Finite Group Databasebeta

← #26233061

Subgroups — #26233061

Order 12032.

1066 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15689858C211yesmaximal, characteristic
2#20252339C212yesmaximal, complemented
3#20244236C214yesmaximal, complemented
4#20244227C212yesmaximal
5#20243379C212yesmaximal, complemented
6C4×#13093943C211yesmaximal, characteristic, complemented
7#20243179C212yesmaximal, complemented
8C47×#1801C211yesmaximal, characteristic, complemented
9#15689858—22no—
10#15689858—22no—
11#15689858—24no—
12C2×#13094021C2211yescharacteristic
13C2×#13094013C2212yes—
14C2×#13094012C412yes—
15C2×#13094009C414yes—
16C2×C4×Dic94C2211yescharacteristic
17C2×C4×D188C412yes—
18C2×#13093986C2212yes—
19#15689252—24no—
20#15689250—22no—
21#15689247—22no—
22#15689146C2214yes—
23C2×#13093943C2212yes—
24C2×#13093940C2214yes—
25C2×#13093940C2214yes—
26C4×#10625329C2212yes—
27M16×Dic47C2214yes—
28#15688892C2211yescharacteristic
29#15688892C2211yescharacteristic
30C2×C4×Q8×C47C2211yescharacteristic
31C47×#423C2212yes—
32C2×#101×C47C2212yes—
33C47×#343C2212yes—
34C22×Dic94C2×C411yescharacteristic
35C22×D188C2×C412yes—
36C22×D188C2×C412yes—
37C22×D188C2312yes—
38C22×C4×D47C2×C414yes—
39C2×#10625329C2312yes—
40C2×#10625329C2312yes—
41#13094021—22no—
42#13094021—22no—
43#13094013—24no—
44#13094013—24no—
45#13094012—24no—
46#13094012—24no—
47#13094009—28no—
48#13094009—28no—
49C4×Dic94D412yes—
50C4×Dic94D412yes—
51C4×Dic94D414yes—
52C4×D188—24no—
53C4×D188—22no—
54C4×D188—22no—
55#13093986—24no—
56#13093986—24no—
57C2×C4⋊Dic47C2×C411yescharacteristic
58C2×Dic47⋊C4C2×C412yes—
59C2×(C2×D47)⋊C4C2×C414yes—
60C2×(C2×D47)⋊C4C2×C414yes—
61C2×(C2×D47)⋊C4C2×C414yes—
62C2×C4×Dic47C2312yes—
63Dic47⋊C8—24no—
64Dic47⋊C8—24no—
65#13093943—24no—
66#13093940—28no—
67#13093940—44no—
68#13093935—22no—
69#13093934—24no—
70C4×C47⋊C8—22no—
71C8×Dic47—24no—
72C22×Q8×C47C2311yescharacteristic
73C2×M16×C47C2314yes—
74C4×Q8×C47—22no—
75C4×Q8×C47—22no—
76C2×C4⋊C4×C47C2×C412yes—
77C2×C4⋊C4×C47C2311yescharacteristic
78C2×C42×C47C2×C412yes—
79C2×C42×C47C2311yescharacteristic
80C4⋊C8×C47—22no—
81C4⋊C8×C47—22no—
82#101×C47—42no—
83C23×D47C22×C414yes—
84C2×Q8×C47—22no—
85C2×Q8×C47—22no—
86C2×Q8×C47—21no—
87C2×Q8×C47—21no—
88C22×C4×C47C22×C412yes—
89C22×C4×C47C22×C411yescharacteristic
90C22×C4×C47C22×C412yes—
91C22×C4×C47C22×C411yescharacteristic
92C22×C4×C47C2411yescharacteristic
93C22×Dic47C22×C412yes—
94C22×Dic47C22×C412yes—
95C2×D188—22no—
96C2×D188—22no—
97C2×D188—24no—
98C2×D188—24no—
99C2×D188—22no—
100C2×D188—22no—
101C2×D188—24no—
102C2×D188—44no—
103C2×D188—82no—
104C2×C4×D47—24no—
105C2×C4×D47—24no—
106C2×C4×D47—84no—
107C2×Dic94—42no—
108C2×Dic94C22⋊C412yes—
109C2×Dic94C22⋊C412yes—
110M16×C47—24no—
111M16×C47—24no—
112C2×C8×C47—24no—
113C4⋊C4×C47—24no—
114C4⋊C4×C47—22no—
115C42×C47—22no—
116C42×C47—22no—
117C42×C47C2×D412yes—
118C42×C47C2×D412yes—
119(C2×D47)⋊C4—28no—
120(C2×D47)⋊C4—24no—
121(C2×D47)⋊C4—24no—
122(C2×D47)⋊C4—28no—
123C4⋊Dic47C22⋊C412yes—
124C4⋊Dic47C22⋊C411yescharacteristic
125C4⋊Dic47C22⋊C411yescharacteristic
126Dic47⋊C4C22⋊C414yes—
127Dic47⋊C4C22⋊C414yes—
128C4×Dic47C2×D414yes—
129C4×Dic47C2×D414yes—
130#10625329—22no—
131#10625329—22no—
132#10625329—22no—
133#10625329#4114yes—
134C2×C47⋊C8—22no—
135C2×C47⋊C8—22no—
136C23×C47C23×C411yescharacteristic
137C22×D47—24no—
138C22×D47—24no—
139C22×D47—24no—
140C22×D47—44no—
141C22×D47—44no—
142Q8×C47—42no—
143Q8×C47—42no—
144C2×C4×C47—22no—
145C2×C4×C47—22no—
146C2×C4×C47—22no—
147C2×C4×C47—22no—
148C2×C4×C47—22no—
149C2×C4×C47—41no—
150C2×C4×C47—42no—
151C2×C4×C47C2×C22⋊C412yes—
152C2×C4×C47C2×C22⋊C411yescharacteristic
153C2×C4×C47C2×C22⋊C411yescharacteristic
154C2×C4×C47C22×D412yes—
155C2×C4×C47C2×#4112yes—
156C2×C4×C47H(5,𝔽2)11yescharacteristic
157C2×C4×C472−1+411yescharacteristic
158C2×Dic47—42no—
159C2×Dic47—42no—
160C2×Dic47C2×C22⋊C412yes—
161C2×Dic47C2×C22⋊C412yes—
162C2×Dic47C2×C22⋊C414yes—
163D188—44no—
164D188—44no—
165D188—44no—
166D188—44no—
167D188—84no—
168C4×D47—84no—
169C4×D47—84no—
170Dic94—24no—
171Dic94—42no—
172C8×C47—44no—
173C47⋊C8—24no—
174C47⋊C8—42no—
175#22539—471nomaximal, Sylow
176C22×C47—21no—
177C22×C47—21no—
178C22×C47C22×C22⋊C411yescharacteristic
179C22×C47C4×#4111yescharacteristic
180C22×C47#52011yescharacteristic
181C2×D47—44no—
182C2×D47—44no—
183C2×D47—44no—
184C2×D47—84no—
185C2×D47—84no—
186C4×C47—21no—
187C4×C47—22no—
188C4×C47—42no—
189C4×C47—42no—
190C4×C47#41312yes—
191C4×C47#54812yes—
192C4×C47#55012yes—
193Dic47—24no—
194Dic47—22no—
195Dic47—22no—
196C2×#559—471no—
197Q8×M16—472no—
198#1967—474no—
199#1965—472no—
200#1801D4711yescharacteristic, complemented
201#1800—472no—
202C4×#101—471no—
203C4×#100—472no—
204C2×C47—21no—
205C2×C47—21no—
206C2×C47#234111yescharacteristic
207C2×C47C2×#41311yescharacteristic
208C2×C47#288811yescharacteristic
209D47—84no—
210#559—942no—
211#559—944no—
212#559—942no—
213C2×#127—471no—
214C2×#124—472no—
215C2×#124—472no—
216C2×#123—474no—
217C2×C4×Q8—471no—
218C2×C4×Q8C2×D4711yescharacteristic
219C2×C4×D4—472no—
220C2×C4×D4—472no—
221#446—942no—
222#446—944no—
223C8×Q8—942no—
224#423—474no—
225#423C2×D4712yes—
226C2×#101—472no—
227C2×#101C2×D4712yes—
228C2×#100—474no—
229C2×#100—474no—
230C4×M16—474no—
231C4×M16—472no—
232#343—471no—
233#343—471no—
234#343C2×D4712yes—
235C47#2253911yescharacteristic, complemented, Sylow
236C22×Q8—471no—
237C22×Q8C22×D4711yescharacteristic
238C22×D4—472no—
239C22×D4—472no—
240C22×D4—472no—
241C23×C4—474no—
242C2×M16—472no—
243C2×M16—472no—
244C2×M16C22×D4714yes—
245#127—942no—
246#127—942no—
247#124—944no—
248#124—944no—
249#124—944no—
250#124—944no—
251#123—948no—
252#123—948no—
253C4×Q8—22no—
254C4×Q8—22no—
255C4×Q8—472no—
256C4×Q8—474no—
257C4×Q8—472no—
258C4×D4—944no—
259C4×D4—942no—
260C4×D4—942no—
261C4×D4—944no—
262C4×D4—944no—
263C2×C4⋊C4—471no—
264C2×C4⋊C4—472no—
265C2×C4⋊C4C4×D4712yes—
266C2×C4⋊C4C22×D4711yescharacteristic
267C2×C22⋊C4—474no—
268C2×C22⋊C4—474no—
269C2×C22⋊C4—474no—
270C2×C42—472no—
271C2×C42C4×D4712yes—
272C2×C42C22×D4711yescharacteristic
273C4⋊C8—22no—
274C4⋊C8—22no—
275C4⋊C8—944no—
276C4⋊C8—944no—
277#101—42no—
278#101—944no—
279#100—948no—
280#100—1884no—
281#97—944no—
282#97—942no—
283C4×C8—944no—
284C4×C8—942no—
285C24—474no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—472no—
291C2×Q8—472no—
292C2×Q8—1882no—
293C2×D4—944no—
294C2×D4—942no—
295C2×D4—942no—
296C2×D4—944no—
297C2×D4—944no—
298C2×D4—942no—
299C2×D4—942no—
300C2×D4—3762no—
301C2×D4—1884no—
302C22×C4—472no—
303C22×C4—472no—
304C22×C4—944no—
305C22×C4—944no—
306C22×C4—3764no—
307C22×C4C2×C4×D4712yes—
308C22×C4C2×C4×D4711yescharacteristic
309C22×C4C2×C4×D4712yes—
310C22×C4C2×C4×D4711yescharacteristic
311C22×C4C23×D4711yescharacteristic
312M16—24no—
313M16—24no—
314M16—474no—
315M16—942no—
316M16—942no—
317M16—942no—
318C2×C8—24no—
319C2×C8—942no—
320C2×C8—942no—
321C4⋊C4—24no—
322C4⋊C4—22no—
323C4⋊C4—474no—
324C4⋊C4—474no—
325C4⋊C4—472no—
326C4⋊C4—471no—
327C4⋊C4—471no—
328C22⋊C4—944no—
329C22⋊C4—944no—
330C22⋊C4—948no—
331C22⋊C4—948no—
332C42—22no—
333C42—22no—
334C42—474no—
335C42—474no—
336C42D4×D4712yes—
337C42D4×D4712yes—
338C23—944no—
339C23—944no—
340C23—944no—
341C23—1884no—
342C23—1884no—
343C23C22×C4×D4711yescharacteristic
344Q8—42no—
345Q8—42no—
346Q8—944no—
347Q8—1882no—
348D4—3764no—
349D4—1884no—
350D4—1884no—
351D4—1884no—
352D4—1884no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—42no—
360C2×C4—474no—
361C2×C4—472no—
362C2×C4—472no—
363C2×C4—3764no—
364C2×C4—3764no—
365C2×C4—1882no—
366C2×C4—1882no—
367C2×C4C22⋊C4×D4711yescharacteristic
368C2×C4C22⋊C4×D4712yes—
369C2×C4C22⋊C4×D4711yescharacteristic
370C2×C4C2×D4×D4712yes—
371C2×C4#41×D4712yes—
372C2×C4#1309406911yescharacteristic
373C2×C4#1309407211yescharacteristic
374C8—44no—
375C8—944no—
376C8—1882no—
377C22—21no—
378C22—21no—
379C22—3764no—
380C22—3764no—
381C22—1884no—
382C22—1884no—
383C22—1884no—
384C22C2×C22⋊C4×D4711yescharacteristic
385C22#1568959111yescharacteristic
386C22#1568960411yescharacteristic
387C4—21no—
388C4—22no—
389C4—42no—
390C4—42no—
391C4—944no—
392C4—942no—
393C4—942no—
394C4#1568910612yes—
395C4#1568978312yes—
396C4#1568979112yes—
397C2—21no—
398C2—21no—
399C2—3764no—
400C2#2024622511yescharacteristic
401C2C2×#1568910611yescharacteristic
402C2#2025186511yescharacteristic
subgroupG/H
centerC22C2×C22⋊C4×D47
commutatorC23×C47C23×C4
FrattiniC22×C4C23×D47
FittingC47×#1801C2
socleC22×C47C22×C22⋊C4
prime radicalC2×#13094021C22
cosocleC22×C4×C47C24
hypercenter#1801D47
hypocenterC47#22539
LCSG ▷ C23×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1801

Complements

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

NH ≅ G/Ntype
2. #20252339C2semidirect
3. #20244236C2semidirect
5. #20243379C2semidirect
6. C4×#13093943C2semidirect
7. #20243179C2semidirect
8. C47×#1801C2semidirect
200. #1801D47semidirect
235. C47#22539semidirect

Maximal subgroups

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

subgroupindex
C2×#156898582
#202523392
#202442362
#202442272
#202433792
C4×#130939432
#202431792
C47×#18012
#2253947