Finite Group Databasebeta

← #26375748

Subgroups — #26375748

Order 12032.

2446 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Q8×D47C211yesmaximal, characteristic
2#20246349C216yesmaximal, complemented
3#20246306C212yesmaximal, complemented
4#20246253C216yesmaximal
5Q8×(C2×D47)⋊C4C212yesmaximal
6#20245865C213yesmaximal, complemented
7#20245862C213yesmaximal, complemented
8#20245784C213yesmaximal
9C4×Q8×Dic47C211yesmaximal, characteristic, complemented
10C4×#13093981C213yesmaximal
11C42×Q8×C47C211yesmaximal, characteristic, complemented
12C22×Q8×D47C2211yescharacteristic
13C2×Q8×Dic47C2212yes—
14C2×Q8×Dic47C2211yescharacteristic
15C2×C4×Dic94C2213yes—
16C2×#13093989C2213yes—
17C4×Q8×D47C4116yescomplemented
18C2×#13093981C2216yes—
19C2×C4⋊C4×D47C2213yes—
20C2×C42×D47C2213yes—
21#15689029C2216yes—
22#15689028C2216yes—
23#15689021C2216yes—
24#15689019C2216yes—
25#15689007C22112yes—
26#15689005C22112yes—
27#15688990C2216yes—
28#15688988C2216yes—
29#15688985C2216yes—
30#15688980C2216yes—
31C4⋊C4×Dic47C2216yes—
32C4×Dic47⋊C4C2216yes—
33C4×Dic47⋊C4C2213yes—
34C4×Dic47⋊C4C2213yes—
35C4×C4⋊Dic47C2213yes—
36C4×(C2×D47)⋊C4C2213yes—
37C4×(C2×D47)⋊C4C2216yes—
38C4×(C2×D47)⋊C4C2211yescharacteristic
39#15688969C2216yes—
40#15688969C2216yes—
41C42×Dic47C2213yes—
42C2×C4×Q8×C47C2212yes—
43C2×C4×Q8×C47C2211yescharacteristic
44C4×C4⋊C4×C47C2213yes—
45C4×C4⋊C4×C47C2216yes—
46C43×C47C2213yes—
47C22×Dic94C2313yes—
48C2×Q8×D47—28no—
49C2×Q8×D47C2×C418yes—
50C22×C4×D47C2311yescharacteristic
51C22×C4×D47C2313yes—
52C22×C4×D47C2313yes—
53Q8×Dic47—28no—
54Q8×Dic47C2×C418yes—
55C4×Dic94C2×C4124yes—
56#13093989C2×C4124yes—
57#13093981—224no—
58C2×C4⋊Dic47C2316yes—
59C2×C4⋊Dic47C2313yes—
60C2×Dic47⋊C4C2316yes—
61C2×Dic47⋊C4C2316yes—
62C2×Dic47⋊C4C2316yes—
63C4⋊C4×D47C2×C4124yes—
64C2×(C2×D47)⋊C4C2316yes—
65C2×(C2×D47)⋊C4C2312yes—
66C2×C4×Dic47C2311yescharacteristic
67C2×C4×Dic47C2313yes—
68C2×C4×Dic47C2316yes—
69C2×C4×Dic47C2316yes—
70C2×C4×Dic47C2313yes—
71C2×C4×Dic47C2313yes—
72C2×C4×Dic47C2313yes—
73C42×D47C2×C4124yes—
74C47⋊#95C2312yes—
75C47⋊#95C2316yes—
76C47⋊#95C2316yes—
77C47⋊#95C2316yes—
78C47⋊#95C23112yes—
79C47⋊#95C23112yes—
80C47⋊#95C2316yes—
81C22×Q8×C47C2311yescharacteristic
82C4×Q8×C47—28no—
83C4×Q8×C47C2×C418yes—
84C2×C4⋊C4×C47C2316yes—
85C2×C4⋊C4×C47C2313yes—
86C2×C42×C47C2313yes—
87C2×C42×C47C2316yes—
88C2×C42×C47C2311yescharacteristic
89C2×C42×C47C2316yes—
90C2×C42×C47C2313yes—
91#95×C47C2316yes—
92C23×D47C2411yescharacteristic
93C2×Q8×C47C22×C414yes—
94C2×Q8×C47#4118yes—
95C22×C4×C47C2416yes—
96C22×C4×C47C2413yes—
97C22×C4×C47C2412yes—
98C22×C4×C47C2413yes—
99C22×C4×C47C2411yescharacteristic
100C22×Dic47C2416yes—
101C22×Dic47C2411yescharacteristic
102C22×Dic47C2413yes—
103C22×Dic47C2413yes—
104C22×Dic47C2412yes—
105Q8×D47—232no—
106C2×C4×D47—212no—
107C2×C4×D47—212no—
108C2×C4×D47—22no—
109C2×C4×D47—22no—
110C2×C4×D47C22×C414yes—
111C2×C4×D47C22×C4112yes—
112C2×C4×D47C22×C4112yes—
113C2×Dic94—212no—
114C2×Dic94C22×C4112yes—
115C4⋊C4×C47—212no—
116C4⋊C4×C47C22×C4112yes—
117C42×C47—212no—
118C42×C47—212no—
119C42×C47—22no—
120C42×C47—26no—
121C42×C47C22×C4112yes—
122(C2×D47)⋊C4—212no—
123(C2×D47)⋊C4#4118yes—
124C4⋊Dic47—212no—
125C4⋊Dic47C22×C4112yes—
126Dic47⋊C4—212no—
127Dic47⋊C4—212no—
128Dic47⋊C4C22×C4124yes—
129C4×Dic47—212no—
130C4×Dic47—22no—
131C4×Dic47—26no—
132C4×Dic47—212no—
133C4×Dic47C22×C4112yes—
134C4×Dic47C22×C4112yes—
135C4×Dic47C22×C4112yes—
136C23×C47C2511yescharacteristic
137C22×D47—22no—
138C22×D47—21no—
139C22×D47—21no—
140C22×D47C23×C412yes—
141C22×D47C2×#4114yes—
142Q8×C47#117116yescomplemented
143C2×C4×C47—26no—
144C2×C4×C47—22no—
145C2×C4×C47—22no—
146C2×C4×C47—26no—
147C2×C4×C47—26no—
148C2×C4×C47—26no—
149C2×C4×C47C23×C416yes—
150C2×C4×C47C23×C416yes—
151C2×C4×C47C23×C412yes—
152C2×C4×C47C2×#4114yes—
153C2×C4×C47C2×#41112yes—
154C2×C4×C472−1+412yes—
155C2×C4×C472−1+412yes—
156C2×Dic47—22no—
157C2×Dic47—26no—
158C2×Dic47—26no—
159C2×Dic47—22no—
160C2×Dic47—21no—
161C2×Dic47—21no—
162C2×Dic47—26no—
163C2×Dic47—26no—
164C2×Dic47—26no—
165C2×Dic47C23×C412yes—
166C2×Dic47C23×C416yes—
167C2×Dic47C23×C416yes—
168C2×Dic47C2×#4114yes—
169C4×D47—224no—
170C4×D47—224no—
171C4×D47—28no—
172Dic94—224no—
173#34128—471nomaximal, Sylow
174C22×C47C24×C411yescharacteristic
175C22×C47C22×#4112yes—
176C22×C47C22×#4112yes—
177C22×C47C2×2−1+411yescharacteristic
178C22×C47C2×2−1+411yescharacteristic
179C2×D47—22no—
180C2×D47—22no—
181C2×D47—24no—
182C2×D47—24no—
183C2×D47C4×#4114yes—
184C4×C47—26no—
185C4×C47—24no—
186C4×C47—46no—
187C4×C47C2×#117112yes—
188C4×C47#51914yes—
189Dic47—26no—
190Dic47—26no—
191Dic47—22no—
192Dic47—24no—
193Dic47—46no—
194C22×C4×Q8—471no—
195#2356—476no—
196#2351—472no—
197#2344—476no—
198Q8×C22⋊C4—472no—
199#2295—473no—
200#2295—473no—
201#2283—473no—
202C42×Q8—471no—
203C42×Q8D4711yescharacteristic, complemented
204C4×#117—473no—
205C2×C47C22×#11711yescharacteristic
206C2×C47C2×C4×#4111yescharacteristic
207C2×C47C2×#51911yescharacteristic
208C2×C47#348212yes—
209C2×C47#348212yes—
210D47—24no—
211C23×Q8—471no—
212C2×C4×Q8—471no—
213C2×C4×Q8—4716no—
214C2×C4×Q8—472no—
215C2×C4×Q8—473no—
216C2×C4×Q8—473no—
217C2×C4×Q8C2×D4712yes—
218C2×C4×Q8C2×D4711yescharacteristic
219C2×#117—476no—
220C22×C4⋊C4—473no—
221C22×C42—473no—
222#391—476no—
223#391—476no—
224#389—476no—
225#389—476no—
226#387—4712no—
227#387—4712no—
228#383—476no—
229#382—476no—
230#381—476no—
231#380—476no—
232C4×C4⋊C4—476no—
233C4×C4⋊C4—473no—
234C4×C4⋊C4—476no—
235C4×C4⋊C4—473no—
236C4×C4⋊C4—473no—
237C4×C4⋊C4C2×D4716yes—
238C4×C4⋊C4C2×D4713yes—
239C4×C22⋊C4—476no—
240C4×C22⋊C4—471no—
241C4×C22⋊C4—473no—
242#376—476no—
243#376—476no—
244C43—473no—
245C43C2×D4713yes—
246C47#3412811yescharacteristic, complemented, Sylow
247C22×Q8—478no—
248C22×Q8—473no—
249C22×Q8—948no—
250C22×Q8C22×D4711yescharacteristic
251C23×C4—473no—
252C23×C4—473no—
253C23×C4—471no—
254C4×Q8—28no—
255C4×Q8—478no—
256C4×Q8—4724no—
257C4×Q8—4724no—
258C4×Q8—948no—
259C4×Q8C4×D4718yes—
260#117—9424no—
261C2×C4⋊C4—476no—
262C2×C4⋊C4—473no—
263C2×C4⋊C4—4724no—
264C2×C4⋊C4—476no—
265C2×C4⋊C4—476no—
266C2×C4⋊C4—476no—
267C2×C4⋊C4C22×D4716yes—
268C2×C4⋊C4C22×D4713yes—
269C2×C22⋊C4—472no—
270C2×C22⋊C4—476no—
271C2×C42—473no—
272C2×C42—473no—
273C2×C42—473no—
274C2×C42—4724no—
275C2×C42—476no—
276C2×C42—476no—
277C2×C42—471no—
278C2×C42—473no—
279C2×C42C22×D4716yes—
280C2×C42C22×D4716yes—
281C2×C42C22×D4711yescharacteristic
282C2×C42C22×D4713yes—
283C2×C42C22×D4713yes—
284#95—472no—
285#95—476no—
286#95—476no—
287#95—476no—
288#95—4712no—
289#95—4712no—
290#95—476no—
291#95C22×D4716yes—
292C24—471no—
293C2×Q8—4712no—
294C2×Q8—9432no—
295C2×Q8—9412no—
296C2×Q8C2×C4×D4714yes—
297C2×Q8#1062535018yes—
298C22×C4—471no—
299C22×C4—473no—
300C22×C4—472no—
301C22×C4—473no—
302C22×C4—474no—
303C22×C4—4712no—
304C22×C4—4712no—
305C22×C4—476no—
306C22×C4—9412no—
307C22×C4—9412no—
308C22×C4—942no—
309C22×C4—942no—
310C22×C4C23×D4716yes—
311C22×C4C23×D4713yes—
312C22×C4C23×D4712yes—
313C22×C4C23×D4713yes—
314C22×C4C23×D4711yescharacteristic
315C4⋊C4—212no—
316C4⋊C4—4724no—
317C4⋊C4—4712no—
318C4⋊C4—9412no—
319C4⋊C4—9412no—
320C4⋊C4—9412no—
321C4⋊C4C2×C4×D47112yes—
322C22⋊C4—478no—
323C22⋊C4—9412no—
324C42—212no—
325C42—212no—
326C42—26no—
327C42—22no—
328C42—4712no—
329C42—4712no—
330C42—4712no—
331C42—9412no—
332C42—9412no—
333C42—942no—
334C42—946no—
335C42C2×C4×D47112yes—
336C23—472no—
337C23—474no—
338C23—942no—
339C23—941no—
340C23—941no—
341C23C24×D4711yescharacteristic
342Q8—9424no—
343Q8#13093981116yescomplemented
344C2×C4—26no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—26no—
348C2×C4—26no—
349C2×C4—26no—
350C2×C4—472no—
351C2×C4—476no—
352C2×C4—476no—
353C2×C4—474no—
354C2×C4—9424no—
355C2×C4—946no—
356C2×C4—942no—
357C2×C4—946no—
358C2×C4—942no—
359C2×C4—941no—
360C2×C4—941no—
361C2×C4—9424no—
362C2×C4—948no—
363C2×C4—946no—
364C2×C4—946no—
365C2×C4—946no—
366C2×C4C22×C4×D4716yes—
367C2×C4C22×C4×D4716yes—
368C2×C4C22×C4×D4712yes—
369C2×C4#41×D4714yes—
370C2×C4C2×#10625350112yes—
371C2×C4#1309407112yes—
372C2×C4#1309407212yes—
373C22—474no—
374C22—942no—
375C22—942no—
376C22—944no—
377C22—944no—
378C22C23×C4×D4711yescharacteristic
379C22C2×#41×D4712yes—
380C22C22×#1062535012yes—
381C22C2×#1309407111yescharacteristic
382C22C2×#1309407211yescharacteristic
383C4—26no—
384C4—24no—
385C4—46no—
386C4—946no—
387C4—946no—
388C4—942no—
389C4—944no—
390C4—1886no—
391C4C2×#13093981112yes—
392C4#1568959914yes—
393C2—944no—
394C2C22×#1309398111yescharacteristic
395C2C4×#41×D4711yescharacteristic
396C2C2×#1568959911yescharacteristic
397C2#2025700512yes—
398C2#2025701012yes—
subgroupG/H
centerC22×C4C23×D47
commutatorC22×C47C24×C4
FrattiniC23C24×D47
FittingC42×Q8×C47C2
socle, cosocleC23×C47C25
prime radicalC23×D47C24
hypercenterC42×Q8D47
hypocenterC47#34128
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C22×C4 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
2. #20246349C2semidirect
3. #20246306C2semidirect
6. #20245865C2semidirect
7. #20245862C2semidirect
9. C4×Q8×Dic47C2semidirect
11. C42×Q8×C47C2semidirect
17. C4×Q8×D47C4semidirect
142. Q8×C47#117semidirect
203. C42×Q8D47semidirect
246. C47#34128semidirect
343. Q8#13093981semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Q8×D472
#202463492
#202463062
#202462532
Q8×(C2×D47)⋊C42
#202458652
#202458622
#202457842
C4×Q8×Dic472
C4×#130939812
C42×Q8×C472
#3412847