Finite Group Databasebeta

← #26131262

Subgroups — #26131262

Order 12032.

746 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13093943C211yesmaximal, characteristic
2#20245974C211yesmaximal, characteristic
3#20241514C214yesmaximal, complemented
4C22×#101×C47C211yesmaximal, characteristic
5C22×#10625329—21no—
6C2×#13093983C411yescharacteristic
7C2×#13093983C411yescharacteristic
8C2×#13093943—22no—
9C2×#13093943—21no—
10C2×#13093943C414yescomplemented
11C2×#13093943C2212yes—
12#15689030—21no—
13#15689030—21no—
14#15689030C412yes—
15#15689030C2212yes—
16#15688986—21no—
17#15688986—21no—
18C23×Q8×C47C2211yescharacteristic
19C22×M16×C47—21no—
20C2×#101×C47—22no—
21C2×#101×C47—21no—
22C2×#101×C47C414yescomplemented
23C2×#101×C47C2212yes—
24C2×#10625329—24no—
25C2×#10625329—22no—
26C2×#10625329—42no—
27C2×#10625329—41no—
28C22×C47⋊C8—41no—
29#13093983—41no—
30#13093983—41no—
31#13093983—42no—
32C2×C4⋊Dic47—21no—
33C2×C4⋊Dic47C2×C412yes—
34C2×C22⋊Dic47—21no—
35C2×C22⋊Dic47—21no—
36C2×C4×Dic47—21no—
37C2×C4×Dic47C2×C412yes—
38#13093943—24no—
39#13093943—22no—
40#13093943C2×C414yes—
41C47⋊#95—22no—
42C47⋊#95—22no—
43C47⋊#95—22no—
44C47⋊#95—22no—
45C22×Q8×C47—41no—
46C22×Q8×C47—41no—
47C22×Q8×C47—42no—
48C22×Q8×C47C2×C412yes—
49C22×Q8×C47C2×C412yes—
50C22×Q8×C47C2×C411yescharacteristic
51C22×Q8×C47D412yes—
52C22×Q8×C47D411yescharacteristic
53C22×Q8×C47D411yescharacteristic
54C22×Q8×C47Q812yes—
55C22×Q8×C47C2311yescharacteristic
56C23×C4×C47C2×C411yescharacteristic
57C23×C4×C47D411yescharacteristic
58C23×C4×C47D411yescharacteristic
59C2×M16×C47—24no—
60C2×M16×C47—22no—
61C2×M16×C47—42no—
62C2×M16×C47—41no—
63C22×C8×C47—41no—
64#101×C47—24no—
65#101×C47—22no—
66#101×C47C2×C414yes—
67C24×C47C22⋊C411yescharacteristic
68C2×Q8×C47—42no—
69C2×Q8×C47—42no—
70C2×Q8×C47—42no—
71C2×Q8×C47—42no—
72C2×Q8×C47—41no—
73C2×Q8×C47—41no—
74C2×Q8×C47—42no—
75C2×Q8×C47—42no—
76C2×Q8×C47—41no—
77C2×Q8×C47—41no—
78C2×Q8×C47—41no—
79C2×Q8×C47—41no—
80C2×Q8×C47—42no—
81C2×Q8×C47—42no—
82C2×Q8×C47—41no—
83C2×Q8×C47—41no—
84C2×Q8×C47C4212yes—
85C2×Q8×C47C22⋊C412yes—
86C2×Q8×C47C22⋊C412yes—
87C2×Q8×C47C4⋊C412yes—
88C2×Q8×C47C4⋊C412yes—
89C2×Q8×C47C4⋊C412yes—
90C2×Q8×C47C22×C411yescharacteristic
91C2×Q8×C47C22×C411yescharacteristic
92C2×Q8×C47C2×D411yescharacteristic
93C2×Q8×C47C2×Q811yescharacteristic
94C22×C4×C47—21no—
95C22×C4×C47—22no—
96C22×C4×C47—21no—
97C22×C4×C47—21no—
98C22×C4×C47—21no—
99C22×C4×C47—22no—
100C22×C4×C47—21no—
101C22×C4×C47—22no—
102C22×C4×C47—21no—
103C22×C4×C47C4212yes—
104C22×C4×C47C22⋊C411yescharacteristic
105C22×C4×C47C22⋊C411yescharacteristic
106C22×C4×C47C22⋊C411yescharacteristic
107C22×C4×C47C22⋊C412yes—
108C22×C4×C47C22⋊C412yes—
109C22×C4×C47C4⋊C412yes—
110C22×C4×C47C4⋊C412yes—
111C22×C4×C47C4⋊C412yes—
112C22×C4×C47C22×C411yescharacteristic
113C22×C4×C47C2×D411yescharacteristic
114C22×C4×C47C2×D411yescharacteristic
115C22×Dic47—22no—
116C22×Dic47—22no—
117M16×C47—24no—
118M16×C47—44no—
119M16×C47—42no—
120C2×C8×C47—44no—
121C2×C8×C47—42no—
122C22⋊Dic47—41no—
123C22⋊Dic47—41no—
124C22⋊Dic47—42no—
125C4⋊Dic47—42no—
126C4⋊Dic47—42no—
127C4×Dic47—42no—
128C4×Dic47—42no—
129#10625329—24no—
130#10625329—44no—
131#10625329—42no—
132C2×C47⋊C8—44no—
133C2×C47⋊C8—42no—
134C23×C47—42no—
135C23×C47#9512yes—
136C23×C47#9911yescharacteristic
137C23×C47#9911yescharacteristic
138C23×C47#9912yes—
139C23×C47C2×C22⋊C411yescharacteristic
140Q8×C47—42no—
141Q8×C47—42no—
142Q8×C47—42no—
143Q8×C47—42no—
144Q8×C47—42no—
145Q8×C47—42no—
146Q8×C47—41no—
147Q8×C47—41no—
148Q8×C47—41no—
149Q8×C47—41no—
150C2×C4×C47—22no—
151C2×C4×C47—22no—
152C2×C4×C47—22no—
153C2×C4×C47—22no—
154C2×C4×C47—22no—
155C2×C4×C47—22no—
156C2×C4×C47—22no—
157C2×C4×C47—22no—
158C2×C4×C47—22no—
159C2×C4×C47—22no—
160C2×C4×C47—22no—
161C2×C4×C47—22no—
162C2×C4×C47—21no—
163C2×C4×C47—21no—
164C2×C4×C47—21no—
165C2×C4×C47—21no—
166C2×C4×C47—21no—
167C2×C4×C47—21no—
168C2×C4×C47—21no—
169C2×C4×C47—21no—
170C2×C4×C47—21no—
171C2×C4×C47—21no—
172C2×C4×C47—21no—
173C2×C4×C47—21no—
174C2×C4×C47#9512yes—
175C2×C4×C47#9512yes—
176C2×C4×C47#9512yes—
177C2×C4×C47C2×C4211yescharacteristic
178C2×C4×C47C2×C22⋊C411yescharacteristic
179C2×C4×C47C2×C22⋊C411yescharacteristic
180C2×C4×C47C2×C4⋊C411yescharacteristic
181C2×C4×C47C2×C4⋊C411yescharacteristic
182C2×C4×C47C2×C4⋊C411yescharacteristic
183C2×Dic47—44no—
184C2×Dic47—44no—
185C2×Dic47—42no—
186C2×Dic47—42no—
187C8×C47—44no—
188C47⋊C8—44no—
189#12586—471nomaximal, Sylow
190C22×C47—22no—
191C22×C47—22no—
192C22×C47—22no—
193C22×C47—42no—
194C22×C47—42no—
195C22×C47#34212yes—
196C22×C47#34212yes—
197C22×C47C2×#9511yescharacteristic
198C22×C47C2×#9911yescharacteristic
199C22×C47C2×#9911yescharacteristic
200C4×C47—21no—
201C4×C47—21no—
202C4×C47—22no—
203C4×C47—22no—
204C4×C47—22no—
205C4×C47—21no—
206C4×C47—21no—
207C4×C47—21no—
208C4×C47—21no—
209Dic47—82no—
210Dic47—82no—
211C22×#101—471no—
212C22×#101D4711yescharacteristic
213#2310—471no—
214#1394—474no—
215C2×C47—21no—
216C2×C47—22no—
217C2×C47—22no—
218C2×C47—21no—
219C2×C47C2×#34211yescharacteristic
220C2×C47#213312yes—
221C23×Q8C2×D4711yescharacteristic
222C22×M16—21no—
223C22×M16—941no—
224C2×#117—471no—
225C2×#117—471no—
226C2×#101—21no—
227C2×#101—22no—
228C2×#101—474no—
229C2×#101—472no—
230C2×#101—941no—
231C2×#101—942no—
232C2×#101Dic4714yescomplemented
233C2×#101C2×D4712yes—
234#391—472no—
235#391—472no—
236#391—941no—
237#391—941no—
238#381—941no—
239#381—941no—
240C47#1258611yescharacteristic, complemented, Sylow
241C22×Q8—41no—
242C22×Q8—41no—
243C22×Q8—42no—
244C22×Q8Dic9412yes—
245C22×Q8C4×D4712yes—
246C22×Q8C4×D4711yescharacteristic
247C22×Q8D18811yescharacteristic
248C22×Q8C2×Dic4712yes—
249C22×Q8C22⋊D4712yes—
250C22×Q8C22⋊D4711yescharacteristic
251C22×Q8C22×D4711yescharacteristic
252C23×C4C4×D4711yescharacteristic
253C23×C4D18811yescharacteristic
254C23×C4C22⋊D4711yescharacteristic
255C2×M16—24no—
256C2×M16—22no—
257C2×M16—41no—
258C2×M16—42no—
259C2×M16—944no—
260C2×M16—942no—
261C2×M16—1881no—
262C2×M16—1882no—
263C22×C8—41no—
264C22×C8—1881no—
265#117—1882no—
266#117—1881no—
267#117—1881no—
268C2×C4⋊C4—472no—
269C2×C4⋊C4—941no—
270C2×C22⋊C4—941no—
271C2×C22⋊C4—941no—
272C2×C42—472no—
273C2×C42—941no—
274#101—24no—
275#101—22no—
276#101—474no—
277#101—944no—
278#101—942no—
279#101C2×Dic4714yes—
280#95—942no—
281#95—942no—
282#95—942no—
283#95—942no—
284C24(C2×D47)⋊C411yescharacteristic
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—41no—
288C2×Q8—41no—
289C2×Q8—42no—
290C2×Q8—42no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8—42no—
294C2×Q8—42no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—42no—
298C2×Q8—42no—
299C2×Q8—41no—
300C2×Q8—41no—
301C2×Q8C4×Dic4712yes—
302C2×Q8Dic47⋊C412yes—
303C2×Q8Dic47⋊C412yes—
304C2×Q8C4⋊Dic4712yes—
305C2×Q8(C2×D47)⋊C412yes—
306C2×Q8C22⋊Dic4712yes—
307C2×Q8C2×Dic9411yescharacteristic
308C2×Q8C2×C4×D4711yescharacteristic
309C2×Q8C22×Dic4711yescharacteristic
310C2×Q8C2×C22⋊D4711yescharacteristic
311C22×C4—22no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—22no—
317C22×C4—21no—
318C22×C4—22no—
319C22×C4—21no—
320C22×C4—942no—
321C22×C4—942no—
322C22×C4C4×Dic4712yes—
323C22×C4Dic47⋊C412yes—
324C22×C4Dic47⋊C412yes—
325C22×C4C4⋊Dic4712yes—
326C22×C4(C2×D47)⋊C411yescharacteristic
327C22×C4(C2×D47)⋊C411yescharacteristic
328C22×C4(C2×D47)⋊C411yescharacteristic
329C22×C4(C2×D47)⋊C412yes—
330C22×C4C22⋊Dic4712yes—
331C22×C4C2×C4×D4711yescharacteristic
332C22×C4C2×D18811yescharacteristic
333C22×C4C2×C22⋊D4711yescharacteristic
334M16—24no—
335M16—42no—
336M16—44no—
337M16—944no—
338M16—1882no—
339M16—1884no—
340C2×C8—42no—
341C2×C8—44no—
342C2×C8—1882no—
343C2×C8—1884no—
344C4⋊C4—1882no—
345C4⋊C4—1882no—
346C22⋊C4—1882no—
347C22⋊C4—1881no—
348C22⋊C4—1881no—
349C42—1882no—
350C42—1882no—
351C23—42no—
352C23C47⋊#9512yes—
353C23#1309393811yescharacteristic
354C23#1309393811yescharacteristic
355C23#1309393812yes—
356C23C2×(C2×D47)⋊C411yescharacteristic
357Q8—42no—
358Q8—42no—
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—42no—
363Q8—41no—
364Q8—41no—
365Q8—41no—
366Q8—41no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—1884no—
392C2×C4—1884no—
393C2×C4—1882no—
394C2×C4—1882no—
395C2×C4C47⋊#9512yes—
396C2×C4C47⋊#9512yes—
397C2×C4C47⋊#9512yes—
398C2×C4C2×C4×Dic4711yescharacteristic
399C2×C4C2×(C2×D47)⋊C411yescharacteristic
400C2×C4C2×C22⋊Dic4711yescharacteristic
401C2×C4C2×Dic47⋊C411yescharacteristic
402C2×C4C2×Dic47⋊C411yescharacteristic
403C2×C4C2×C4⋊Dic4711yescharacteristic
404C8—44no—
405C8—1884no—
406C22—22no—
407C22—22no—
408C22—22no—
409C22—42no—
410C22—42no—
411C22#1568888912yes—
412C22#1568888912yes—
413C22C2×C47⋊#9511yescharacteristic
414C22C2×#1309393811yescharacteristic
415C22C2×#1309393811yescharacteristic
416C4—21no—
417C4—21no—
418C4—22no—
419C4—22no—
420C4—22no—
421C4—21no—
422C4—21no—
423C4—21no—
424C4—21no—
425C4—3762no—
426C4—3762no—
427C2—21no—
428C2—21no—
429C2—22no—
430C2—22no—
431C2C2×#1568888911yescharacteristic
432C2#2024507512yes—
subgroupG/H
centerC22C2×#13093938
commutatorC2×C4×C47C2×C42
FrattiniC22×Q8C22×D47
FittingC22×#101×C47C2
socleC22×C47C2×#99
prime radicalC24×C47C22⋊C4
cosocleC22×Q8×C47C23
hypercenterC22×#101D47
hypocenterC47#12586
LCSG ▷ C2×C4×C47 ▷ C22×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ C22×#101

Complements

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

NH ≅ G/Ntype
3. #20241514C2semidirect
10. C2×#13093943C4semidirect
22. C2×#101×C47C4semidirect
232. C2×#101Dic47semidirect
240. C47#12586semidirect

Maximal subgroups

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

subgroupindex
C22×#130939432
#202459742
#202415142
C22×#101×C472
#1258647