Finite Group Databasebeta

← #27154358

Subgroups — #27154358

Order 12416.

354 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20261448C211yesmaximal, characteristic
2#20261440C211yesmaximal, characteristic, complemented
3#20261139C211yesmaximal, characteristic, complemented
4#20261079C211yesmaximal, characteristic, complemented
5#20261076C211yesmaximal, characteristic, complemented
6#20261050C211yesmaximal, characteristic, complemented
7#20261048C211yesmaximal, characteristic, complemented
8#20261029C211yesmaximal, characteristic, complemented
9#20261024C211yesmaximal, characteristic, complemented
10#20260982C211yesmaximal, characteristic, complemented
11#20260981C211yesmaximal, characteristic, complemented
12#20260822C211yesmaximal, characteristic, complemented
13#20260811C211yesmaximal, characteristic, complemented
14#20260774C211yesmaximal, characteristic, complemented
15#428×C97C211yesmaximal, characteristic, complemented
16C2×#13095500C2211yescharacteristic
17C2×#13095494C2211yescharacteristic
18#15690809C2211yescharacteristic
19#15690809C2211yescharacteristic
20#15690800—21no—
21#15690798—21no—
22#15690796—21no—
23#15690790C2211yescharacteristic
24#15690789—21no—
25#15690789C2211yescharacteristic
26#15690788—21no—
27#15690785C2211yescharacteristic
28#15690784C2211yescharacteristic
29#15690783—21no—
30#15690775C2211yescharacteristic
31#15690772C2211yescharacteristic
32C4×C22⋊D97—21no—
33C4×C22⋊D97C2211yescharacteristic
34#15690768C2211yescharacteristic
35#15690767—21no—
36C4⋊C4×D97C2211yescharacteristic
37#15690744C2211yescharacteristic
38#15690743C2211yescharacteristic
39#15690743C2211yescharacteristic
40#15690743C2211yescharacteristic
41#15690741C2211yescharacteristic
42#15690739C2211yescharacteristic
43#15690735C2211yescharacteristic
44#15690733C2211yescharacteristic
45#15690733C2211yescharacteristic
46#15690732C2211yescharacteristic
47#15690730C2211yescharacteristic
48#15690730C2211yescharacteristic
49C22⋊#11715895C2211yescharacteristic
50#15690722C2211yescharacteristic
51C8×Dic97C2211yescharacteristic
52C2×M16×C97C2211yescharacteristic
53#117×C97C2211yescharacteristic
54C2×C4⋊C4×C97C2211yescharacteristic
55#107×C97C2211yescharacteristic
56#107×C97C2211yescharacteristic
57#106×C97C2211yescharacteristic
58#106×C97C2211yescharacteristic
59C2×Dic194—21no—
60C2×Dic194—21no—
61C2×Dic194C2311yescharacteristic
62C2×D388—21no—
63C2×D388C2311yescharacteristic
64C2×C22⋊D97—21no—
65C2×C22⋊D97D411yescharacteristic
66C2×C4×D97—21no—
67C2×C4×D97—21no—
68C2×C4×D97C2311yescharacteristic
69#13095500—22no—
70#13095494—22no—
71C2×#11715895C2311yescharacteristic
72C2×#11715895C2311yescharacteristic
73C4⋊Dic97—21no—
74C4⋊Dic97C2311yescharacteristic
75C4⋊Dic97C2311yescharacteristic
76#13095490—21no—
77#13095490—21no—
78#13095490—21no—
79#13095490—21no—
80#13095490—21no—
81#13095490D411yescharacteristic
82C22⋊Dic97—21no—
83C22⋊Dic97D411yescharacteristic
84#13095488—21no—
85#13095488—21no—
86#13095488—21no—
87#13095488—21no—
88#13095488—21no—
89#13095488D411yescharacteristic
90C4×Dic97—21no—
91C4×Dic97—21no—
92C4×Dic97C2311yescharacteristic
93C22×C4×C97—21no—
94C22×C4×C97C2311yescharacteristic
95M16×C97—22no—
96C2×C8×C97C2311yescharacteristic
97C2×C8×C97C2311yescharacteristic
98C4⋊C4×C97—21no—
99C4⋊C4×C97C2311yescharacteristic
100C4⋊C4×C97C2311yescharacteristic
101C4⋊C4×C97C2311yescharacteristic
102C4⋊C4×C97C2311yescharacteristic
103C22⋊C4×C97—21no—
104C42×C97—21no—
105C23×C97C2×D411yescharacteristic
106C22×D97—21no—
107C22×D97C2×D411yescharacteristic
108C2×C4×C97—21no—
109C2×C4×C97—21no—
110C2×C4×C97—21no—
111C2×C4×C97—21no—
112C2×C4×C97—21no—
113C2×C4×C97—21no—
114C2×C4×C97—81no—
115C2×C4×C97C2×D411yescharacteristic
116C2×C4×C97C2411yescharacteristic
117C22⋊D97—22no—
118C22⋊D97—81no—
119C2×Dic97—21no—
120C2×Dic97—21no—
121C2×Dic97—21no—
122C2×Dic97C2×D411yescharacteristic
123C2×Dic97C2×D411yescharacteristic
124C2×Dic97C2×D411yescharacteristic
125D388—21no—
126D388—21no—
127D388—81no—
128C4×D97—21no—
129C4×D97—21no—
130C4×D97—42no—
131C4×D97—81no—
132Dic194—21no—
133Dic194—21no—
134Dic194—81no—
135Dic194—81no—
136C8×C97—21no—
137C8×C97#4112yes—
138#11715895—21no—
139#11715895—21no—
140C22×C97—21no—
141C22×C97—21no—
142C22×C97—21no—
143C22×C97C22×D411yescharacteristic
144C2×D97—21no—
145C2×D97—21no—
146C2×D97—21no—
147C2×D97—41no—
148C2×D97—41no—
149C2×D97—41no—
150C4×C97—21no—
151C4×C97—41no—
152C4×C97—41no—
153C4×C97—41no—
154C4×C97—41no—
155C4×C97C2×#4111yescharacteristic
156C4×C972−1+411yescharacteristic
157Dic97—21no—
158Dic97—21no—
159Dic97—41no—
160Dic97—41no—
161Dic97—41no—
162Dic97#13712yes—
163C2×C97—41no—
164C2×C97#54711yescharacteristic
165C2×C97C2×#13711yescharacteristic
166C2×C97#57611yescharacteristic
167D97—41no—
168D97—81no—
169#3357—971nomaximal, Sylow
170C97#335711yescharacteristic, complemented, Sylow
171#555—971no—
172#554—971no—
173#500—971no—
174#485—971no—
175#485—971no—
176#480—971no—
177#480—971no—
178#477—971no—
179#476—971no—
180#468—971no—
181#468—971no—
182C4×SD16—971no—
183#436—971no—
184#428—971no—
185#428D9711yescharacteristic, complemented
186C2×SD16—971no—
187C2×M16—971no—
188C2×M16C2×D9711yescharacteristic
189C4⋊Q8—971no—
190C4⋊Q8—971no—
191#125—1941no—
192#124—1941no—
193#124—1941no—
194C22⋊Q8—971no—
195C22⋊Q8—971no—
196C22⋊Q8—1941no—
197C22⋊Q8—1941no—
198#121—971no—
199#121—971no—
200#121—1941no—
201C4×Q8—971no—
202C4×D4—971no—
203C4×D4—971no—
204C4×D4—1941no—
205#117—971no—
206#117—1941no—
207#117C2×D9711yescharacteristic
208C2×C4⋊C4—971no—
209C2×C4⋊C4C2×D9711yescharacteristic
210#107—971no—
211#107—971no—
212#107—971no—
213#107—971no—
214#107C2×D9711yescharacteristic
215#107C2×D9711yescharacteristic
216#106—971no—
217#106C2×D9711yescharacteristic
218#106C2×D9711yescharacteristic
219C4⋊C8—971no—
220#103—971no—
221#103—971no—
222#103—971no—
223#102—971no—
224#102—971no—
225#102—971no—
226C22⋊C8—971no—
227C22⋊C8—971no—
228C4×C8—971no—
229C2×Q8—971no—
230C2×Q8—1941no—
231C2×Q8—1941no—
232C2×D4—971no—
233C2×D4—971no—
234C2×D4—1941no—
235C2×D4—1941no—
236C22×C4—21no—
237C22×C4—971no—
238C22×C4—1941no—
239C22×C4—1941no—
240C22×C4C22×D9711yescharacteristic
241SD16—1942no—
242M16—22no—
243M16—1942no—
244C2×C8—971no—
245C2×C8—971no—
246C2×C8C22×D9711yescharacteristic
247C2×C8C22×D9711yescharacteristic
248C4⋊C4—21no—
249C4⋊C4—971no—
250C4⋊C4—971no—
251C4⋊C4—971no—
252C4⋊C4—1941no—
253C4⋊C4—1941no—
254C4⋊C4—1941no—
255C4⋊C4—1941no—
256C4⋊C4—1941no—
257C4⋊C4—1941no—
258C4⋊C4C22×D9711yescharacteristic
259C4⋊C4C22×D9711yescharacteristic
260C4⋊C4C22×D9711yescharacteristic
261C4⋊C4C22×D9711yescharacteristic
262C22⋊C4—21no—
263C22⋊C4—971no—
264C22⋊C4—971no—
265C22⋊C4—1941no—
266C22⋊C4—1941no—
267C22⋊C4—1941no—
268C22⋊C4—1941no—
269C22⋊C4—1941no—
270C22⋊C4—1941no—
271C42—21no—
272C42—971no—
273C42—1941no—
274C42—1941no—
275C23—971no—
276C23—1941no—
277C23D4×D9711yescharacteristic
278Q8—7761no—
279Q8—7761no—
280Q8—1941no—
281Q8—1941no—
282D4—7761no—
283D4—7761no—
284D4—1942no—
285D4—1941no—
286D4—1941no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—81no—
294C2×C4—7761no—
295C2×C4—971no—
296C2×C4—971no—
297C2×C4—971no—
298C2×C4—3882no—
299C2×C4—1941no—
300C2×C4—1941no—
301C2×C4—1941no—
302C2×C4—1941no—
303C2×C4—1941no—
304C2×C4D4×D9711yescharacteristic
305C2×C4C23×D9711yescharacteristic
306C8—21no—
307C8—1941no—
308C8—1941no—
309C8#1309551112yes—
310C22—21no—
311C22—21no—
312C22—21no—
313C22—3881no—
314C22—3881no—
315C22—3881no—
316C22—1941no—
317C22—1941no—
318C22—1941no—
319C22C2×D4×D9711yescharacteristic
320C4—21no—
321C4—41no—
322C4—41no—
323C4—41no—
324C4—41no—
325C4—972no—
326C4—3881no—
327C4—3881no—
328C4—3881no—
329C4—1941no—
330C4—1941no—
331C4C2×#1309551111yescharacteristic
332C4#1569085811yescharacteristic
333C2—41no—
334C2—7761no—
335C2—3881no—
336C2#2026138611yescharacteristic
337C2#137×D9711yescharacteristic
338C2#2026153911yescharacteristic
subgroupG/H
centerC22C2×D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
Fitting#428×C97C2
socleC22×C97C22×D4
prime radical#20261448C2
hypercenter#428D97
hypocenterC97#3357
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #428

Complements

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

NH ≅ G/Ntype
2. #20261440C2semidirect
3. #20261139C2semidirect
4. #20261079C2semidirect
5. #20261076C2semidirect
6. #20261050C2semidirect
7. #20261048C2semidirect
8. #20261029C2semidirect
9. #20261024C2semidirect
10. #20260982C2semidirect
11. #20260981C2semidirect
12. #20260822C2semidirect
13. #20260811C2semidirect
14. #20260774C2semidirect
15. #428×C97C2semidirect
170. C97#3357semidirect
185. #428D97semidirect

Maximal subgroups

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

subgroupindex
#202614482
#202614402
#202611392
#202610792
#202610762
#202610502
#202610482
#202610292
#202610242
#202609822
#202609812
#202608222
#202608112
#202607742
#428×C972
#335797