Finite Group Databasebeta

← #27153147

Subgroups — #27153147

Order 12416.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261340C211yesmaximal, characteristic, complemented
2#20261042C211yesmaximal, characteristic, complemented
3#20261035C211yesmaximal, characteristic, complemented
4#20261022C211yesmaximal, characteristic, complemented
5#20261012C211yesmaximal, characteristic, complemented
6#20260986C211yesmaximal, characteristic, complemented
7#20260983C211yesmaximal, characteristic, complemented
8#20260969C211yesmaximal, characteristic, complemented
9#20260966C211yesmaximal, characteristic, complemented
10#20260947C211yesmaximal, characteristic, complemented
11#20260947C211yesmaximal, characteristic, complemented
12#20260935C211yesmaximal, characteristic, complemented
13#20260930C211yesmaximal, characteristic, complemented
14#20260762C211yesmaximal, characteristic, complemented
15#542×C97C211yesmaximal, characteristic, complemented
16C2×#13095498C2211yescharacteristic
17C2×#13095498C2211yescharacteristic
18C2×#13095496C2211yescharacteristic
19C2×#13095496C2211yescharacteristic
20C2×#13095495C2211yescharacteristic
21C22×#11715895C2211yescharacteristic
22#15690807C2211yescharacteristic
23#15690802C2211yescharacteristic
24#15690795—21no—
25#15690795—21no—
26#15690784C2211yescharacteristic
27#15690784C2211yescharacteristic
28#15690783—21no—
29#15690783—21no—
30C4×D388C2211yescharacteristic
31C4×D388C2211yescharacteristic
32#15690743C2211yescharacteristic
33#15690743C2211yescharacteristic
34#15690741C2211yescharacteristic
35#15690741C2211yescharacteristic
36C4⋊#11715895C2211yescharacteristic
37C4⋊#11715895C2211yescharacteristic
38C4⋊#11715895C2211yescharacteristic
39C4⋊#11715895C2211yescharacteristic
40#15690734C2211yescharacteristic
41#15690734C2211yescharacteristic
42#15690731C2211yescharacteristic
43#15690731C2211yescharacteristic
44#15690730C2211yescharacteristic
45#15690730C2211yescharacteristic
46#15690730C2211yescharacteristic
47#15690730C2211yescharacteristic
48C4⋊Q8×C97C2211yescharacteristic
49#126×C97—21no—
50#124×C97—21no—
51#124×C97C2211yescharacteristic
52#123×C97—21no—
53C22⋊Q8×C97—21no—
54C22⋊Q8×C97C2211yescharacteristic
55#121×C97C2211yescharacteristic
56C4×Q8×C97C2211yescharacteristic
57C4×D4×C97C2211yescharacteristic
58#117×C97C2211yescharacteristic
59C2×D388—21no—
60C2×D388—21no—
61C2×D388C2311yescharacteristic
62C2×D388C2311yescharacteristic
63C2×C22⋊D97—21no—
64C2×C22⋊D97—21no—
65C2×C4×D97—21no—
66C2×C4×D97—21no—
67#13095498—41no—
68#13095498—41no—
69#13095496—41no—
70#13095496—41no—
71#13095495—41no—
72C2×#11715895—41no—
73C2×#11715895C2311yescharacteristic
74C2×#11715895C2311yescharacteristic
75C2×#11715895C2311yescharacteristic
76C2×#11715895C2311yescharacteristic
77C4⋊Dic97C2311yescharacteristic
78C4⋊Dic97C2311yescharacteristic
79#13095490—21no—
80#13095490—21no—
81#13095488—21no—
82#13095488—21no—
83#13095488—21no—
84#13095488—21no—
85C2×Q8×C97—21no—
86C2×Q8×C97C2311yescharacteristic
87C2×D4×C97—21no—
88C2×D4×C97C2311yescharacteristic
89C22×C4×C97—21no—
90C22×C4×C97C2311yescharacteristic
91C4⋊C4×C97—21no—
92C4⋊C4×C97—21no—
93C4⋊C4×C97—21no—
94C4⋊C4×C97D411yescharacteristic
95C4⋊C4×C97D411yescharacteristic
96C4⋊C4×C97C2311yescharacteristic
97C4⋊C4×C97C2311yescharacteristic
98C22⋊C4×C97—21no—
99C22⋊C4×C97—21no—
100C22⋊C4×C97—21no—
101C22⋊C4×C97—21no—
102C22⋊C4×C97—21no—
103C22⋊C4×C97D411yescharacteristic
104C22⋊C4×C97D411yescharacteristic
105C42×C97—21no—
106C42×C97C2311yescharacteristic
107C42×C97C2311yescharacteristic
108C23×C97—21no—
109C23×C97C2×D411yescharacteristic
110C22×D97—21no—
111C22×D97—21no—
112Q8×C97—21no—
113Q8×C97—21no—
114Q8×C97—81no—
115D4×C97—21no—
116D4×C97—21no—
117D4×C97—81no—
118C2×C4×C97—21no—
119C2×C4×C97—21no—
120C2×C4×C97—21no—
121C2×C4×C97—21no—
122C2×C4×C97—21no—
123C2×C4×C97—81no—
124C2×C4×C97C2×D411yescharacteristic
125C2×C4×C97C2×D411yescharacteristic
126C2×C4×C97C2×D411yescharacteristic
127C2×C4×C97C2×D411yescharacteristic
128C2×C4×C97C2×D411yescharacteristic
129C2×C4×C97C2411yescharacteristic
130C22⋊D97—81no—
131C22⋊D97—81no—
132C2×Dic97—21no—
133C2×Dic97—21no—
134D388—21no—
135D388—21no—
136D388—21no—
137D388—21no—
138D388—81no—
139D388—81no—
140C4×D97—81no—
141C4×D97—81no—
142#11715895—21no—
143#11715895—21no—
144#11715895—21no—
145#11715895—21no—
146C22×C97—21no—
147C22×C97—21no—
148C22×C97—21no—
149C22×C97—41no—
150C22×C97—41no—
151C22×C97—41no—
152C22×C97C22×D411yescharacteristic
153C2×D97—41no—
154C2×D97—41no—
155C2×D97—41no—
156C2×D97—41no—
157C2×D97—41no—
158C2×D97—41no—
159C4×C97—21no—
160C4×C97—21no—
161C4×C97—21no—
162C4×C97—21no—
163C4×C97—21no—
164C4×C97—41no—
165C4×C97—41no—
166C4×C97—41no—
167C4×C97H(5,𝔽2)11yescharacteristic
168C4×C972−1+411yescharacteristic
169Dic97—41no—
170Dic97—41no—
171C2×C97—41no—
172C2×C97—81no—
173C2×C97#53711yescharacteristic
174C2×C97#57611yescharacteristic
175C2×C97#57711yescharacteristic
176D97—81no—
177D97—81no—
178#3257—971nomaximal, Sylow
179C97#325711yescharacteristic, complemented, Sylow
180#542D9711yescharacteristic, complemented
181#540—971no—
182#479—971no—
183#478—971no—
184#476—971no—
185#474—971no—
186#469—971no—
187#468—971no—
188#466—971no—
189#465—971no—
190#463—971no—
191#463—971no—
192#460—971no—
193#459—971no—
194#424—971no—
195C2×SD16—971no—
196C2×SD16—971no—
197C2×D8—971no—
198C2×D8—971no—
199C2×M16—971no—
200C22×C8—971no—
201C4⋊Q8C2×D9711yescharacteristic
202#127—971no—
203#126—21no—
204#125—971no—
205#124—21no—
206#124C2×D9711yescharacteristic
207#123—21no—
208#123—1941no—
209#123—1941no—
210C22⋊Q8—21no—
211C22⋊Q8C2×D9711yescharacteristic
212#121—971no—
213#121—971no—
214#121—1941no—
215#121—1941no—
216#121C2×D9711yescharacteristic
217C4×Q8C2×D9711yescharacteristic
218C4×D4—971no—
219C4×D4—971no—
220C4×D4C2×D9711yescharacteristic
221#117C2×D9711yescharacteristic
222#107—971no—
223#107—971no—
224#106—971no—
225#106—971no—
226C4⋊C8—971no—
227C4⋊C8—971no—
228C4⋊C8—971no—
229C4⋊C8—971no—
230#103—971no—
231#103—971no—
232#102—971no—
233#102—971no—
234#102—971no—
235#102—971no—
236#102—971no—
237#102—971no—
238C2×Q8—21no—
239C2×Q8C22×D9711yescharacteristic
240C2×D4—21no—
241C2×D4—971no—
242C2×D4—971no—
243C2×D4—1941no—
244C2×D4—1941no—
245C2×D4—1941no—
246C2×D4—1941no—
247C2×D4C22×D9711yescharacteristic
248C22×C4—21no—
249C22×C4—1941no—
250C22×C4—1941no—
251C22×C4C22×D9711yescharacteristic
252SD16—3881no—
253SD16—3881no—
254D8—3881no—
255D8—3881no—
256M16—3881no—
257C2×C8—971no—
258C2×C8—971no—
259C2×C8—971no—
260C2×C8—971no—
261C2×C8—3881no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—971no—
266C4⋊C4—971no—
267C4⋊C4—1941no—
268C4⋊C4—1941no—
269C4⋊C4C22⋊D9711yescharacteristic
270C4⋊C4C22⋊D9711yescharacteristic
271C4⋊C4C22×D9711yescharacteristic
272C4⋊C4C22×D9711yescharacteristic
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—1941no—
279C22⋊C4—1941no—
280C22⋊C4—1941no—
281C22⋊C4—1941no—
282C22⋊C4C22⋊D9711yescharacteristic
283C22⋊C4C22⋊D9711yescharacteristic
284C42—21no—
285C42C22×D9711yescharacteristic
286C42C22×D9711yescharacteristic
287C23—21no—
288C23—1941no—
289C23—1941no—
290C23C2×C22⋊D9711yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—81no—
294D4—21no—
295D4—21no—
296D4—81no—
297D4—7761no—
298D4—7761no—
299D4—7761no—
300D4—7761no—
301D4—1941no—
302D4—1941no—
303D4—1941no—
304D4—1941no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—81no—
311C2×C4—7761no—
312C2×C4—7761no—
313C2×C4—1941no—
314C2×C4—1941no—
315C2×C4C2×C22⋊D9711yescharacteristic
316C2×C4C2×C22⋊D9711yescharacteristic
317C2×C4C2×C22⋊D9711yescharacteristic
318C2×C4C2×C22⋊D9711yescharacteristic
319C2×C4C2×C22⋊D9711yescharacteristic
320C2×C4C23×D9711yescharacteristic
321C8—1941no—
322C8—1941no—
323C8—1941no—
324C8—1941no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—3881no—
332C22—3881no—
333C22—3881no—
334C22—3881no—
335C22—3881no—
336C22—3881no—
337C22C22×C22⋊D9711yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—41no—
344C4—41no—
345C4—41no—
346C4—3881no—
347C4—3881no—
348C4#1569085611yescharacteristic
349C4#1569085711yescharacteristic
350C2—41no—
351C2—81no—
352C2—7761no—
353C2—7761no—
354C2#2026131311yescharacteristic
355C2#2026153811yescharacteristic
356C2#2026154311yescharacteristic
subgroupG/H
centerC22C22×C22⋊D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#542×C97C2
socleC22×C97C22×D4
hypercenter#542D97
hypocenterC97#3257
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #20261340C2semidirect
2. #20261042C2semidirect
3. #20261035C2semidirect
4. #20261022C2semidirect
5. #20261012C2semidirect
6. #20260986C2semidirect
7. #20260983C2semidirect
8. #20260969C2semidirect
9. #20260966C2semidirect
10. #20260947C2semidirect
11. #20260947C2semidirect
12. #20260935C2semidirect
13. #20260930C2semidirect
14. #20260762C2semidirect
15. #542×C97C2semidirect
179. C97#3257semidirect
180. #542D97semidirect

Maximal subgroups

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

subgroupindex
#202613402
#202610422
#202610352
#202610222
#202610122
#202609862
#202609832
#202609692
#202609662
#202609472
#202609352
#202609302
#202607622
#542×C972
#325797