Finite Group Databasebeta

← #26225644

Subgroups — #26225644

Order 12032.

1354 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689673C211yesmaximal, characteristic
2#20245786C211yesmaximal, characteristic, complemented
3#20244318C214yesmaximal, complemented
4#20244212C212yesmaximal, complemented
5#20244208C212yesmaximal, complemented
6#20243329C214yesmaximal, complemented
7C2×C47×#432C211yesmaximal, characteristic, complemented
8C22×#10625350C2212yes—
9#15689673—24no—
10#15689673C418yescomplemented
11C2×#13094024C2211yescharacteristic
12C2×#13094021C2211yescharacteristic
13C2×#13094012C2212yes—
14C2×#13093998—21no—
15C2×#13093998—21no—
16C2×C4×D188—21no—
17C2×C4×D188—21no—
18C2×#13093949C2218yes—
19C2×#13093946C2218yes—
20#15688991—21no—
21#15688989C2212yes—
22#15688986—21no—
23#15688983C2212yes—
24C4×C4⋊Dic47C2212yes—
25C4×C22⋊Dic47—21no—
26#15688969—21no—
27C22×C42×C47C2211yescharacteristic
28C47×#432—24no—
29C2×C4⋊C8×C47C2212yes—
30C2×C22⋊C8×C47C2212yes—
31C2×#97×C47C2212yes—
32C2×#10625350—24no—
33C2×#10625350—48no—
34C2×#10625350C2×C418yes—
35C22×Dic94C2312yes—
36C22×D188—21no—
37C22×D188—21no—
38C22×D188C2312yes—
39C22×C22⋊D47—22no—
40C22×C4×D47—22no—
41#13094024—22no—
42#13094024C2×C414yes—
43#13094021—22no—
44#13094021C2×C414yes—
45#13094012—24no—
46#13094012C2×C418yes—
47#13093998—28no—
48#13093998—28no—
49C4×D188—28no—
50C4×D188—28no—
51C2×C4⋊Dic47—21no—
52C2×C4⋊Dic47—21no—
53C2×C4⋊Dic47C2314yes—
54C2×C22⋊Dic47—22no—
55C2×(C2×D47)⋊C4—22no—
56C2×(C2×D47)⋊C4—22no—
57C2×C4×Dic47—22no—
58#13093949—48no—
59#13093946—48no—
60C47⋊#95—22no—
61C47⋊#95—22no—
62C47⋊#95—22no—
63C47⋊#95—22no—
64C23×C4×C47D411yescharacteristic
65C23×C4×C47D411yescharacteristic
66C23×C4×C47C2311yescharacteristic
67C22×C8×C47C2314yes—
68C2×C42×C47C2×C414yes—
69C2×C42×C47D411yescharacteristic
70C2×C42×C47D411yescharacteristic
71C2×C42×C47D414yes—
72C2×C42×C47C2311yescharacteristic
73C2×C42×C47C2311yescharacteristic
74C4⋊C8×C47—24no—
75C22⋊C8×C47—24no—
76#97×C47—24no—
77C24×C47C2×D411yescharacteristic
78C23×D47—22no—
79C22×C4×C47—24no—
80C22×C4×C47—42no—
81C22×C4×C47—42no—
82C22×C4×C47C22⋊C414yes—
83C22×C4×C47C22⋊C414yes—
84C22×C4×C47C22×C412yes—
85C22×C4×C47C2×D411yescharacteristic
86C22×C4×C47C2×D411yescharacteristic
87C22×C4×C47C2×D411yescharacteristic
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D412yes—
91C22×C4×C47C2411yescharacteristic
92C2×C22⋊D47—28no—
93C2×C22⋊D47—48no—
94C22×Dic47—22no—
95C22×Dic47—24no—
96#10625350—416no—
97#10625350—416no—
98C2×D188—28no—
99C2×D188—24no—
100C2×D188—24no—
101C2×D188—22no—
102C2×D188—44no—
103C2×D188—44no—
104C2×D188C22×C414yes—
105C2×C4×D47—28no—
106C2×C4×D47—48no—
107C2×Dic94—28no—
108C2×Dic94—22no—
109C2×Dic94C22×C414yes—
110C2×C8×C47—24no—
111C2×C8×C47—24no—
112C2×C8×C47—24no—
113C42×C47C22⋊C414yes—
114C42×C47C22⋊C414yes—
115C42×C47C22×C412yes—
116C42×C47C22×C412yes—
117C42×C47C2×D412yes—
118C42×C47C2×D412yes—
119C22⋊Dic47—44no—
120(C2×D47)⋊C4—28no—
121(C2×D47)⋊C4—28no—
122C4⋊Dic47—24no—
123C4⋊Dic47—24no—
124C4⋊Dic47—28no—
125C4×Dic47—44no—
126C23×C47—22no—
127C23×C47—22no—
128C23×C47—22no—
129C23×C47C2×C22⋊C412yes—
130C23×C47C22×D411yescharacteristic
131C22×D47—24no—
132C22×D47—44no—
133C22×D47—44no—
134C22×D47—44no—
135C2×C4×C47—22no—
136C2×C4×C47—22no—
137C2×C4×C47—22no—
138C2×C4×C47—22no—
139C2×C4×C47—22no—
140C2×C4×C47—24no—
141C2×C4×C47—24no—
142C2×C4×C47—44no—
143C2×C4×C47—44no—
144C2×C4×C47C2×C22⋊C412yes—
145C2×C4×C47C2×C22⋊C412yes—
146C2×C4×C47C2×C22⋊C412yes—
147C2×C4×C47C2×C22⋊C412yes—
148C2×C4×C47C2×C22⋊C412yes—
149C2×C4×C47C23×C411yescharacteristic
150C2×C4×C47C22×D411yescharacteristic
151C2×C4×C47H(5,𝔽2)14yes—
152C22⋊D47—416no—
153C2×Dic47—24no—
154C2×Dic47—44no—
155C2×Dic47—44no—
156C2×Dic47—44no—
157C2×Dic47—44no—
158D188—28no—
159D188—48no—
160D188—48no—
161D188—44no—
162C4×D47—416no—
163Dic94—28no—
164Dic94—44no—
165C8×C47—44no—
166#21812—471nomaximal, Sylow
167C22×C47—22no—
168C22×C47—22no—
169C22×C47—22no—
170C22×C47—44no—
171C22×C47C22×C22⋊C411yescharacteristic
172C22×C47#53512yes—
173C22×C47#57512yes—
174C22×C47#57512yes—
175C2×D47—44no—
176C2×D47—44no—
177C2×D47—44no—
178C2×D47—88no—
179C4×C47—22no—
180C4×C47—22no—
181C4×C47—22no—
182C4×C47—22no—
183C4×C47—22no—
184C4×C47#51812yes—
185C4×C47#53512yes—
186Dic47—44no—
187Dic47—84no—
188C2×#532—471no—
189C2×#432D4711yescharacteristic, complemented
190#2283—471no—
191#1983—474no—
192#1961—472no—
193#1960—472no—
194#1788—474no—
195C2×C47—42no—
196C2×C47#230911yescharacteristic
197C2×C47#289611yescharacteristic
198C2×C47#289611yescharacteristic
199C2×C47#321914yes—
200D47—84no—
201C22×#41—472no—
202#532—478no—
203#532—944no—
204C2×C4⋊Q8—471no—
205C2×#127—471no—
206C2×#124—472no—
207C2×#121—941no—
208C2×#121—941no—
209C2×C4×D4—941no—
210C2×C4×D4—941no—
211C22×C42C2×D4711yescharacteristic
212#432—24no—
213C2×C4⋊C8C2×D4712yes—
214C2×#103—478no—
215C2×#102—478no—
216C2×C22⋊C8C2×D4712yes—
217C2×#97C2×D4712yes—
218#383—941no—
219#382—472no—
220#381—941no—
221#380—472no—
222C4×C4⋊C4—472no—
223C4×C22⋊C4—941no—
224#376—941no—
225C47#2181211yescharacteristic, complemented, Sylow
226C2×#41—478no—
227C2×#41—944no—
228C2×#41—1888no—
229C22×Q8—472no—
230C22×D4—472no—
231C22×D4—942no—
232C22×D4—941no—
233C22×D4—941no—
234C23×C4—942no—
235C23×C4D18811yescharacteristic
236C23×C4D18811yescharacteristic
237C23×C4C22×D4711yescharacteristic
238C22×C8C22×D4714yes—
239C4⋊Q8—474no—
240C4⋊Q8—942no—
241#127—474no—
242#127—942no—
243#124—478no—
244#124—944no—
245#121—948no—
246#121—948no—
247C4×D4—948no—
248C4×D4—948no—
249C2×C4⋊C4—474no—
250C2×C4⋊C4—941no—
251C2×C4⋊C4—941no—
252C2×C22⋊C4—942no—
253C2×C22⋊C4—942no—
254C2×C22⋊C4—942no—
255C2×C42—942no—
256C2×C42C4×D4714yes—
257C2×C42D18811yescharacteristic
258C2×C42D18811yescharacteristic
259C2×C42C22⋊D4714yes—
260C2×C42C22×D4711yescharacteristic
261C2×C42C22×D4711yescharacteristic
262C4⋊C8—24no—
263#103—1888no—
264#102—1888no—
265C22⋊C8—24no—
266#97—24no—
267#95—942no—
268#95—942no—
269#95—942no—
270#95—942no—
271C24—942no—
272C24C2×D18811yescharacteristic
273#41—18816no—
274#41—18816no—
275C2×Q8—474no—
276C2×Q8—942no—
277C2×Q8—948no—
278C2×D4—474no—
279C2×D4—948no—
280C2×D4—944no—
281C2×D4—944no—
282C2×D4—942no—
283C2×D4—948no—
284C2×D4—1884no—
285C2×D4—1884no—
286C2×D4—1888no—
287C22×C4—24no—
288C22×C4—42no—
289C22×C4—42no—
290C22×C4—942no—
291C22×C4—948no—
292C22×C4—944no—
293C22×C4—1888no—
294C22×C4(C2×D47)⋊C414yes—
295C22×C4(C2×D47)⋊C414yes—
296C22×C4C2×C4×D4712yes—
297C22×C4C2×D18811yescharacteristic
298C22×C4C2×D18811yescharacteristic
299C22×C4C2×D18811yescharacteristic
300C22×C4C2×D18811yescharacteristic
301C22×C4C2×D18811yescharacteristic
302C22×C4C2×C22⋊D4712yes—
303C22×C4C23×D4711yescharacteristic
304C2×C8—24no—
305C2×C8—24no—
306C2×C8—24no—
307C4⋊C4—944no—
308C4⋊C4—944no—
309C4⋊C4—948no—
310C22⋊C4—948no—
311C22⋊C4—948no—
312C22⋊C4—1884no—
313C42—1884no—
314C42(C2×D47)⋊C414yes—
315C42(C2×D47)⋊C414yes—
316C42C2×C4×D4712yes—
317C42C2×C4×D4712yes—
318C42C2×C22⋊D4712yes—
319C42C2×C22⋊D4712yes—
320C23—22no—
321C23—22no—
322C23—22no—
323C23—944no—
324C23—1884no—
325C23—1884no—
326C23—1884no—
327C23C2×(C2×D47)⋊C412yes—
328C23C22×D18811yescharacteristic
329Q8—948no—
330Q8—1884no—
331D4—948no—
332D4—1888no—
333D4—1888no—
334D4—18816no—
335D4—1884no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—24no—
342C2×C4—24no—
343C2×C4—44no—
344C2×C4—44no—
345C2×C4—944no—
346C2×C4—18816no—
347C2×C4—1884no—
348C2×C4—1884no—
349C2×C4—1884no—
350C2×C4—1884no—
351C2×C4C2×(C2×D47)⋊C412yes—
352C2×C4C2×(C2×D47)⋊C412yes—
353C2×C4C2×(C2×D47)⋊C412yes—
354C2×C4C2×(C2×D47)⋊C412yes—
355C2×C4C2×(C2×D47)⋊C412yes—
356C2×C4C22×C4×D4711yescharacteristic
357C2×C4C22×C22⋊D4711yescharacteristic
358C2×C4#1309407014yes—
359C8—44no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—44no—
364C22—3768no—
365C22—1884no—
366C22—1884no—
367C22—1884no—
368C22C22×(C2×D47)⋊C411yescharacteristic
369C22#1568969212yes—
370C22#1568992312yes—
371C22#1568992312yes—
372C4—22no—
373C4—22no—
374C4—22no—
375C4—22no—
376C4—22no—
377C4—3764no—
378C4—1884no—
379C4#1568959712yes—
380C4#1568968712yes—
381C2—42no—
382C2—3764no—
383C2#2024596811yescharacteristic
384C2#2025190811yescharacteristic
385C2#2025190811yescharacteristic
386C2#2025442714yes—
subgroupG/H
centerC23C22×D188
commutatorC2×C4×C47C23×C4
FrattiniC22×C4C23×D47
FittingC2×C47×#432C2
socleC23×C47C22×D4
prime radicalC2×#15689673C2
cosocleC22×C4×C47C24
hypercenterC2×#432D47
hypocenterC47#21812
LCSG ▷ C2×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#432

Complements

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

NH ≅ G/Ntype
2. #20245786C2semidirect
3. #20244318C2semidirect
4. #20244212C2semidirect
5. #20244208C2semidirect
6. #20243329C2semidirect
7. C2×C47×#432C2semidirect
10. #15689673C4semidirect
189. C2×#432D47semidirect
225. C47#21812semidirect

Maximal subgroups

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

subgroupindex
C2×#156896732
#202457862
#202443182
#202442122
#202442082
#202433292
C2×C47×#4322
#2181247