Finite Group Databasebeta

← #26376467

Subgroups — #26376467

Order 12032.

2250 conjugacy classes of subgroups.

The subgroup lattice has 390 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#20246261C216yesmaximal
5Q8×(C2×D47)⋊C4C212yesmaximal
6#20246040C213yesmaximal, complemented
7#20246034C213yesmaximal, complemented
8#20245784C213yesmaximal
9#20245780C213yesmaximal
10#20245753C211yesmaximal, characteristic, complemented
11C47×#2276C211yesmaximal, characteristic, complemented
12C22×Q8×D47C2211yescharacteristic
13C2×Q8×Dic47C2212yes—
14C2×Q8×Dic47C2211yescharacteristic
15C2×C4×Dic94C2213yes—
16C2×#13093989C2213yes—
17C4×Q8×D47C4116yescomplemented
18C2×C4⋊C4×D47C2213yes—
19C2×C42×D47C2213yes—
20#15689029C2216yes—
21#15689028C2216yes—
22#15689021C2216yes—
23#15689019C2216yes—
24#15689007C22112yes—
25#15689005C22112yes—
26#15688990C2216yes—
27#15688990C2216yes—
28#15688988C2216yes—
29#15688987C2213yes—
30#15688987C2213yes—
31#15688985C2211yescharacteristic
32#15688985C2213yes—
33#15688982C2216yes—
34#15688980C2216yes—
35C4⋊C4×Dic47C2216yes—
36C4×Dic47⋊C4C2216yes—
37C4×C4⋊Dic47C2213yes—
38C4×(C2×D47)⋊C4C2216yes—
39C4×(C2×D47)⋊C4C2216yes—
40#15688969C2213yes—
41#15688969C2216yes—
42C2×C4×Q8×C47C2212yes—
43C2×C4×Q8×C47C2211yescharacteristic
44C4×C4⋊C4×C47C2213yes—
45C4×C4⋊C4×C47C2216yes—
46C47×#376C2213yes—
47C22×Dic94C2313yes—
48C2×Q8×D47—28no—
49C2×Q8×D47C2×C418yes—
50C22×C4×D47C2313yes—
51C22×C4×D47C2311yescharacteristic
52C22×C4×D47C2313yes—
53Q8×Dic47—28no—
54Q8×Dic47C2×C418yes—
55C4×Dic94C2×C4124yes—
56#13093989C2×C4124yes—
57C2×C4⋊Dic47C2316yes—
58C2×C4⋊Dic47C2313yes—
59C2×Dic47⋊C4C2316yes—
60C2×Dic47⋊C4C2316yes—
61C2×Dic47⋊C4C2316yes—
62C4⋊C4×D47C2×C4124yes—
63C2×(C2×D47)⋊C4C2316yes—
64C2×(C2×D47)⋊C4C2312yes—
65C2×C4×Dic47C2316yes—
66C2×C4×Dic47C2316yes—
67C2×C4×Dic47C2313yes—
68C2×C4×Dic47C2313yes—
69C2×C4×Dic47C2313yes—
70C42×D47C2×C4124yes—
71C47⋊#95C2311yescharacteristic
72C47⋊#95C2316yes—
73C47⋊#95C2313yes—
74C47⋊#95C2316yes—
75C47⋊#95C23112yes—
76C47⋊#95C2316yes—
77C47⋊#95C2316yes—
78C47⋊#95C23112yes—
79C47⋊#95C2312yes—
80C22×Q8×C47C2311yescharacteristic
81C4×Q8×C47—28no—
82C4×Q8×C47C2×C418yes—
83C2×C4⋊C4×C47C2316yes—
84C2×C4⋊C4×C47C2313yes—
85C2×C42×C47C2316yes—
86C2×C42×C47C2316yes—
87C2×C42×C47C2313yes—
88#95×C47C2316yes—
89#95×C47C2311yescharacteristic
90#95×C47C2313yes—
91C23×D47C2411yescharacteristic
92C2×Q8×C47C22×C414yes—
93C2×Q8×C47#4118yes—
94C22×C4×C47C2416yes—
95C22×C4×C47C2412yes—
96C22×C4×C47C2413yes—
97C22×C4×C47C2411yescharacteristic
98C22×C4×C47C2413yes—
99C22×Dic47C2412yes—
100C22×Dic47C2416yes—
101C22×Dic47C2411yescharacteristic
102C22×Dic47C2413yes—
103C22×Dic47C2413yes—
104Q8×D47—232no—
105C2×C4×D47—24no—
106C2×C4×D47—212no—
107C2×C4×D47—43no—
108C2×C4×D47—43no—
109C2×C4×D47C22×C414yes—
110C2×C4×D47C22×C4112yes—
111C2×C4×D47C22×C4112yes—
112C2×Dic94—212no—
113C2×Dic94C22×C4112yes—
114C4⋊C4×C47—212no—
115C4⋊C4×C47C22×C4112yes—
116C42×C47—212no—
117C42×C47—46no—
118C42×C47C22×C4112yes—
119(C2×D47)⋊C4—212no—
120(C2×D47)⋊C4—24no—
121C4⋊Dic47—212no—
122C4⋊Dic47C22×C4112yes—
123Dic47⋊C4—46no—
124Dic47⋊C4—46no—
125Dic47⋊C4C22×C4124yes—
126C4×Dic47—212no—
127C4×Dic47—46no—
128C4×Dic47C22×C4112yes—
129C4×Dic47C22×C4112yes—
130C4×Dic47C22×C4112yes—
131C23×C47C2511yescharacteristic
132C22×D47—22no—
133C22×D47—21no—
134C22×D47—21no—
135C22×D47C23×C412yes—
136C22×D472−1+414yes—
137Q8×C47#117116yes—
138C2×C4×C47—22no—
139C2×C4×C47—23no—
140C2×C4×C47—23no—
141C2×C4×C47—22no—
142C2×C4×C47—26no—
143C2×C4×C47—26no—
144C2×C4×C47—22no—
145C2×C4×C47—22no—
146C2×C4×C47—26no—
147C2×C4×C47C23×C412yes—
148C2×C4×C47C23×C416yes—
149C2×C4×C47C23×C416yes—
150C2×C4×C47C2×#41112yes—
151C2×Dic47—26no—
152C2×Dic47—22no—
153C2×Dic47—23no—
154C2×Dic47—23no—
155C2×Dic47—22no—
156C2×Dic47—26no—
157C2×Dic47—22no—
158C2×Dic47—26no—
159C2×Dic47—26no—
160C2×Dic47—22no—
161C2×Dic47C23×C412yes—
162C2×Dic47C23×C416yes—
163C2×Dic47C23×C416yes—
164C4×D47—28no—
165C4×D47—224no—
166C4×D47—412no—
167Dic94—224no—
168#34192—471nomaximal, Sylow
169C22×C47C24×C411yescharacteristic
170C22×C47C22×#4112yes—
171C22×C47C2×2−1+412yes—
172C22×C47#58511yescharacteristic
173C22×C47#58511yescharacteristic
174C2×D47—22no—
175C2×D47—22no—
176C2×D47—24no—
177C2×D47—24no—
178C2×D47#51914yes—
179C4×C47—22no—
180C4×C47—26no—
181C4×C47—46no—
182C4×C47—42no—
183C4×C47C2×#117112yes—
184Dic47—26no—
185Dic47—22no—
186Dic47—26no—
187Dic47—42no—
188Dic47—46no—
189C22×C4×Q8—471no—
190#2356—476no—
191#2351—472no—
192#2345—476no—
193Q8×C22⋊C4—472no—
194#2320—473no—
195#2320—473no—
196#2283—473no—
197#2283—473no—
198#2276—471no—
199#2276D4711yescharacteristic, complemented
200C2×C47C22×#11711yescharacteristic
201C2×C47C2×#51911yescharacteristic
202C2×C47#342811yescharacteristic
203C2×C47#351612yes—
204C2×C47#351612yes—
205D47—24no—
206C23×Q8—471no—
207C2×C4×Q8—471no—
208C2×C4×Q8—4716no—
209C2×C4×Q8—472no—
210C2×C4×Q8—473no—
211C2×C4×Q8—473no—
212C2×C4×Q8C2×D4712yes—
213C2×C4×Q8C2×D4711yescharacteristic
214C22×C4⋊C4—473no—
215C22×C42—473no—
216#391—476no—
217#391—476no—
218#389—476no—
219#389—476no—
220#387—4712no—
221#387—4712no—
222#383—476no—
223#383—476no—
224#382—476no—
225#382—473no—
226#382—473no—
227#381—471no—
228#381—473no—
229#380—476no—
230#380—476no—
231C4×C4⋊C4—476no—
232C4×C4⋊C4—473no—
233C4×C4⋊C4—476no—
234C4×C4⋊C4C2×D4713yes—
235C4×C4⋊C4C2×D4716yes—
236C4×C22⋊C4—476no—
237C4×C22⋊C4—476no—
238#376—473no—
239#376—476no—
240#376C2×D4713yes—
241C47#3419211yescharacteristic, complemented, Sylow
242C22×Q8—478no—
243C22×Q8—473no—
244C22×Q8—948no—
245C22×Q8C22×D4711yescharacteristic
246C23×C4—473no—
247C23×C4—471no—
248C23×C4—473no—
249C4×Q8—28no—
250C4×Q8—478no—
251C4×Q8—4724no—
252C4×Q8—4724no—
253C4×Q8—948no—
254C4×Q8C4×D4718yes—
255C2×C4⋊C4—476no—
256C2×C4⋊C4—473no—
257C2×C4⋊C4—4724no—
258C2×C4⋊C4—476no—
259C2×C4⋊C4—476no—
260C2×C4⋊C4—476no—
261C2×C4⋊C4C22×D4716yes—
262C2×C4⋊C4C22×D4713yes—
263C2×C22⋊C4—476no—
264C2×C22⋊C4—472no—
265C2×C42—473no—
266C2×C42—473no—
267C2×C42—473no—
268C2×C42—4724no—
269C2×C42—476no—
270C2×C42—476no—
271C2×C42C22×D4716yes—
272C2×C42C22×D4716yes—
273C2×C42C22×D4713yes—
274#95—471no—
275#95—476no—
276#95—476no—
277#95—473no—
278#95—4712no—
279#95—4712no—
280#95—476no—
281#95—476no—
282#95—472no—
283#95C22×D4711yescharacteristic
284#95C22×D4716yes—
285#95C22×D4713yes—
286C24—471no—
287C2×Q8—4712no—
288C2×Q8—9432no—
289C2×Q8—9412no—
290C2×Q8C2×C4×D4714yes—
291C2×Q8#1062535018yes—
292C22×C4—471no—
293C22×C4—473no—
294C22×C4—473no—
295C22×C4—474no—
296C22×C4—4712no—
297C22×C4—4712no—
298C22×C4—472no—
299C22×C4—476no—
300C22×C4—944no—
301C22×C4—9412no—
302C22×C4—1883no—
303C22×C4—1883no—
304C22×C4C23×D4716yes—
305C22×C4C23×D4712yes—
306C22×C4C23×D4711yescharacteristic
307C22×C4C23×D4713yes—
308C22×C4C23×D4713yes—
309C4⋊C4—212no—
310C4⋊C4—4724no—
311C4⋊C4—4712no—
312C4⋊C4—9412no—
313C4⋊C4—1886no—
314C4⋊C4—1886no—
315C4⋊C4C2×C4×D47112yes—
316C22⋊C4—9412no—
317C22⋊C4—944no—
318C42—212no—
319C42—46no—
320C42—4712no—
321C42—4712no—
322C42—4712no—
323C42—9412no—
324C42—1886no—
325C42C2×C4×D47112yes—
326C23—472no—
327C23—474no—
328C23—942no—
329C23—941no—
330C23—941no—
331C23C24×D4711yescharacteristic
332Q8—9424no—
333Q8#13093981116yes—
334C2×C4—22no—
335C2×C4—23no—
336C2×C4—23no—
337C2×C4—22no—
338C2×C4—26no—
339C2×C4—26no—
340C2×C4—22no—
341C2×C4—26no—
342C2×C4—22no—
343C2×C4—472no—
344C2×C4—476no—
345C2×C4—476no—
346C2×C4—948no—
347C2×C4—946no—
348C2×C4—942no—
349C2×C4—943no—
350C2×C4—943no—
351C2×C4—9424no—
352C2×C4—942no—
353C2×C4—946no—
354C2×C4—942no—
355C2×C4—946no—
356C2×C4—942no—
357C2×C4—946no—
358C2×C4—18812no—
359C2×C4C22×C4×D4712yes—
360C2×C4C22×C4×D4716yes—
361C2×C4C22×C4×D4716yes—
362C2×C4C2×#10625350112yes—
363C22—474no—
364C22—942no—
365C22—942no—
366C22—944no—
367C22—944no—
368C22C23×C4×D4711yescharacteristic
369C22C22×#1062535012yes—
370C222−1+4×D4712yes—
371C22#1568995811yescharacteristic
372C22#1568996011yescharacteristic
373C4—22no—
374C4—26no—
375C4—46no—
376C4—42no—
377C4—946no—
378C4—942no—
379C4—946no—
380C4—1882no—
381C4—1886no—
382C4C2×#13093981112yes—
383C2—944no—
384C2C22×#1309398111yescharacteristic
385C2#519×D4711yescharacteristic
386C2#2025658811yescharacteristic
387C2#2025737312yes—
388C2#2025738312yes—
subgroupG/H
center, FrattiniC23C24×D47
commutatorC22×C47C24×C4
FittingC47×#2276C2
socle, cosocleC23×C47C25
prime radicalC23×D47C24
hypercenter#2276D47
hypocenterC47#34192
LCSG ▷ C22×C47 ▷ C47
UCSC1 ◁ C23 ◁ #2276

Complements

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

NH ≅ G/Ntype
2. #20246349C2semidirect
3. #20246306C2semidirect
6. #20246040C2semidirect
7. #20246034C2semidirect
10. #20245753C2semidirect
11. C47×#2276C2semidirect
17. C4×Q8×D47C4semidirect
199. #2276D47semidirect
241. C47#34192semidirect

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
#202462612
Q8×(C2×D47)⋊C42
#202460402
#202460342
#202457842
#202457802
#202457532
C47×#22762
#3419247