Finite Group Databasebeta

← #26245403

Subgroups — #26245403

Order 12032.

766 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
1#20251978C211yesmaximal, characteristic
2#20251977C211yesmaximal, characteristic
3C2×#15689159C211yesmaximal, characteristic
4C2×#15689151C211yesmaximal, characteristic
5#20244902C214yesmaximal, complemented
6#20244897C214yesmaximal, complemented
7#20243126C211yesmaximal, characteristic
8C2×#15688887C211yesmaximal, characteristic
9C47×#1742C211yesmaximal, characteristic
10C22×#10625329C2211yescharacteristic
11C23×C47⋊C8C2211yescharacteristic
12C2×#13093983C2211yescharacteristic
13C2×#13093983C2211yescharacteristic
14#15689166—22no—
15#15689165—22no—
16#15689165C2214yes—
17#15689163C2214yes—
18#15689161—22no—
19#15689159—22no—
20#15689159—22no—
21#15689159—22no—
22C2×#13093957C2212yes—
23C2×#13093955C2212yes—
24#15689151—22no—
25#15689151C2212yes—
26#15689151C2212yes—
27C2×Dic47⋊C8—22no—
28#15688891C2214yes—
29#15688887C2212yes—
30#15688887C2212yes—
31C22×M16×C47C2211yescharacteristic
32C22×M16×C47C2211yescharacteristic
33C2×#117×C47C2211yescharacteristic
34C47×#343C2214yes—
35C2×#10625329—21no—
36C2×#10625329—22no—
37C2×#10625329Q814yes—
38C2×#10625329C2312yes—
39C22×C47⋊C8—21no—
40C22×C47⋊C8—22no—
41C22×C47⋊C8—22no—
42C22×C47⋊C8D412yes—
43C22×C47⋊C8D412yes—
44C22×C47⋊C8C2311yescharacteristic
45C22×C47⋊C8C2311yescharacteristic
46#13093983—21no—
47#13093983—21no—
48#13093983—21no—
49#13093983—21no—
50#13093983—21no—
51#13093983—21no—
52#13093983D412yes—
53#13093983C2312yes—
54C2×C4⋊Dic47D412yes—
55C2×C4⋊Dic47C2312yes—
56C2×C22⋊Dic47—21no—
57C2×C22⋊Dic47—21no—
58C2×C4×Dic47—21no—
59C2×C4×Dic47—21no—
60#13093958—24no—
61#13093957—24no—
62#13093957—24no—
63#13093956—24no—
64#13093955—24no—
65#13093955—24no—
66Dic47⋊C8—24no—
67Dic47⋊C8—24no—
68C23×C4×C47C2311yescharacteristic
69C2×M16×C47—21no—
70C2×M16×C47—22no—
71C2×M16×C47—21no—
72C2×M16×C47—22no—
73C2×M16×C47—22no—
74C2×M16×C47Q814yes—
75C2×M16×C47C2312yes—
76C2×M16×C47C2312yes—
77C22×C8×C47—21no—
78C22×C8×C47—21no—
79#117×C47—21no—
80#117×C47—21no—
81#117×C47—21no—
82#117×C47—21no—
83C2×C4⋊C4×C47C2312yes—
84C2×C22⋊C4×C47—21no—
85C2×C42×C47—21no—
86C24×C47#4111yescharacteristic
87C22×C4×C47—22no—
88C22×C4×C47C2×D411yescharacteristic
89C22×C4×C47C2×D411yescharacteristic
90C22×C4×C47C2×D411yescharacteristic
91C22×C4×C47C2×D411yescharacteristic
92C22×C4×C47C2×Q812yes—
93C22×C4×C47C2×Q812yes—
94C22×C4×C47#4111yescharacteristic
95C22×C4×C47#4112yes—
96C22×C4×C47#4112yes—
97C22×C4×C47C2411yescharacteristic
98C22×Dic47—22no—
99C22×Dic47—22no—
100M16×C47—24no—
101M16×C47—22no—
102M16×C47—22no—
103M16×C47—22no—
104M16×C47—42no—
105M16×C47C2×Q814yes—
106C2×C8×C47—24no—
107C2×C8×C47—22no—
108C2×C8×C47—22no—
109C2×C8×C47—42no—
110C4⋊C4×C47—22no—
111C4⋊C4×C47—22no—
112C22⋊C4×C47—41no—
113C22⋊C4×C47—41no—
114C42×C47—41no—
115C42×C47—41no—
116C22⋊Dic47—22no—
117C22⋊Dic47—22no—
118C22⋊Dic47—41no—
119C22⋊Dic47—41no—
120C4⋊Dic47—22no—
121C4⋊Dic47—22no—
122C4⋊Dic47—22no—
123C4⋊Dic47C2×D414yes—
124C4×Dic47—22no—
125C4×Dic47—22no—
126C4×Dic47—41no—
127C4×Dic47—41no—
128#10625329—24no—
129#10625329—22no—
130#10625329C2×Q814yes—
131C2×C47⋊C8—24no—
132C2×C47⋊C8—24no—
133C2×C47⋊C8—24no—
134C2×C47⋊C8—22no—
135C2×C47⋊C8—21no—
136C2×C47⋊C8—21no—
137C2×C47⋊C8—21no—
138C2×C47⋊C8—21no—
139C2×C47⋊C8C2×D412yes—
140C2×C47⋊C8C2×D412yes—
141C23×C47—42no—
142C23×C47#12111yescharacteristic
143C23×C47#12111yescharacteristic
144C23×C47#12512yes—
145C23×C47#12512yes—
146C23×C47C2×#4111yescharacteristic
147C2×C4×C47—42no—
148C2×C4×C47—82no—
149C2×C4×C47#12111yescharacteristic
150C2×C4×C47#12111yescharacteristic
151C2×C4×C47C22⋊Q812yes—
152C2×C4×C47C22⋊Q812yes—
153C2×C4×C47C22⋊Q812yes—
154C2×C4×C47C22⋊Q812yes—
155C2×C4×C47#12312yes—
156C2×C4×C47#12312yes—
157C2×C4×C47#12512yes—
158C2×C4×C47#12512yes—
159C2×C4×C47C4⋊Q812yes—
160C2×C4×C47C4⋊Q812yes—
161C2×C4×C47C22×D411yescharacteristic
162C2×C4×C47C22×D411yescharacteristic
163C2×C4×C47C22×Q811yescharacteristic
164C2×C4×C47C22×Q811yescharacteristic
165C2×C4×C47C2×#4111yescharacteristic
166C2×C4×C47C2×#4111yescharacteristic
167C2×Dic47—24no—
168C2×Dic47—44no—
169C2×Dic47—42no—
170C2×Dic47—82no—
171C8×C47—24no—
172C8×C47—42no—
173C47⋊C8—24no—
174C47⋊C8—22no—
175C47⋊C8—22no—
176#23674—471nomaximal, Sylow
177C22×C47—22no—
178C22×C47—22no—
179C22×C47—22no—
180C22×C47—42no—
181C22×C47—42no—
182C22×C47#39812yes—
183C22×C47#39812yes—
184C22×C47C2×#12111yescharacteristic
185C22×C47C2×#12511yescharacteristic
186C22×C47C2×#12511yescharacteristic
187C4×C47—82no—
188C4×C47#39812yes—
189C4×C47#39812yes—
190C4×C47C2×C22⋊Q811yescharacteristic
191C4×C47C2×C22⋊Q811yescharacteristic
192C4×C47C2×#12311yescharacteristic
193C4×C47C2×C4⋊Q811yescharacteristic
194Dic47—44no—
195Dic47—82no—
196#2911—471no—
197#2911—471no—
198C2×#427—471no—
199C2×#424—471no—
200#2090—474no—
201#2089—474no—
202#1742—471no—
203#1742D4711yescharacteristic
204C2×#341—471no—
205C2×C47—21no—
206C2×C47—21no—
207C2×C47—22no—
208C2×C47—22no—
209C2×C47C2×#39811yescharacteristic
210C2×C47#306712yes—
211C22×M16—471no—
212C22×M16C2×D4711yescharacteristic
213C22×M16C2×D4711yescharacteristic
214C23×C8—471no—
215C2×#117—471no—
216C2×#117—471no—
217C2×#117C2×D4711yescharacteristic
218#428—474no—
219#428—474no—
220#428—942no—
221#428—942no—
222#427—942no—
223#427—942no—
224#427—942no—
225#427—942no—
226C2×#107—472no—
227C2×#106—472no—
228#424—472no—
229#424—472no—
230#424—942no—
231C2×C4⋊C8—942no—
232#343—474no—
233#343C2×D4714yes—
234#341—472no—
235#341—472no—
236C47#2367411yescharacteristic, complemented, Sylow
237C23×C4C22×D4711yescharacteristic
238C2×M16—21no—
239C2×M16—22no—
240C2×M16—21no—
241C2×M16—22no—
242C2×M16—22no—
243C2×M16—474no—
244C2×M16—472no—
245C2×M16—942no—
246C2×M16—941no—
247C2×M16Dic9414yes—
248C2×M16C22×D4712yes—
249C2×M16C22×D4712yes—
250C22×C8—21no—
251C22×C8—21no—
252C22×C8—472no—
253C22×C8—472no—
254C22×C8—471no—
255C22×C8—471no—
256C22×C8—942no—
257C22×C8—941no—
258C22×C8—942no—
259#117—21no—
260#117—21no—
261#117—21no—
262#117—21no—
263#117—472no—
264#117—472no—
265#117—941no—
266#117—941no—
267#117—941no—
268#117—941no—
269#117—941no—
270#117—941no—
271C2×C4⋊C4—472no—
272C2×C4⋊C4—472no—
273C2×C4⋊C4C22×D4712yes—
274C2×C22⋊C4—21no—
275C2×C22⋊C4—941no—
276C2×C22⋊C4—941no—
277C2×C42—21no—
278C2×C42—941no—
279C2×C42—941no—
280#107—944no—
281#107—944no—
282#107—944no—
283#106—944no—
284#106—944no—
285#106—944no—
286C4⋊C8—944no—
287C4⋊C8—944no—
288C24#1062535011yescharacteristic
289C22×C4—22no—
290C22×C4—942no—
291C22×C4—942no—
292C22×C4C2×Dic9412yes—
293C22×C4#1062535011yescharacteristic
294C22×C4#1062535012yes—
295C22×C4D4×D4711yescharacteristic
296C22×C4D4×D4711yescharacteristic
297C22×C4D4×D4711yescharacteristic
298C22×C4D4×D4711yescharacteristic
299C22×C4#1062535212yes—
300C22×C4Q8×D4712yes—
301C22×C4C23×D4711yescharacteristic
302M16—24no—
303M16—22no—
304M16—22no—
305M16—22no—
306M16—42no—
307M16—474no—
308M16—944no—
309M16—942no—
310M16C2×Dic9414yes—
311C2×C8—24no—
312C2×C8—22no—
313C2×C8—22no—
314C2×C8—42no—
315C2×C8—472no—
316C2×C8—472no—
317C2×C8—944no—
318C2×C8—944no—
319C2×C8—944no—
320C2×C8—942no—
321C2×C8—941no—
322C2×C8—941no—
323C2×C8—941no—
324C2×C8—941no—
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—474no—
328C4⋊C4—942no—
329C4⋊C4—942no—
330C4⋊C4—942no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—942no—
334C22⋊C4—942no—
335C22⋊C4—1881no—
336C22⋊C4—1881no—
337C42—41no—
338C42—41no—
339C42—942no—
340C42—942no—
341C42—1881no—
342C42—1881no—
343C23—42no—
344C23#1309399711yescharacteristic
345C23#1309399711yescharacteristic
346C23#1309401412yes—
347C23#1309401612yes—
348C23C2×#1062535011yescharacteristic
349C2×C4—42no—
350C2×C4—82no—
351C2×C4—944no—
352C2×C4—3762no—
353C2×C4—1882no—
354C2×C4—1884no—
355C2×C4#1309399711yescharacteristic
356C2×C4#1309399711yescharacteristic
357C2×C4#1309400112yes—
358C2×C4#1309400112yes—
359C2×C4#1309400212yes—
360C2×C4#1309400212yes—
361C2×C4#1309400612yes—
362C2×C4#1309400612yes—
363C2×C4#1309401412yes—
364C2×C4#1309401612yes—
365C2×C4#1309402312yes—
366C2×C4#1309402312yes—
367C2×C4C2×D4×D4711yescharacteristic
368C2×C4C2×D4×D4711yescharacteristic
369C2×C4C2×Q8×D4711yescharacteristic
370C2×C4C22×Dic9411yescharacteristic
371C2×C4C2×#1062535211yescharacteristic
372C2×C4C2×#1062535011yescharacteristic
373C8—24no—
374C8—42no—
375C8—944no—
376C8—942no—
377C8—942no—
378C22—22no—
379C22—22no—
380C22—22no—
381C22—42no—
382C22—42no—
383C22#1568905712yes—
384C22#1568905712yes—
385C22C2×#1309399711yescharacteristic
386C22C2×#1309401411yescharacteristic
387C22C2×#1309401611yescharacteristic
388C4—82no—
389C4—3762no—
390C4—1884no—
391C4#1568905712yes—
392C4#1568905712yes—
393C4C2×#1309400111yescharacteristic
394C4C2×#1309400211yescharacteristic
395C4C2×#1309400611yescharacteristic
396C4C2×#1309402311yescharacteristic
397C2—22no—
398C2—22no—
399C2—21no—
400C2—21no—
401C2C2×#1568905711yescharacteristic
402C2#2025310712yes—
subgroupG/H
centerC2×C4#13093997
commutator, cosocleC22×C4×C47C24
FrattiniC22×C4C23×D47
FittingC47×#1742C2
socleC22×C47C2×#121
prime radicalC24×C47#41
hypercenter#1742D47
hypocenterC47#23674
LCSG ▷ C22×C4×C47 ▷ C2×C47 ▷ C47
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1742

Complements

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

NH ≅ G/Ntype
5. #20244902C2semidirect
6. #20244897C2semidirect
236. C47#23674semidirect

Maximal subgroups

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

subgroupindex
#202519782
#202519772
C2×#156891592
C2×#156891512
#202449022
#202448972
#202431262
C2×#156888872
C47×#17422
#2367447