Finite Group Databasebeta

← #26201618

Subgroups — #26201618

Order 12032.

644 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20254516C211yesmaximal, characteristic
2#20252860C211yesmaximal, characteristic
3#20252469C211yesmaximal, characteristic, complemented
4#20242806C214yesmaximal, complemented
5#20242772C214yesmaximal, complemented
6#20242158C211yesmaximal, characteristic, complemented
7#20242069C211yesmaximal, characteristic, complemented
8#20242038C211yesmaximal, characteristic
9C2×C47×#477C211yesmaximal, characteristic, complemented
10C2×#13094055—21no—
11#15689673C2211yescharacteristic
12#15689519C2212yes—
13#15689471C2212yes—
14#15689441—22no—
15#15689410—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#15688873C2214yes—
27#15688872C2214yes—
28#15688870C2212yes—
29#15688866C2212yes—
30C2×C4⋊Q8×C47C2211yescharacteristic
31C2×C4×Q8×C47C2211yescharacteristic
32C47×#477—22no—
33C47×#477C2214yes—
34C2×#107×C47—21no—
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#13093955—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
65C22×C8×C47—21no—
66C4⋊Q8×C47—21no—
67C4⋊Q8×C47—21no—
68C4⋊Q8×C47—21no—
69C4⋊Q8×C47C2312yes—
70C4×Q8×C47—21no—
71C4×Q8×C47—21no—
72C4×Q8×C47D412yes—
73C4×Q8×C47C2312yes—
74C2×C4⋊C4×C47—21no—
75C2×C4⋊C4×C47—21no—
76C2×C4⋊C4×C47—21no—
77C2×C4⋊C4×C47D411yescharacteristic
78C2×C42×C47—21no—
79C2×C42×C47C2311yescharacteristic
80#107×C47—24no—
81C4⋊C8×C47—21no—
82C4⋊C8×C47C2312yes—
83#103×C47—24no—
84C2×Q8×C47—21no—
85C2×Q8×C47—22no—
86C2×Q8×C47—22no—
87C2×Q8×C47—41no—
88C2×Q8×C47—82no—
89C2×Q8×C47SD1614yes—
90C2×Q8×C47C2×D412yes—
91C22×C4×C47—21no—
92C22×C4×C47—21no—
93C22×C4×C47—41no—
94C22×C4×C47C2×D411yescharacteristic
95C22×C4×C47C2×D411yescharacteristic
96C22×C4×C47#4111yescharacteristic
97C2×C22⋊D47—41no—
98#10625350—42no—
99#10625350—42no—
100C2×D188—21no—
101C2×D188—21no—
102C2×D188—21no—
103C2×C4×D47—41no—
104C2×Dic94—21no—
105C2×C8×C47—22no—
106C2×C8×C47—42no—
107C4⋊C4×C47—22no—
108C4⋊C4×C47—21no—
109C4⋊C4×C47—22no—
110C4⋊C4×C47—22no—
111C4⋊C4×C47—22no—
112C4⋊C4×C47—41no—
113C4⋊C4×C47—41no—
114C4⋊C4×C47C2×D412yes—
115C42×C47—22no—
116C42×C47—22no—
117C42×C47C2×D411yescharacteristic
118C42×C47C2×D411yescharacteristic
119C42×C47#4111yescharacteristic
120C42×C47C2411yescharacteristic
121#10625337—44no—
122#10625336—44no—
123(C2×D47)⋊C4—41no—
124(C2×D47)⋊C4—41no—
125C4⋊Dic47—21no—
126C4⋊Dic47—21no—
127#10625329—42no—
128C2×C47⋊C8—22no—
129C2×C47⋊C8—22no—
130C2×C47⋊C8C2×D414yes—
131C23×C47#12111yescharacteristic
132C22×D47—41no—
133Q8×C47—22no—
134Q8×C47—24no—
135Q8×C47—82no—
136Q8×C47—82no—
137Q8×C47C2×SD1614yes—
138C2×C4×C47—22no—
139C2×C4×C47—24no—
140C2×C4×C47—22no—
141C2×C4×C47—22no—
142C2×C4×C47—41no—
143C2×C4×C47—42no—
144C2×C4×C47—81no—
145C2×C4×C47—82no—
146C2×C4×C47#12111yescharacteristic
147C2×C4×C47#12111yescharacteristic
148C2×C4×C47#12111yescharacteristic
149C2×C4×C47C2×SD1612yes—
150C2×C4×C47#13512yes—
151C2×C4×C47C22×D411yescharacteristic
152C2×C4×C47C22×D411yescharacteristic
153C2×C4×C47C2×#4111yescharacteristic
154C22⋊D47—82no—
155C2×Dic47—41no—
156D188—41no—
157D188—41no—
158D188—42no—
159D188—81no—
160C4×D47—82no—
161Dic94—41no—
162Dic94—41no—
163C8×C47—42no—
164C47⋊C8—24no—
165C47⋊C8—42no—
166C47⋊C8—42no—
167#19403—471nomaximal, Sylow
168C22×C47—21no—
169C22×C47—21no—
170C22×C47#46512yes—
171C22×C47C2×#12111yescharacteristic
172C2×D47—81no—
173C2×D47—81no—
174C2×D47—81no—
175C4×C47—21no—
176C4×C47—24no—
177C4×C47—21no—
178C4×C47—21no—
179C4×C47—42no—
180C4×C47—82no—
181C4×C47#46512yes—
182C4×C47C22×SD1611yescharacteristic
183C4×C47C2×#13511yescharacteristic
184C4×C47#57812yes—
185Dic47—81no—
186#3227—471no—
187C2×#477D4711yescharacteristic, complemented
188#3037—471no—
189#2982—471no—
190#1678—474no—
191#1672—474no—
192#1570—471no—
193#1548—471no—
194#1540—471no—
195C2×C47—22no—
196C2×C47#304411yescharacteristic
197C2×C47C2×#46511yescharacteristic
198C2×C47#306811yescharacteristic
199D47—161no—
200C2×#137—941no—
201#532—471no—
202C2×C4⋊Q8C2×D4711yescharacteristic
203C2×C4×Q8C2×D4711yescharacteristic
204#498—472no—
205#486—472no—
206#481—942no—
207#477—22no—
208#477C2×D4714yes—
209#476—942no—
210#464—942no—
211#462—472no—
212#460—472no—
213C8×Q8—474no—
214C8×Q8—942no—
215#431—471no—
216#431—471no—
217#431—942no—
218C2×#107—21no—
219#423—471no—
220C2×C4⋊C8C2×D4711yescharacteristic
221#417—941no—
222C2×#103—21no—
223#334—474no—
224#333—474no—
225#332—472no—
226#331—472no—
227C47#1940311yescharacteristic, complemented, Sylow
228C2×#41—941no—
229C22×Q8—21no—
230C22×Q8C22⋊D4711yescharacteristic
231#137—1884no—
232C2×Q16—942no—
233C2×SD16—942no—
234C2×M16—941no—
235C22×C8—21no—
236C4⋊Q8—21no—
237C4⋊Q8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—471no—
240C4⋊Q8C22×D4712yes—
241#127—471no—
242#124—941no—
243#121—941no—
244#121—941no—
245C4×Q8—21no—
246C4×Q8—21no—
247C4×Q8C22⋊D4712yes—
248C4×Q8C22×D4712yes—
249C4×D4—941no—
250C4×D4—941no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4C22⋊D4711yescharacteristic
255C2×C42—21no—
256C2×C42C22×D4711yescharacteristic
257#107—24no—
258#106—944no—
259C4⋊C8—21no—
260C4⋊C8—472no—
261C4⋊C8—472no—
262C4⋊C8—472no—
263C4⋊C8—944no—
264C4⋊C8—941no—
265C4⋊C8C22×D4712yes—
266#103—24no—
267#103—942no—
268#102—944no—
269#102—942no—
270C22⋊C8—472no—
271C22⋊C8—941no—
272C4×C8—472no—
273C4×C8—944no—
274C4×C8—941no—
275#41—1882no—
276#41—1882no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—41no—
281C2×Q8—82no—
282C2×Q8—941no—
283C2×Q8#1062533614yes—
284C2×Q8C2×C22⋊D4712yes—
285C2×D4—941no—
286C2×D4—941no—
287C2×D4—941no—
288C2×D4—1881no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4—1881no—
293C22×C4D4×D4711yescharacteristic
294C22×C4#1062535211yescharacteristic
295C22×C4C2×C22⋊D4711yescharacteristic
296Q16—1884no—
297SD16—1884no—
298M16—1882no—
299C2×C8—22no—
300C2×C8—42no—
301C2×C8—474no—
302C2×C8—942no—
303C2×C8—942no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—41no—
310C4⋊C4—41no—
311C4⋊C4—941no—
312C4⋊C4—941no—
313C4⋊C4C2×C22⋊D4712yes—
314C22⋊C4—1881no—
315C22⋊C4—1881no—
316C42—22no—
317C42—22no—
318C42D4×D4711yescharacteristic
319C42#1062535211yescharacteristic
320C42C2×C22⋊D4711yescharacteristic
321C42C23×D4711yescharacteristic
322C23—1881no—
323C23#1309399611yescharacteristic
324Q8—24no—
325Q8—22no—
326Q8—82no—
327Q8—82no—
328Q8—1881no—
329Q8—1881no—
330Q8C2×#1062533614yes—
331D4—3761no—
332D4—3762no—
333D4—1881no—
334D4—1881no—
335D4—1882no—
336C2×C4—24no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—42no—
342C2×C4—81no—
343C2×C4—82no—
344C2×C4—3762no—
345C2×C4—1881no—
346C2×C4#1309399611yescharacteristic
347C2×C4#1309399611yescharacteristic
348C2×C4#1309399611yescharacteristic
349C2×C4C2×#1062533612yes—
350C2×C4#1309404412yes—
351C2×C4C2×D4×D4711yescharacteristic
352C2×C4C22×C22⋊D4711yescharacteristic
353C2×C4C2×#1062535211yescharacteristic
354C8—42no—
355C8—944no—
356C8—1882no—
357C8—1882no—
358C22—21no—
359C22—21no—
360C22—3761no—
361C22—3761no—
362C22—3761no—
363C22#1568935412yes—
364C22C2×#1309399611yescharacteristic
365C4—24no—
366C4—21no—
367C4—21no—
368C4—21no—
369C4—42no—
370C4—82no—
371C4—3761no—
372C4#1568935412yes—
373C4C22×#1062533611yescharacteristic
374C4C2×#1309404411yescharacteristic
375C4#1568993612yes—
376C2—22no—
377C2—7521no—
378C2#2025291511yescharacteristic
379C2C2×#1568935411yescharacteristic
380C2#2025311311yescharacteristic
subgroupG/H
centerC22C2×#13093996
commutator, cosocleC42×C47C24
FrattiniC42C23×D47
FittingC2×C47×#477C2
socleC22×C47C2×#121
prime radical#15689673C22
hypercenterC2×#477D47
hypocenterC47#19403
LCSG ▷ C42×C47 ▷ C22×C47 ▷ C47
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#477

Complements

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

NH ≅ G/Ntype
3. #20252469C2semidirect
4. #20242806C2semidirect
5. #20242772C2semidirect
6. #20242158C2semidirect
7. #20242069C2semidirect
9. C2×C47×#477C2semidirect
187. C2×#477D47semidirect
227. C47#19403semidirect

Maximal subgroups

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

subgroupindex
#202545162
#202528602
#202524692
#202428062
#202427722
#202421582
#202420692
#202420382
C2×C47×#4772
#1940347