Finite Group Databasebeta

← #26376464

Subgroups — #26376464

Order 12032.

2250 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Q8×Dic47C211yesmaximal, characteristic
2#20246352C216yesmaximal, complemented
3#20246306C212yesmaximal, complemented
4#20246267C213yesmaximal
5#20246263C213yesmaximal
6Q8×C22⋊Dic47C211yesmaximal, characteristic
7#20246033C216yesmaximal, complemented
8#20245783C213yesmaximal
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
18C22×C4⋊Dic47C2213yes—
19C22×C4×Dic47C2213yes—
20#15689030C2216yes—
21#15689021C2216yes—
22#15689019C2216yes—
23#15689008C22112yes—
24#15689006C22112yes—
25#15688990C2216yes—
26#15688990C2216yes—
27#15688988C2216yes—
28#15688987C2216yes—
29#15688984C2211yescharacteristic
30#15688984C2213yes—
31#15688983C2213yes—
32#15688981C2213yes—
33#15688981C2213yes—
34C4⋊C4×Dic47C2216yes—
35C4⋊C4×Dic47C2216yes—
36C4×Dic47⋊C4C2216yes—
37C4×C4⋊Dic47C2216yes—
38C22⋊C4×Dic47C2216yes—
39C4×C22⋊Dic47C2213yes—
40#15688969C2216yes—
41#15688968C2216yes—
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—
54C2×C4⋊Dic47C2×C4124yes—
55C2×C4⋊Dic47C2316yes—
56C2×C4⋊Dic47C2313yes—
57C2×C4⋊Dic47C2316yes—
58C2×C4⋊Dic47C2313yes—
59C2×Dic47⋊C4C2316yes—
60C2×Dic47⋊C4C2316yes—
61C2×C22⋊Dic47C2311yescharacteristic
62C2×C22⋊Dic47C2313yes—
63C2×C4×Dic47C2×C4124yes—
64C2×C4×Dic47C2316yes—
65C2×C4×Dic47C23112yes—
66C2×C4×Dic47C2313yes—
67C2×C4×Dic47C2313yes—
68C2×C4×Dic47C2313yes—
69C2×C4×Dic47C2313yes—
70C47⋊#95C2316yes—
71C47⋊#95C2316yes—
72C47⋊#95C2312yes—
73C47⋊#95C2316yes—
74C47⋊#95C2316yes—
75C47⋊#95C2316yes—
76C47⋊#95C2316yes—
77C47⋊#95C23112yes—
78C47⋊#95C2312yes—
79C22×Q8×C47—28no—
80C22×Q8×C47C2×C418yes—
81C22×Q8×C47C2313yes—
82C22×Q8×C47C2311yescharacteristic
83C23×C4×C47C2313yes—
84C4×Q8×C47—28no—
85C2×C4⋊C4×C47C2316yes—
86C2×C22⋊C4×C47C2313yes—
87C2×C22⋊C4×C47C2311yescharacteristic
88C2×C42×C47C2316yes—
89#95×C47C2316yes—
90#95×C47C2316yes—
91C24×C47C2411yescharacteristic
92C2×Q8×C47—232no—
93C2×Q8×C47—212no—
94C2×Q8×C47C22×C4112yes—
95C2×Q8×C47C22×C414yes—
96C2×Q8×C47#4118yes—
97C22×C4×C47—212no—
98C22×C4×C47C22×C4112yes—
99C22×C4×C47C2412yes—
100C22×C4×C47C2416yes—
101C22×C4×C47C2413yes—
102C22×C4×C47C2413yes—
103C22×Dic47—22no—
104C22×Dic47—22no—
105C22×Dic47—46no—
106C22×Dic47C22×C414yes—
107C22×Dic47C22×C4112yes—
108C22×Dic47C2412yes—
109C22×Dic47C2416yes—
110C22×Dic47C2411yescharacteristic
111C22×Dic47C2411yescharacteristic
112C22×Dic47C2413yes—
113C22×Dic47C2413yes—
114C4⋊C4×C47—212no—
115C22⋊C4×C47—26no—
116C22⋊C4×C47—22no—
117C42×C47—212no—
118C22⋊Dic47—22no—
119C22⋊Dic47—26no—
120C4⋊Dic47—212no—
121C4⋊Dic47C22×C4124yes—
122C4⋊Dic47C22×C4112yes—
123C4⋊Dic47C22×C4112yes—
124Dic47⋊C4—46no—
125Dic47⋊C4—46no—
126C4×Dic47—212no—
127C4×Dic47—412no—
128C4×Dic47C22×C4112yes—
129C4×Dic47C22×C4112yes—
130C4×Dic47C22×C4112yes—
131C4×Dic47C22×C4112yes—
132C23×C47—22no—
133C23×C47—22no—
134C23×C47C23×C412yes—
135C23×C472−1+412yes—
136C23×C472−1+412yes—
137C23×C47C2511yescharacteristic
138Q8×C47—224no—
139Q8×C47#117116yes—
140C2×C4×C47—26no—
141C2×C4×C47—224no—
142C2×C4×C47—22no—
143C2×C4×C47—26no—
144C2×C4×C47—26no—
145C2×C4×C47—22no—
146C2×C4×C47—26no—
147C2×C4×C47—22no—
148C2×C4×C47C23×C416yes—
149C2×C4×C47C23×C416yes—
150C2×C4×C47C2×#41112yes—
151C2×Dic47—28no—
152C2×Dic47—21no—
153C2×Dic47—21no—
154C2×Dic47—21no—
155C2×Dic47—21no—
156C2×Dic47—26no—
157C2×Dic47—26no—
158C2×Dic47—22no—
159C2×Dic47—26no—
160C2×Dic47—22no—
161C2×Dic47—26no—
162C2×Dic47—26no—
163C2×Dic47—22no—
164C2×Dic47—412no—
165C2×Dic47C23×C412yes—
166C2×Dic47C23×C412yes—
167C2×Dic47C23×C416yes—
168C2×Dic47C23×C416yes—
169#34192—471nomaximal, Sylow
170C22×C47—22no—
171C22×C47—22no—
172C22×C47—24no—
173C22×C47—24no—
174C22×C47#51914yes—
175C22×C47C24×C411yescharacteristic
176C22×C47C22×#4112yes—
177C22×C47C2×2−1+411yescharacteristic
178C22×C47C2×2−1+411yescharacteristic
179C22×C47#58512yes—
180C4×C47—26no—
181C4×C47—46no—
182C4×C47—42no—
183C4×C47C2×#117112yes—
184Dic47—22no—
185Dic47—22no—
186Dic47—26no—
187Dic47—26no—
188Dic47—42no—
189Dic47—46no—
190C22×C4×Q8—471no—
191#2356—476no—
192#2351—472no—
193#2345—473no—
194#2345—473no—
195Q8×C22⋊C4—471no—
196Q8×C22⋊C4D4711yescharacteristic
197#2320—476no—
198#2283—473no—
199#2283—473no—
200#2276—472no—
201C2×C47—24no—
202C2×C47C22×#11711yescharacteristic
203C2×C47C2×#51911yescharacteristic
204C2×C47#342811yescharacteristic
205C2×C47#351612yes—
206C2×C47#351612yes—
207C23×Q8C2×D4711yescharacteristic
208C2×C4×Q8—4716no—
209C2×C4×Q8—473no—
210C2×C4×Q8—471no—
211C2×C4×Q8—473no—
212C2×C4×Q8—471no—
213C2×C4×Q8—472no—
214C2×C4×Q8C2×D4712yes—
215C22×C4⋊C4—473no—
216C22×C42—473no—
217#391—476no—
218#391C2×D4716yes—
219#389—476no—
220#389—476no—
221#387—4712no—
222#387—4712no—
223#383—476no—
224#383—476no—
225#382—476no—
226#382—476no—
227#381—471no—
228#381—473no—
229#380—473no—
230#380—473no—
231#380—473no—
232#380C2×D4713yes—
233C4×C4⋊C4—476no—
234C4×C4⋊C4—476no—
235C4×C4⋊C4—476no—
236C4×C4⋊C4—476no—
237C4×C22⋊C4—473no—
238C4×C22⋊C4—476no—
239C4×C22⋊C4C2×D4713yes—
240#376—476no—
241#376—476no—
242C47#3419211yescharacteristic, complemented, Sylow
243C22×Q8—28no—
244C22×Q8C4×D4718yes—
245C22×Q8C22×D4713yes—
246C22×Q8C22×D4711yescharacteristic
247C23×C4—471no—
248C23×C4—473no—
249C23×C4C22×D4713yes—
250C4×Q8—28no—
251C4×Q8—4724no—
252C4×Q8—4724no—
253C4×Q8—478no—
254C4×Q8—478no—
255C4×Q8—948no—
256C2×C4⋊C4—4724no—
257C2×C4⋊C4—473no—
258C2×C4⋊C4—476no—
259C2×C4⋊C4—473no—
260C2×C4⋊C4—476no—
261C2×C4⋊C4—476no—
262C2×C4⋊C4—476no—
263C2×C4⋊C4C22×D4716yes—
264C2×C22⋊C4—471no—
265C2×C22⋊C4—473no—
266C2×C22⋊C4C22×D4713yes—
267C2×C22⋊C4C22×D4711yescharacteristic
268C2×C42—4724no—
269C2×C42—473no—
270C2×C42—473no—
271C2×C42—473no—
272C2×C42—473no—
273C2×C42—476no—
274C2×C42—4712no—
275C2×C42C22×D4716yes—
276#95—472no—
277#95—476no—
278#95—476no—
279#95—476no—
280#95—476no—
281#95—476no—
282#95—4712no—
283#95—476no—
284#95—472no—
285#95C22×D4716yes—
286#95C22×D4716yes—
287C24C23×D4711yescharacteristic
288C2×Q8—232no—
289C2×Q8—212no—
290C2×Q8C2×C4×D47112yes—
291C2×Q8C2×C4×D4714yes—
292C2×Q8#1062535218yes—
293C22×C4—212no—
294C22×C4—474no—
295C22×C4—471no—
296C22×C4—471no—
297C22×C4—473no—
298C22×C4—473no—
299C22×C4—4712no—
300C22×C4—472no—
301C22×C4—476no—
302C22×C4—942no—
303C22×C4—942no—
304C22×C4—1886no—
305C22×C4C2×C4×D47112yes—
306C22×C4C23×D4716yes—
307C22×C4C23×D4712yes—
308C22×C4C23×D4713yes—
309C22×C4C23×D4713yes—
310C4⋊C4—212no—
311C4⋊C4—4724no—
312C4⋊C4—4712no—
313C4⋊C4—4712no—
314C4⋊C4—9412no—
315C4⋊C4—1886no—
316C4⋊C4—1886no—
317C22⋊C4—26no—
318C22⋊C4—22no—
319C22⋊C4—942no—
320C22⋊C4—946no—
321C42—212no—
322C42—4712no—
323C42—4712no—
324C42—4712no—
325C42—4712no—
326C42—9412no—
327C42—18812no—
328C23—22no—
329C23—22no—
330C23C22×C4×D4712yes—
331C23#1309407112yes—
332C23#1309407212yes—
333C23C24×D4711yescharacteristic
334Q8—224no—
335Q8#13093980116yes—
336C2×C4—26no—
337C2×C4—224no—
338C2×C4—22no—
339C2×C4—26no—
340C2×C4—26no—
341C2×C4—22no—
342C2×C4—26no—
343C2×C4—22no—
344C2×C4—472no—
345C2×C4—472no—
346C2×C4—476no—
347C2×C4—476no—
348C2×C4—948no—
349C2×C4—941no—
350C2×C4—941no—
351C2×C4—941no—
352C2×C4—941no—
353C2×C4—946no—
354C2×C4—946no—
355C2×C4—942no—
356C2×C4—946no—
357C2×C4—942no—
358C2×C4—946no—
359C2×C4—946no—
360C2×C4—942no—
361C2×C4—18812no—
362C2×C4C22×C4×D4716yes—
363C2×C4C22×C4×D4716yes—
364C2×C4C2×#10625352112yes—
365C22—22no—
366C22—22no—
367C22—24no—
368C22—24no—
369C22#1568959914yes—
370C22C23×C4×D4711yescharacteristic
371C22C22×#1062535212yes—
372C22C2×#1309407111yescharacteristic
373C22C2×#1309407211yescharacteristic
374C22#1568996012yes—
375C4—26no—
376C4—46no—
377C4—42no—
378C4—942no—
379C4—942no—
380C4—946no—
381C4—946no—
382C4—1882no—
383C4—1886no—
384C4C2×#13093980112yes—
385C2—24no—
386C2C22×#1309398011yescharacteristic
387C2C2×#1568959911yescharacteristic
388C2#2025659011yescharacteristic
389C2#2025737012yes—
390C2#2025738212yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingQ8×C22⋊C4×C47C2
socle, cosocleC23×C47C25
prime radicalC24×C47C24
hypercenterQ8×C22⋊C4D47
hypocenterC47#34192
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ Q8×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. #20246352C2semidirect
3. #20246306C2semidirect
7. #20246033C2semidirect
10. #20245753C2semidirect
12. C2×Q8×Dic47C4semidirect
242. C47#34192semidirect

Maximal subgroups

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

subgroupindex
C22×Q8×Dic472
#202463522
#202463062
#202462672
#202462632
Q8×C22⋊Dic472
#202460332
#202457832
#202457822
#202457532
Q8×C22⋊C4×C472
#3419247