Finite Group Databasebeta

← #26376510

Subgroups — #26376510

Order 12032.

2350 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic47C211yesmaximal, characteristic
2#20246344C216yesmaximal, complemented
3Q8×Dic47⋊C4C212yesmaximal, complemented
4#20246259C213yesmaximal
5#20246255C213yesmaximal
6Q8×C22⋊Dic47C211yesmaximal, characteristic
7#20246033C216yesmaximal, complemented
8#20245790C213yesmaximal
9#20245782C213yesmaximal
10#20245753C212yesmaximal, complemented
11Q8×C22⋊C4×C47C211yesmaximal, characteristic
12C2×Q8×Dic47C4116yescomplemented
13C2×Q8×Dic47C2212yes—
14C2×Q8×Dic47C2213yes—
15C2×Q8×Dic47C2211yescharacteristic
16C2×Q8×Dic47C2213yes—
17C2×Q8×Dic47C2211yescharacteristic
18C2×#13093980C2216yes—
19C22×C4⋊Dic47C2213yes—
20C22×C4×Dic47C2213yes—
21#15689030C2216yes—
22#15689021C2216yes—
23#15689019C2216yes—
24#15689008C22112yes—
25#15689006C22112yes—
26#15688992C2216yes—
27#15688990C2216yes—
28#15688987C2216yes—
29#15688987C2216yes—
30#15688984C2213yes—
31#15688984C2213yes—
32#15688983C2213yes—
33#15688981C2211yescharacteristic
34#15688981C2213yes—
35C4⋊C4×Dic47C2216yes—
36C4⋊C4×Dic47C2216yes—
37C4×Dic47⋊C4C2216yes—
38C4×C4⋊Dic47C2216yes—
39#15688974C2216yes—
40C4×C22⋊Dic47C2213yes—
41#15688969C2216yes—
42C23×Q8×C47C2211yescharacteristic
43C2×C4×Q8×C47C2212yes—
44C47×#391C2216yes—
45C47×#380C2213yes—
46C4×C22⋊C4×C47C2213yes—
47C23×Dic47C2311yescharacteristic
48C23×Dic47C2313yes—
49Q8×Dic47—28no—
50Q8×Dic47C2×C4124yes—
51Q8×Dic47C2×C4124yes—
52Q8×Dic47C2×C418yes—
53Q8×Dic47C2×C418yes—
54#13093980—412no—
55C2×C4⋊Dic47C2×C4124yes—
56C2×C4⋊Dic47C2316yes—
57C2×C4⋊Dic47C2313yes—
58C2×C4⋊Dic47C2316yes—
59C2×C4⋊Dic47C2313yes—
60C2×Dic47⋊C4C2316yes—
61C2×Dic47⋊C4C2312yes—
62C2×Dic47⋊C4C23112yes—
63C2×C22⋊Dic47C2311yescharacteristic
64C2×C22⋊Dic47C2313yes—
65C2×C4×Dic47C2×C4124yes—
66C2×C4×Dic47C2316yes—
67C2×C4×Dic47C23112yes—
68C2×C4×Dic47C2313yes—
69C2×C4×Dic47C2313yes—
70C2×C4×Dic47C2313yes—
71C2×C4×Dic47C2313yes—
72C47⋊#95C2312yes—
73C47⋊#95C2316yes—
74C47⋊#95C2316yes—
75C47⋊#95C2316yes—
76C47⋊#95C2316yes—
77C47⋊#95C2316yes—
78C47⋊#95C2316yes—
79C47⋊#95C2316yes—
80C22×Q8×C47—28no—
81C22×Q8×C47C2×C418yes—
82C22×Q8×C47C2313yes—
83C22×Q8×C47C2311yescharacteristic
84C23×C4×C47C2313yes—
85C4×Q8×C47—28no—
86C2×C4⋊C4×C47C2316yes—
87C2×C22⋊C4×C47C2311yescharacteristic
88C2×C22⋊C4×C47C2313yes—
89C2×C42×C47C2316yes—
90#95×C47C2316yes—
91#95×C47C2316yes—
92C24×C47C2411yescharacteristic
93C2×Q8×C47—232no—
94C2×Q8×C47—212no—
95C2×Q8×C47C22×C4112yes—
96C2×Q8×C47C22×C414yes—
97C2×Q8×C47#4118yes—
98C22×C4×C47—212no—
99C22×C4×C47C22×C4112yes—
100C22×C4×C47C2412yes—
101C22×C4×C47C2416yes—
102C22×C4×C47C2413yes—
103C22×C4×C47C2413yes—
104C22×Dic47—26no—
105C22×Dic47—26no—
106C22×Dic47—24no—
107C22×Dic47C22×C414yes—
108C22×Dic47C22×C4112yes—
109C22×Dic47C2412yes—
110C22×Dic47C2416yes—
111C22×Dic47C2411yescharacteristic
112C22×Dic47C2413yes—
113C22×Dic47C2413yes—
114C22×Dic47C2411yescharacteristic
115C4⋊C4×C47—212no—
116C22⋊C4×C47—26no—
117C22⋊C4×C47—22no—
118C42×C47—212no—
119C22⋊Dic47—26no—
120C22⋊Dic47—22no—
121C4⋊Dic47—212no—
122C4⋊Dic47C22×C4124yes—
123C4⋊Dic47C22×C4112yes—
124C4⋊Dic47C22×C4112yes—
125Dic47⋊C4—212no—
126Dic47⋊C4—24no—
127Dic47⋊C4—412no—
128C4×Dic47—212no—
129C4×Dic47—412no—
130C4×Dic47C22×C4112yes—
131C4×Dic47C22×C4112yes—
132C4×Dic47C22×C4112yes—
133C4×Dic47C22×C4112yes—
134C23×C47—22no—
135C23×C47—22no—
136C23×C47C23×C412yes—
137C23×C47H(5,𝔽2)12yes—
138C23×C472−1+412yes—
139C23×C47C2511yescharacteristic
140Q8×C47—224no—
141Q8×C47#117116yescomplemented
142C2×C4×C47—26no—
143C2×C4×C47—224no—
144C2×C4×C47—22no—
145C2×C4×C47—26no—
146C2×C4×C47—26no—
147C2×C4×C47—26no—
148C2×C4×C47—22no—
149C2×C4×C47—22no—
150C2×C4×C47C23×C416yes—
151C2×C4×C47C23×C416yes—
152C2×C4×C47C2×#41112yes—
153C2×Dic47—224no—
154C2×Dic47—23no—
155C2×Dic47—23no—
156C2×Dic47—22no—
157C2×Dic47—23no—
158C2×Dic47—23no—
159C2×Dic47—28no—
160C2×Dic47—22no—
161C2×Dic47—26no—
162C2×Dic47—26no—
163C2×Dic47—26no—
164C2×Dic47—22no—
165C2×Dic47—22no—
166C2×Dic47C23×C412yes—
167C2×Dic47C23×C416yes—
168C2×Dic47C23×C416yes—
169C2×Dic47C23×C412yes—
170C2×Dic472−1+414yes—
171#34196—471nomaximal, Sylow
172C22×C47—22no—
173C22×C47—22no—
174C22×C47—24no—
175C22×C47—24no—
176C22×C47#52014yes—
177C22×C47C24×C411yescharacteristic
178C22×C47C22×#4112yes—
179C22×C47C2×H(5,𝔽2)11yescharacteristic
180C22×C47C2×2−1+412yes—
181C22×C47C2×2−1+411yescharacteristic
182C4×C47—26no—
183C4×C47—46no—
184C4×C47—42no—
185C4×C47C2×#117112yes—
186Dic47—22no—
187Dic47—26no—
188Dic47—26no—
189Dic47—46no—
190Dic47—42no—
191Dic47#51914yes—
192C22×C4×Q8—471no—
193#2355—476no—
194Q8×C4⋊C4—472no—
195#2344—473no—
196#2344—473no—
197Q8×C22⋊C4—471no—
198Q8×C22⋊C4D4711yescharacteristic
199#2320—476no—
200#2285—473no—
201#2283—473no—
202#2276—472no—
203C2×C47—24no—
204C2×C47C22×#11711yescharacteristic
205C2×C47C2×#51911yescharacteristic
206C2×C47C2×#52011yescharacteristic
207C2×C47#352212yes—
208C2×C47#352312yes—
209C23×Q8C2×D4711yescharacteristic
210C2×C4×Q8—4716no—
211C2×C4×Q8—473no—
212C2×C4×Q8—471no—
213C2×C4×Q8—473no—
214C2×C4×Q8—471no—
215C2×C4×Q8—472no—
216C2×C4×Q8C2×D4712yes—
217C2×#117—476no—
218C22×C4⋊C4—473no—
219C22×C42—473no—
220#391—476no—
221#391C2×D4716yes—
222#389—476no—
223#389—476no—
224#387—4712no—
225#387—4712no—
226#384—476no—
227#383—476no—
228#382—476no—
229#382—476no—
230#381—473no—
231#381—473no—
232#380—473no—
233#380—471no—
234#380—473no—
235#380C2×D4713yes—
236C4×C4⋊C4—476no—
237C4×C4⋊C4—476no—
238C4×C4⋊C4—476no—
239C4×C4⋊C4—476no—
240C4×C4⋊C4—476no—
241C4×C22⋊C4—473no—
242C4×C22⋊C4C2×D4713yes—
243#376—476no—
244C47#3419611yescharacteristic, complemented, Sylow
245C22×Q8—28no—
246C22×Q8C4×D4718yes—
247C22×Q8C22×D4713yes—
248C22×Q8C22×D4711yescharacteristic
249C23×C4—473no—
250C23×C4—471no—
251C23×C4C22×D4713yes—
252C4×Q8—28no—
253C4×Q8—4724no—
254C4×Q8—4724no—
255C4×Q8—478no—
256C4×Q8—478no—
257C4×Q8—948no—
258#117—18812no—
259C2×C4⋊C4—4724no—
260C2×C4⋊C4—473no—
261C2×C4⋊C4—476no—
262C2×C4⋊C4—473no—
263C2×C4⋊C4—476no—
264C2×C4⋊C4—472no—
265C2×C4⋊C4—476no—
266C2×C4⋊C4—4712no—
267C2×C4⋊C4C22×D4716yes—
268C2×C22⋊C4—473no—
269C2×C22⋊C4—471no—
270C2×C22⋊C4C22×D4713yes—
271C2×C22⋊C4C22×D4711yescharacteristic
272C2×C42—4724no—
273C2×C42—473no—
274C2×C42—473no—
275C2×C42—473no—
276C2×C42—473no—
277C2×C42—476no—
278C2×C42—4712no—
279C2×C42C22×D4716yes—
280#95—472no—
281#95—476no—
282#95—476no—
283#95—476no—
284#95—476no—
285#95—476no—
286#95—476no—
287#95—476no—
288#95C22×D4716yes—
289#95C22×D4716yes—
290C24C23×D4711yescharacteristic
291C2×Q8—232no—
292C2×Q8—212no—
293C2×Q8C2×C4×D47112yes—
294C2×Q8C2×C4×D4714yes—
295C2×Q8#1062535218yes—
296C22×C4—212no—
297C22×C4—471no—
298C22×C4—474no—
299C22×C4—4712no—
300C22×C4—471no—
301C22×C4—473no—
302C22×C4—473no—
303C22×C4—472no—
304C22×C4—476no—
305C22×C4—944no—
306C22×C4—946no—
307C22×C4—946no—
308C22×C4C2×C4×D47112yes—
309C22×C4C23×D4712yes—
310C22×C4C23×D4716yes—
311C22×C4C23×D4713yes—
312C22×C4C23×D4713yes—
313C4⋊C4—212no—
314C4⋊C4—4724no—
315C4⋊C4—4712no—
316C4⋊C4—4712no—
317C4⋊C4—944no—
318C4⋊C4—9412no—
319C4⋊C4—9412no—
320C4⋊C4—18812no—
321C22⋊C4—26no—
322C22⋊C4—22no—
323C22⋊C4—946no—
324C22⋊C4—942no—
325C42—212no—
326C42—4712no—
327C42—4712no—
328C42—4712no—
329C42—4712no—
330C42—9412no—
331C42—18812no—
332C23—22no—
333C23—22no—
334C23C22×C4×D4712yes—
335C23#1309407012yes—
336C23#1309407212yes—
337C23C24×D4711yescharacteristic
338Q8—224no—
339Q8#13093980116yescomplemented
340C2×C4—26no—
341C2×C4—224no—
342C2×C4—22no—
343C2×C4—26no—
344C2×C4—26no—
345C2×C4—26no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—472no—
349C2×C4—474no—
350C2×C4—472no—
351C2×C4—476no—
352C2×C4—476no—
353C2×C4—948no—
354C2×C4—9424no—
355C2×C4—943no—
356C2×C4—943no—
357C2×C4—942no—
358C2×C4—943no—
359C2×C4—943no—
360C2×C4—942no—
361C2×C4—946no—
362C2×C4—946no—
363C2×C4—946no—
364C2×C4—942no—
365C2×C4—942no—
366C2×C4C22×C4×D4716yes—
367C2×C4C22×C4×D4716yes—
368C2×C4C2×#10625352112yes—
369C22—22no—
370C22—22no—
371C22—24no—
372C22—24no—
373C22#1568960514yes—
374C22C23×C4×D4711yescharacteristic
375C22C22×#1062535212yes—
376C22C2×#1309407011yescharacteristic
377C222−1+4×D4712yes—
378C22C2×#1309407211yescharacteristic
379C4—26no—
380C4—46no—
381C4—42no—
382C4—474no—
383C4—942no—
384C4—946no—
385C4—946no—
386C4—1886no—
387C4—1882no—
388C4C2×#13093980112yes—
389C2—24no—
390C2C22×#1309398011yescharacteristic
391C2#519×D4711yescharacteristic
392C2C2×#1568960511yescharacteristic
393C2#2025746412yes—
394C2#2025747512yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingQ8×C22⋊C4×C47C2
socle, cosocleC23×C47C25
prime radicalC24×C47C24
hypercenterQ8×C22⋊C4D47
hypocenterC47#34196
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20246344C2semidirect
3. Q8×Dic47⋊C4C2semidirect
7. #20246033C2semidirect
10. #20245753C2semidirect
12. C2×Q8×Dic47C4semidirect
141. Q8×C47#117semidirect
244. C47#34196semidirect
339. Q8#13093980semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic472
#202463442
Q8×Dic47⋊C42
#202462592
#202462552
Q8×C22⋊Dic472
#202460332
#202457902
#202457822
#202457532
Q8×C22⋊C4×C472
#3419647