Finite Group Databasebeta

← #27085956

Subgroups — #27085956

Order 12032.

1834 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20257201C212yesmaximal
2#20253932C216yesmaximal, complemented
3#20253920C213yesmaximal, complemented
4#20253918C216yesmaximal, complemented
5#20253916C213yesmaximal, complemented
6#20253900C213yesmaximal, complemented
7C2×#15689520C213yesmaximal, complemented
8#20251983C211yesmaximal, characteristic
9#20251981C212yesmaximal
10#131×Dic47C211yesmaximal, characteristic
11C47×#2912C211yesmaximal, characteristic
12C2×#13094031C2211yescharacteristic
13Q8×Dic94—22no—
14#15689849—22no—
15#15689832—26no—
16D4×Dic94—26no—
17#15689754C2216yes—
18#15689751C2216yes—
19#15689676C2216yes—
20C2×#13094023C2216yes—
21#15689606C2212yes—
22#41×Dic47C2211yescharacteristic
23#15689531C22112yes—
24#15689530C22112yes—
25#15689527C2216yes—
26#15689524C2216yes—
27#15689523C2213yes—
28#15689523C2213yes—
29#15689520C2213yes—
30#15689520C2216yes—
31#15689520C2216yes—
32#15689520C2216yes—
33#15689520C2211yescharacteristic
34#15689166C2216yes—
35#15689165C2216yes—
36#15689163C2216yes—
37#15689161C2213yes—
38#15689159C2216yes—
39C2×#13093956C2213yes—
40C2×#13093955C2216yes—
41#15689081C2213yes—
42#15689078C2213yes—
43M16×Dic47C2213yes—
44C2×C8×Dic47C2213yes—
45C2×#131×C47C2211yescharacteristic
46C47×#520C2212yes—
47C47×#428C2216yes—
48C47×#427C2213yes—
49C2×#106×C47C2213yes—
50C22×Dic94—26no—
51#13094031Q818yescomplemented
52C2×#10625329C2313yes—
53C22×C47⋊C8C2313yes—
54#13094024—26no—
55#13094023—26no—
56#13094023—26no—
57#13094023—26no—
58#13094023C23112yes—
59#13094023C2316yes—
60#13094023C2312yes—
61#13094016—26no—
62#13094015—26no—
63#13094015C2316yes—
64#13094015C2316yes—
65#13094003—212no—
66#13094001—26no—
67#13094001—26no—
68#13094001—212no—
69Q8×Dic47—22no—
70Q8×Dic47D412yes—
71C4×Dic94—26no—
72C4×Dic94—26no—
73D4×Dic47—26no—
74D4×Dic47D416yes—
75#13093983C2316yes—
76#13093983C2313yes—
77C2×C4⋊Dic47C2316yes—
78C2×Dic47⋊C4—26no—
79C2×C4×Dic47C2313yes—
80#13093958C2316yes—
81#13093957C2316yes—
82#13093957C2316yes—
83#13093956C2313yes—
84#13093956C2316yes—
85#13093956C2311yescharacteristic
86#13093955C2316yes—
87#13093955C23112yes—
88#13093955C2312yes—
89#13093934C2316yes—
90C8×Dic47C2311yescharacteristic
91C8×Dic47C2313yes—
92C8×Dic47C2313yes—
93C8×Dic47C2313yes—
94C2×#41×C47C2311yescharacteristic
95#131×C47Q818yescomplemented
96C2×M16×C47C2313yes—
97C22×C8×C47C2313yes—
98C4×Q8×C47—22no—
99C4×D4×C47—26no—
100#117×C47C2316yes—
101C2×C4⋊C4×C47C2316yes—
102#107×C47C2316yes—
103#106×C47C2313yes—
104#106×C47C2316yes—
105#106×C47C2311yescharacteristic
106#41×C47C2×Q814yes—
107#41×C47C2×Q814yes—
108C2×Q8×C47C2×D411yescharacteristic
109C2×D4×C47C2×D413yes—
110C22×C4×C47—26no—
111C22×C4×C47C2413yes—
112C22×Dic47—26no—
113C22×Dic47C2×D416yes—
114C2×Dic94—22no—
115C2×Dic94—26no—
116C2×Dic94—212no—
117C2×Dic94—812no—
118M16×C47C2×Q8112yescomplemented
119C2×C8×C47C2×Q8112yescomplemented
120C2×C8×C47C2411yescharacteristic
121C2×C8×C47C2413yes—
122C4⋊C4×C47—26no—
123C4⋊C4×C47C2416yes—
124C4⋊C4×C47C2412yes—
125C22⋊C4×C47—26no—
126C42×C47—26no—
127C22⋊Dic47—26no—
128C22⋊Dic47C2×D416yes—
129C4⋊Dic47—26no—
130C4⋊Dic47—26no—
131C4⋊Dic47—26no—
132C4⋊Dic47C2×D416yes—
133C4⋊Dic47C2416yes—
134C4⋊Dic47C2412yes—
135Dic47⋊C4—26no—
136Dic47⋊C4—212no—
137Dic47⋊C4—22no—
138Dic47⋊C4—26no—
139Dic47⋊C4—26no—
140C4×Dic47—26no—
141C4×Dic47C2×D416yes—
142C4×Dic47C2411yescharacteristic
143C4×Dic47C2413yes—
144#10625329C2×Q8112yescomplemented
145C2×C47⋊C8C2×Q8112yescomplemented
146C2×C47⋊C8C2411yescharacteristic
147C2×C47⋊C8C2413yes—
148C23×C47C22×D413yes—
149Q8×C47C4⋊Q814yes—
150D4×C47C4⋊Q8112yes—
151C2×C4×C47—22no—
152C2×C4×C47—26no—
153C2×C4×C47—86no—
154C2×C4×C47C22×D413yes—
155C2×C4×C47C22×Q816yes—
156C2×C4×C47C22×Q816yes—
157C2×C4×C47C2511yescharacteristic
158C2×Dic47—212no—
159C2×Dic47—22no—
160C2×Dic47—26no—
161C2×Dic47—86no—
162C2×Dic47C22×D416yes—
163C2×Dic47C22×D412yes—
164Dic94—44no—
165Dic94—412no—
166Dic94—812no—
167C8×C47C22×Q812yes—
168C8×C47C22×Q816yes—
169C47⋊C8C22×Q812yes—
170C47⋊C8C22×Q816yes—
171#60665—471nomaximal, Sylow
172C22×C47—26no—
173C22×C47C2×C4⋊Q816yes—
174C22×C47C23×D411yescharacteristic
175C4×C47—46no—
176C4×C47—42no—
177C4×C47C2×C4⋊Q816yes—
178C4×C47C23×Q811yescharacteristic
179C4×C47C23×Q811yescharacteristic
180Dic47—26no—
181Dic47—46no—
182Dic47—42no—
183Dic47#57714yes—
184#3503—472no—
185#3165—476no—
186#3163—476no—
187#3163—473no—
188#3163—473no—
189#3160—473no—
190C2×#498—473no—
191#2912—471no—
192#2912—472no—
193#2912D4711yescharacteristic
194C4×#131—471no—
195C2×C47—26no—
196C2×C47C22×C4⋊Q811yescharacteristic
197C2×C47C2×#57712yes—
198C2×#131—471no—
199C2×#131C2×D4711yescharacteristic
200Q82—942no—
201#557—942no—
202#554—946no—
203D4×Q8—946no—
204#544—476no—
205#544—476no—
206#533—476no—
207C2×C4⋊Q8—476no—
208#520—472no—
209#520C2×D4712yes—
210C4×#41—471no—
211#501—4712no—
212#501—4712no—
213#500—476no—
214#499—476no—
215#499—473no—
216#499—473no—
217#498—473no—
218#498—476no—
219#498—476no—
220#498—476no—
221#498—471no—
222#428—476no—
223#428—476no—
224#428—476no—
225#428C2×D4716yes—
226#427—473no—
227#427—476no—
228#427C2×D4713yes—
229C2×#106—473no—
230C2×#106—476no—
231C2×#106C2×D4713yes—
232#405—473no—
233#405—473no—
234C4×M16—473no—
235C2×C4×C8—473no—
236C47#6066511yescharacteristic, complemented, Sylow
237C2×#41C22×D4711yescharacteristic
238C22×Q8—946no—
239#131—478no—
240#131Dic9418yescomplemented
241C2×M16—473no—
242C2×M16C22×D4713yes—
243C22×C8—473no—
244C22×C8C22×D4713yes—
245C4⋊Q8—476no—
246C4⋊Q8—4712no—
247C4⋊Q8—472no—
248C4⋊Q8—946no—
249C4⋊Q8—946no—
250C4⋊Q8—946no—
251C4⋊Q8—946no—
252#125—476no—
253#125—476no—
254#125—946no—
255#125—946no—
256C22⋊Q8—9412no—
257C22⋊Q8—946no—
258C22⋊Q8—946no—
259C22⋊Q8—9412no—
260C4×Q8—22no—
261C4×Q8—472no—
262C4×Q8—946no—
263C4×Q8—946no—
264C4×Q8—942no—
265C4×D4—26no—
266C4×D4—476no—
267C4×D4—946no—
268#117—473no—
269#117—476no—
270#117C22×D4716yes—
271C2×C4⋊C4—476no—
272C2×C4⋊C4—946no—
273C2×C4⋊C4C22×D4716yes—
274C2×C42—473no—
275#107—476no—
276#107—476no—
277#107—476no—
278#107C22×D4716yes—
279#106—473no—
280#106—476no—
281#106—471no—
282#106—476no—
283#106—4712no—
284#106—472no—
285#106C22×D4713yes—
286#106C22×D4716yes—
287#106C22×D4711yescharacteristic
288#97—476no—
289C4×C8—471no—
290C4×C8—473no—
291C4×C8—473no—
292C4×C8—473no—
293#41C2×Dic9414yes—
294#41Q8×D4714yes—
295C2×Q8—9412no—
296C2×Q8—942no—
297C2×Q8—946no—
298C2×Q8—37612no—
299C2×Q8D4×D4711yescharacteristic
300C2×D4D4×D4713yes—
301C22×C4—26no—
302C22×C4—476no—
303C22×C4—946no—
304C22×C4C23×D4713yes—
305M16—4712no—
306M16C2×Dic94112yescomplemented
307C2×C8—4712no—
308C2×C8—471no—
309C2×C8—473no—
310C2×C8C2×Dic94112yescomplemented
311C2×C8C23×D4711yescharacteristic
312C2×C8C23×D4713yes—
313C4⋊C4—26no—
314C4⋊C4—476no—
315C4⋊C4—476no—
316C4⋊C4—472no—
317C4⋊C4—946no—
318C4⋊C4—9412no—
319C4⋊C4—942no—
320C4⋊C4—946no—
321C4⋊C4—946no—
322C4⋊C4—946no—
323C4⋊C4—946no—
324C4⋊C4—946no—
325C4⋊C4C23×D4716yes—
326C4⋊C4C23×D4712yes—
327C22⋊C4—26no—
328C22⋊C4—476no—
329C22⋊C4—946no—
330C42—26no—
331C42—476no—
332C42—471no—
333C42—473no—
334C42—946no—
335C23C2×D4×D4713yes—
336Q8—37612no—
337Q8—1884no—
338Q8—18812no—
339Q8#1309402314yes—
340D4#13094023112yes—
341C2×C4—22no—
342C2×C4—26no—
343C2×C4—86no—
344C2×C4—472no—
345C2×C4—476no—
346C2×C4—9412no—
347C2×C4—942no—
348C2×C4—946no—
349C2×C4—3766no—
350C2×C4C2×D4×D4713yes—
351C2×C4C2×Q8×D4716yes—
352C2×C4C22×Dic9416yes—
353C2×C4C24×D4711yescharacteristic
354C8—472no—
355C8—476no—
356C8C22×Dic9412yes—
357C8C22×Dic9416yes—
358C22—26no—
359C22C2×#1309402316yes—
360C22C22×D4×D4711yescharacteristic
361C4—46no—
362C4—42no—
363C4—474no—
364C4—946no—
365C4—1886no—
366C4—1882no—
367C4C2×#1309402316yes—
368C4C22×Q8×D4711yescharacteristic
369C4C23×Dic9411yescharacteristic
370C2—26no—
371C2C22×#1309402311yescharacteristic
372C2#577×D4712yes—
subgroupG/H
centerC22C22×D4×D47
commutator, cosocleC2×C4×C47C25
FrattiniC2×C4C24×D47
FittingC47×#2912C2
socleC22×C47C23×D4
prime radicalC2×#41×C47C23
hypercenter#2912D47
hypocenterC47#60665
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×#41 ◁ #2912

Complements

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

NH ≅ G/Ntype
2. #20253932C2semidirect
3. #20253920C2semidirect
4. #20253918C2semidirect
5. #20253916C2semidirect
6. #20253900C2semidirect
7. C2×#15689520C2semidirect
51. #13094031Q8semidirect
95. #131×C47Q8semidirect
118. M16×C47C2×Q8semidirect
119. C2×C8×C47C2×Q8semidirect
144. #10625329C2×Q8semidirect
145. C2×C47⋊C8C2×Q8semidirect
236. C47#60665semidirect
240. #131Dic94semidirect
306. M16C2×Dic94semidirect
310. C2×C8C2×Dic94semidirect

Maximal subgroups

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

subgroupindex
#202572012
#202539322
#202539202
#202539182
#202539162
#202539002
C2×#156895202
#202519832
#202519812
#131×Dic472
C47×#29122
#6066547