes—
48C2×Q8×D47—24no—
49C2×Q8×D47—24no—
50C2×Q8×D47C2×C418yes—
51C22×C4×D47C2311yescharacteristic
52C22×C4×D47C2313yes—
53C22×C4×D47C2313yes—
54Q8×Dic47—28no—
55Q8×Dic47C2×C418yes—
56C4×Dic94C2×C4124yes—
57#13093989C2×C4124yes—
58#13093982—224no—
59C2×C4⋊Dic47C2316yes—
60C2×C4⋊Dic47C2316yes—
61C2×C4⋊Dic47C2313yes—
62C2×Dic47⋊C4C2316yes—
63C4⋊C4×D47C2×C4124yes—
64C2×(C2×D47)⋊C4C2316yes—
65C2×(C2×D47)⋊C4C2312yes—
66C2×C4×Dic47C2312yes—
67C2×C4×Dic47C2316yes—
68C2×C4×Dic47C2316yes—
69C2×C4×Dic47C2316yes—
70C2×C4×Dic47C2316yes—
71C2×C4×Dic47C2313yes—
72C2×C4×Dic47C2313yes—
73C2×C4×Dic47C2313yes—
74C42×D47C2×C4124yes—
75C47⋊#95C2311yescharacteristic
76C47⋊#95C2313yes—
77C47⋊#95C2316yes—
78C47⋊#95C2316yes—
79C47⋊#95C23112yes—
80C47⋊#95C2316yes—
81C47⋊#95C23112yes—
82C22×Q8×C47C2311yescharacteristic
83C4×Q8×C47—28no—
84C4×Q8×C47C2×C418yes—
85C2×C4⋊C4×C47C2316yes—
86C2×C4⋊C4×C47C2316yes—
87C2×C4⋊C4×C47C2313yes—
88C2×C42×C47C2316yes—
89C2×C42×C47C2313yes—
90#95×C47C2311yescharacteristic
91#95×C47C2313yes—
92#95×C47C2316yes—
93C23×D47C2411yescharacteristic
94C2×Q8×C47C22×C414yes—
95C2×Q8×C47#4114yes—
96C2×Q8×C47#4114yes—
97C22×C4×C47C2416yes—
98C22×C4×C47C2411yescharacteristic
99C22×C4×C47C2413yes—
100C22×C4×C47C2412yes—
101C22×C4×C47C2413yes—
102C22×Dic47C2416yes—
103C22×Dic47C2413yes—
104C22×Dic47C2413yes—
105C22×Dic47C2412yes—
106C22×Dic47C2411yescharacteristic
107Q8×D47—232no—
108C2×C4×D47—212no—
109C2×C4×D47—26no—
110C2×C4×D47—26no—
111C2×C4×D47—24no—
112C2×C4×D47C22×C414yes—
113C2×C4×D47C22×C4112yes—
114C2×C4×D47C22×C4112yes—
115C2×Dic94—26no—
116C2×Dic94—26no—
117C2×Dic94C22×C4112yes—
118C4⋊C4×C47—212no—
119C4⋊C4×C47—212no—
120C4⋊C4×C47C22×C4112yes—
121C42×C47—212no—
122C42×C47C22×C4112yes—
123(C2×D47)⋊C4—212no—
124(C2×D47)⋊C4#4118yes—
125C4⋊Dic47—212no—
126C4⋊Dic47—212no—
127C4⋊Dic47C22×C4112yes—
128Dic47⋊C4C22×C4124yes—
129C4×Dic47—212no—
130C4×Dic47—212no—
131C4×Dic47—212no—
132C4×Dic47—24no—
133C4×Dic47—212no—
134C4×Dic47C22×C4112yes—
135C4×Dic47C22×C4112yes—
136C4×Dic47C22×C4112yes—
137C23×C47C2511yescharacteristic
138C22×D47—21no—
139C22×D47—21no—
140C22×D47—22no—
141C22×D47C23×C412yes—
142C22×D47C2×#4114yes—
143Q8×C47#117116yescomplemented
144C2×C4×C47—26no—
145C2×C4×C47—22no—
146C2×C4×C47—22no—
147C2×C4×C47—22no—
148C2×C4×C47—26no—
149C2×C4×C47—26no—
150C2×C4×C47—26no—
151C2×C4×C47C23×C412yes—
152C2×C4×C47C23×C416yes—
153C2×C4×C47C23×C416yes—
154C2×C4×C47C2×#4114yes—
155C2×C4×C47C2×#4116yes—
156C2×C4×C47C2×#4116yes—
157C2×Dic47—22no—
158C2×Dic47—26no—
159C2×Dic47—23no—
160C2×Dic47—23no—
161C2×Dic47—22no—
162C2×Dic47—26no—
163C2×Dic47—26no—
164C2×Dic47—26no—
165C2×Dic47C23×C416yes—
166C2×Dic47C23×C416yes—
167C2×Dic47C23×C412yes—
168C2×Dic47C2×#4114yes—
169C2×Dic472−1+414yes—
170C4×D47—224no—
171C4×D47—224no—
172C4×D47—28no—
173Dic94—224no—
174#34128—471nomaximal, Sylow
175C22×C47C24×C411yescharacteristic
176C22×C47C22×#4112yes—
177C22×C47C22×#4111yescharacteristic
178C22×C47C22×#4111yescharacteristic
179C22×C47C2×2−1+412yes—
180C2×D47—22no—
181C2×D47—22no—
182C2×D47—24no—
183C2×D47—24no—
184C2×D47C4×#4114yes—
185C4×C47—26no—
186C4×C47—22no—
187C4×C47—24no—
188C4×C47—46no—
189C4×C47C2×#117112yes—
190Dic47—26no—
191Dic47—26no—
192Dic47—24no—
193Dic47—46no—
194Dic47#51914yes—
195C22×C4×Q8—471no—
196#2356—473no—
197#2356—473no—
198#2351—471no—
199#2351D4711yescharacteristic, complemented
200#2344—476no—
201Q8×C22⋊C4—472no—
202#2295—476no—
203#2283—473no—
204C42×Q8—472no—
205C4×#117—473no—
206C2×C47C22×#11711yescharacteristic
207C2×C47C2×C4×#4111yescharacteristic
208C2×C47C2×#51911yescharacteristic
209C2×C47#348212yes—
210C2×C47#348212yes—
211D47—24no—
212C23×Q8—471no—
213C2×C4×Q8—471no—
214C2×C4×Q8—4716no—
215C2×C4×Q8—472no—
216C2×C4×Q8—473no—
217C2×C4×Q8—473no—
218C2×C4×Q8C2×D4712yes—
219C2×C4×Q8C2×D4711yescharacteristic
220C2×#117—476no—
221C22×C4⋊C4—473no—
222C22×C42—473no—
223#391—476no—
224#391—476no—
225#389—476no—
226#389C2×D4716yes—
227#387—4712no—
228#387—4712no—
229#383—476no—
230#382—473no—
231#382C2×D4713yes—
232#381—476no—
233#380—476no—
234C4×C4⋊C4—476no—
235C4×C4⋊C4—476no—
236C4×C4⋊C4—476no—
237C4×C4⋊C4—476no—
238C4×C4⋊C4—473no—
239C4×C4⋊C4C2×D4713yes—
240C4×C22⋊C4—476no—
241C4×C22⋊C4—471no—
242C4×C22⋊C4—473no—
243#376—473no—
244#376—473no—
245#376—476no—
246C43—476no—
247C47#3412811yescharacteristic, complemented, Sylow
248C22×Q8—478no—
249C22×Q8—473no—
250C22×Q8—944no—
251C22×Q8—944no—
252C22×Q8C22×D4711yescharacteristic
253C23×C4—473no—
254C23×C4—473no—
255C23×C4—471no—
256C4×Q8—28no—
257C4×Q8—478no—
258C4×Q8—4724no—
259C4×Q8—4724no—
260C4×Q8—948no—
261C4×Q8C4×D4718yes—
262#117—9424no—
263C2×C4⋊C4—476no—
264C2×C4⋊C4—473no—
265C2×C4⋊C4—4724no—
266C2×C4⋊C4—476no—
267C2×C4⋊C4—476no—
268C2×C4⋊C4C22×D4716yes—
269C2×C4⋊C4C22×D4716yes—
270C2×C4⋊C4C22×D4713yes—
271C2×C22⋊C4—472no—
272C2×C22⋊C4—476no—
273C2×C42—473no—
274C2×C42—473no—
275C2×C42—473no—
276C2×C42—4724no—
277C2×C42—476no—
278C2×C42—476no—
279C2×C42—476no—
280C2×C42—472no—
281C2×C42—476no—
282C2×C42C22×D4716yes—
283C2×C42C22×D4713yes—
284#95—471no—
285#95—473no—
286#95—476no—
287#95—476no—
288#95—4712no—
289#95—476no—
290#95—4712no—
291#95C22×D4711yescharacteristic
292#95C22×D4713yes—
293#95C22×D4716yes—
294C24—471no—
295C2×Q8—4712no—
296C2×Q8—9432no—
297C2×Q8—946no—
298C2×Q8—946no—
299C2×Q8C2×C4×D4714yes—
300C2×Q8#1062535214yes—
301C2×Q8#1062535414yes—
302C22×C4—471no—
303C22×C4—473no—
304C22×C4—472no—
305C22×C4—473no—
306C22×C4—474no—
307C22×C4—4712no—
308C22×C4—4712no—
309C22×C4—476no—
310C22×C4—944no—
311C22×C4—9412no—
312C22×C4—946no—
313C22×C4—946no—
314C22×C4C23×D4716yes—
315C22×C4C23×D4711yescharacteristic
316C22×C4C23×D4713yes—
317C22×C4C23×D4712yes—
318C22×C4C23×D4713yes—
319C4⋊C4—212no—
320C4⋊C4—212no—
321C4⋊C4—4724no—
322C4⋊C4—4712no—
323C4⋊C4—9412no—
324C4⋊C4—9412no—
325C4⋊C4C2×C4×D47112yes—
326C22⋊C4—478no—
327C22⋊C4—9412no—
328C42—212no—
329C42—4712no—
330C42—4712no—
331C42—4712no—
332C42—944no—
333C42—9412no—
334C42—9412no—
335C42—9412no—
336C42—9412no—
337C42C2×C4×D47112yes—
338C23—472no—
339C23—474no—
340C23—941no—
341C23—941no—
342C23—942no—
343C23C24×D4711yescharacteristic
344Q8—9424no—
345Q8#13093982116yescomplemented
346C2×C4—26no—
347C2×C4—22no—
348C2×C4—22no—
349C2×C4—22no—
350C2×C4—26no—
351C2×C4—26no—
352C2×C4—26no—
353C2×C4—472no—
354C2×C4—474no—
355C2×C4—476no—
356C2×C4—476no—
357C2×C4—474no—
358C2×C4—948no—
359C2×C4—9424no—
360C2×C4—946no—
361C2×C4—943no—
362C2×C4—942no—
363C2×C4—943no—
364C2×C4—942no—
365C2×C4—9424no—
366C2×C4—946no—
367C2×C4—946no—
368C2×C4—946no—
369C2×C4C22×C4×D4712yes—
370C2×C4C22×C4×D4716yes—
371C2×C4C22×C4×D4716yes—
372C2×C4#41×D4714yes—
373C2×C4C2×#1062535216yes—
374C2×C4C2×#1062535416yes—
375C22—474no—
376C22—942no—
377C22—942no—
378C22—944no—
379C22—944no—
380C22C23×C4×D4711yescharacteristic
381C22C2×#41×D4712yes—
382C22C22×#1062535211yescharacteristic
383C22C22×#1062535411yescharacteristic
384C222−1+4×D4712yes—
385C4—26no—
386C4—22no—
387C4—24no—
388C4—46no—
389C4—474no—
390C4—946no—
391C4—946no—
392C4—944no—
393C4—1886no—
394C4C2×#13093982112yes—
395C2—944no—
396C2C22×#1309398211yescharacteristic
397C2C4×#41×D4711yescharacteristic
398C2#519×D4711yescharacteristic
399C2#2025700412yes—
400C2#2025700812yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2351C2
socle, cosocleC23×C47C25
prime radicalC23×D47C24
hypercenter#2351D47
hypocenterC47#34128
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2351

Complements

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

NH ≅ G/Ntype
2. #20246355C2semidirect
3. #20246351C2semidirect
4. #20246309C2semidirect
7. #20245859C2semidirect
9. C4×Q8×Dic47C2semidirect
11. C47×#2351C2semidirect
17. C4×Q8×D47C4semidirect
143. Q8×C47#117semidirect
199. #2351D47semidirect
247. C47#34128semidirect
345. Q8#13093982semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D472
#202463552
#202463512
#202463092
#202462562
Q8×(C2×D47)⋊C42
#202458592
#202457812
C4×Q8×Dic472
C4×#130939822
C47×#23512
#3412847