Finite Group Databasebeta

← #26201627

Subgroups — #26201627

Order 12032.

676 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254509C211yesmaximal, characteristic
2#20252860C211yesmaximal, characteristic
3#20252469C211yesmaximal, characteristic, complemented
4#20242834C214yesmaximal, complemented
5#20242780C214yesmaximal, complemented
6#20242162C211yesmaximal, characteristic, complemented
7#20242060C211yesmaximal, characteristic, complemented
8#20242038C211yesmaximal, characteristic
9C2×C47×#462C211yesmaximal, characteristic, complemented
10C2×#13094055—21no—
11#15689673C2211yescharacteristic
12#15689526C2212yes—
13#15689471C2212yes—
14#15689436—22no—
15#15689400—22no—
16#15689345—22no—
17#15689333C2212yes—
18#15689323C2212yes—
19Q8×C47⋊C8—22no—
20Q8×C47⋊C8C2214yes—
21#15689176—22no—
22#15689176C2211yescharacteristic
23#15689176C2211yescharacteristic
24#15689147C2211yescharacteristic
25#15689122—21no—
26#15688875C2214yes—
27#15688870C2212yes—
28#15688869C2214yes—
29#15688866C2212yes—
30C22×Q16×C47—21no—
31C2×C4⋊Q8×C47C2211yescharacteristic
32C2×C4×Q8×C47C2211yescharacteristic
33C47×#462—22no—
34C47×#462C2214yes—
35C2×C4⋊C8×C47C2211yescharacteristic
36C2×#103×C47—21no—
37C2×#10625350—21no—
38#13094055—44no—
39C2×#10625337—22no—
40C2×#10625336—22no—
41C2×#10625329—21no—
42#13094024C2311yescharacteristic
43#13094021C2311yescharacteristic
44#13094012—21no—
45#13093998—21no—
46#13093998—21no—
47C4×D188—21no—
48C4×D188—21no—
49#13093957—24no—
50C4⋊(C47⋊C8)—21no—
51C4⋊(C47⋊C8)—24no—
52C4⋊(C47⋊C8)D412yes—
53C4⋊(C47⋊C8)C2312yes—
54C4⋊(C47⋊C8)C2312yes—
55#13093947—22no—
56#13093944—22no—
57#13093944—24no—
58C22⋊(C47⋊C8)—21no—
59C22⋊(C47⋊C8)D412yes—
60C4×C47⋊C8—21no—
61C4×C47⋊C8—24no—
62C4×C47⋊C8C2312yes—
63C22×Q8×C47—21no—
64C22×Q8×C47D411yescharacteristic
65C2×Q16×C47—24no—
66C2×Q16×C47—44no—
67C22×C8×C47—21no—
68C4⋊Q8×C47—21no—
69C4⋊Q8×C47—21no—
70C4⋊Q8×C47—21no—
71C4⋊Q8×C47C2312yes—
72C4×Q8×C47—21no—
73C4×Q8×C47—21no—
74C4×Q8×C47D412yes—
75C4×Q8×C47C2312yes—
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47—21no—
78C2×C4⋊C4×C47—21no—
79C2×C4⋊C4×C47D411yescharacteristic
80C2×C42×C47—21no—
81C2×C42×C47C2311yescharacteristic
82C4⋊C8×C47—21no—
83C4⋊C8×C47C2312yes—
84#103×C47—24no—
85C2×Q8×C47—21no—
86C2×Q8×C47—22no—
87C2×Q8×C47—22no—
88C2×Q8×C47—41no—
89C2×Q8×C47—42no—
90C2×Q8×C47—42no—
91C2×Q8×C47D814yes—
92C2×Q8×C47C2×D412yes—
93C22×C4×C47—21no—
94C22×C4×C47—21no—
95C22×C4×C47—41no—
96C22×C4×C47C2×D411yescharacteristic
97C22×C4×C47C2×D411yescharacteristic
98C22×C4×C47#4111yescharacteristic
99C2×C22⋊D47—41no—
100#10625350—42no—
101#10625350—42no—
102C2×D188—21no—
103C2×D188—21no—
104C2×D188—21no—
105C2×C4×D47—41no—
106C2×Dic94—21no—
107Q16×C47—48no—
108C2×C8×C47—22no—
109C2×C8×C47—42no—
110C4⋊C4×C47—22no—
111C4⋊C4×C47—21no—
112C4⋊C4×C47—22no—
113C4⋊C4×C47—22no—
114C4⋊C4×C47—22no—
115C4⋊C4×C47—41no—
116C4⋊C4×C47—41no—
117C4⋊C4×C47C2×D412yes—
118C42×C47—22no—
119C42×C47—22no—
120C42×C47C2×D411yescharacteristic
121C42×C47C2×D411yescharacteristic
122C42×C47#4111yescharacteristic
123C42×C47C2411yescharacteristic
124#10625337—44no—
125#10625336—44no—
126(C2×D47)⋊C4—41no—
127(C2×D47)⋊C4—41no—
128C4⋊Dic47—21no—
129C4⋊Dic47—21no—
130#10625329—42no—
131C2×C47⋊C8—22no—
132C2×C47⋊C8—22no—
133C2×C47⋊C8C2×D414yes—
134C23×C47#12111yescharacteristic
135C22×D47—41no—
136Q8×C47—22no—
137Q8×C47—24no—
138Q8×C47—42no—
139Q8×C47—42no—
140Q8×C47—82no—
141Q8×C47C2×D814yes—
142C2×C4×C47—22no—
143C2×C4×C47—24no—
144C2×C4×C47—22no—
145C2×C4×C47—22no—
146C2×C4×C47—41no—
147C2×C4×C47—42no—
148C2×C4×C47—81no—
149C2×C4×C47—82no—
150C2×C4×C47#12111yescharacteristic
151C2×C4×C47#12111yescharacteristic
152C2×C4×C47#12111yescharacteristic
153C2×C4×C47C2×D812yes—
154C2×C4×C47#13512yes—
155C2×C4×C47C22×D411yescharacteristic
156C2×C4×C47C22×D411yescharacteristic
157C2×C4×C47C2×#4111yescharacteristic
158C22⋊D47—82no—
159C2×Dic47—41no—
160D188—41no—
161D188—41no—
162D188—42no—
163D188—81no—
164C4×D47—82no—
165Dic94—41no—
166Dic94—41no—
167C8×C47—42no—
168C47⋊C8—24no—
169C47⋊C8—42no—
170C47⋊C8—42no—
171#19404—471nomaximal, Sylow
172C22×C47—21no—
173C22×C47—21no—
174C22×C47#46612yes—
175C22×C47C2×#12111yescharacteristic
176C2×D47—81no—
177C2×D47—81no—
178C2×D47—81no—
179C4×C47—21no—
180C4×C47—24no—
181C4×C47—21no—
182C4×C47—21no—
183C4×C47—42no—
184C4×C47—82no—
185C4×C47#46612yes—
186C4×C47C22×D811yescharacteristic
187C4×C47C2×#13511yescharacteristic
188C4×C47#57812yes—
189Dic47—81no—
190#3226—471no—
191#3037—471no—
192C2×#462D4711yescharacteristic, complemented
193#2982—471no—
194#1682—474no—
195#1674—474no—
196#1571—471no—
197#1546—471no—
198#1540—471no—
199C2×C47—22no—
200C2×C47#304411yescharacteristic
201C2×C47C2×#46611yescharacteristic
202C2×C47#306911yescharacteristic
203D47—161no—
204C2×#137—941no—
205C22×Q16—21no—
206#532—471no—
207C2×C4⋊Q8C2×D4711yescharacteristic
208C2×C4×Q8C2×D4711yescharacteristic
209#500—472no—
210#486—472no—
211#480—942no—
212#474—942no—
213#464—942no—
214#462—22no—
215#462—472no—
216#462C2×D4714yes—
217#460—472no—
218C8×Q8—474no—
219C8×Q8—942no—
220#431—471no—
221#431—471no—
222#431—942no—
223#423—471no—
224C2×C4⋊C8C2×D4711yescharacteristic
225#417—941no—
226C2×#103—21no—
227#335—474no—
228#332—474no—
229#332—472no—
230#331—472no—
231C47#1940411yescharacteristic, complemented, Sylow
232C2×#41—941no—
233C22×Q8—21no—
234C22×Q8C22⋊D4711yescharacteristic
235#137—1884no—
236C2×Q16—24no—
237C2×Q16—44no—
238C2×Q16—942no—
239C2×SD16—942no—
240C2×M16—941no—
241C22×C8—21no—
242C4⋊Q8—21no—
243C4⋊Q8—21no—
244C4⋊Q8—21no—
245C4⋊Q8—471no—
246C4⋊Q8C22×D4712yes—
247#127—471no—
248#124—941no—
249#121—941no—
250#121—941no—
251C4×Q8—21no—
252C4×Q8—21no—
253C4×Q8C22⋊D4712yes—
254C4×Q8C22×D4712yes—
255C4×D4—941no—
256C4×D4—941no—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4C22⋊D4711yescharacteristic
261C2×C42—21no—
262C2×C42C22×D4711yescharacteristic
263#107—944no—
264C4⋊C8—21no—
265C4⋊C8—472no—
266C4⋊C8—472no—
267C4⋊C8—472no—
268C4⋊C8—944no—
269C4⋊C8—941no—
270C4⋊C8C22×D4712yes—
271#103—24no—
272#103—942no—
273#102—944no—
274#102—942no—
275C22⋊C8—472no—
276C22⋊C8—941no—
277C4×C8—472no—
278C4×C8—944no—
279C4×C8—941no—
280#41—1882no—
281#41—1882no—
282C2×Q8—21no—
283C2×Q8—22no—
284C2×Q8—22no—
285C2×Q8—41no—
286C2×Q8—42no—
287C2×Q8—42no—
288C2×Q8—941no—
289C2×Q8#1062533414yes—
290C2×Q8C2×C22⋊D4712yes—
291C2×D4—941no—
292C2×D4—941no—
293C2×D4—941no—
294C2×D4—1881no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—41no—
298C22×C4—1881no—
299C22×C4D4×D4711yescharacteristic
300C22×C4#1062535211yescharacteristic
301C22×C4C2×C22⋊D4711yescharacteristic
302Q16—48no—
303Q16—1884no—
304SD16—1884no—
305M16—1882no—
306C2×C8—22no—
307C2×C8—42no—
308C2×C8—474no—
309C2×C8—942no—
310C2×C8—942no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—22no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—941no—
319C4⋊C4—941no—
320C4⋊C4C2×C22⋊D4712yes—
321C22⋊C4—1881no—
322C22⋊C4—1881no—
323C42—22no—
324C42—22no—
325C42D4×D4711yescharacteristic
326C42#1062535211yescharacteristic
327C42C2×C22⋊D4711yescharacteristic
328C42C23×D4711yescharacteristic
329C23—1881no—
330C23#1309399611yescharacteristic
331Q8—24no—
332Q8—22no—
333Q8—42no—
334Q8—42no—
335Q8—82no—
336Q8—1881no—
337Q8—1881no—
338Q8C2×#1062533414yes—
339D4—3761no—
340D4—3762no—
341D4—1881no—
342D4—1881no—
343D4—1882no—
344C2×C4—24no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—81no—
351C2×C4—82no—
352C2×C4—3762no—
353C2×C4—1881no—
354C2×C4#1309399611yescharacteristic
355C2×C4#1309399611yescharacteristic
356C2×C4#1309399611yescharacteristic
357C2×C4C2×#1062533412yes—
358C2×C4#1309404412yes—
359C2×C4C2×D4×D4711yescharacteristic
360C2×C4C22×C22⋊D4711yescharacteristic
361C2×C4C2×#1062535211yescharacteristic
362C8—42no—
363C8—944no—
364C8—1882no—
365C8—1882no—
366C22—21no—
367C22—21no—
368C22—3761no—
369C22—3761no—
370C22—3761no—
371C22#1568935712yes—
372C22C2×#1309399611yescharacteristic
373C4—24no—
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—82no—
379C4—3761no—
380C4#1568935712yes—
381C4C22×#1062533411yescharacteristic
382C4C2×#1309404411yescharacteristic
383C4#1568993612yes—
384C2—22no—
385C2—7521no—
386C2#2025291511yescharacteristic
387C2C2×#1568935711yescharacteristic
388C2#2025312211yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#462C2
socleC22×C47C2×#121
prime radical#15689673C22
hypercenterC2×#462D47
hypocenterC47#19404
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#462

Complements

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

NH ≅ G/Ntype
3. #20252469C2semidirect
4. #20242834C2semidirect
5. #20242780C2semidirect
6. #20242162C2semidirect
7. #20242060C2semidirect
9. C2×C47×#462C2semidirect
192. C2×#462D47semidirect
231. C47#19404semidirect

Maximal subgroups

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

subgroupindex
#202545092
#202528602
#202524692
#202428342
#202427802
#202421622
#202420602
#202420382
C2×C47×#4622
#1940447