Finite Group Databasebeta

← #27183117

Subgroups — #27183117

Order 12928.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265836C211yesmaximal, characteristic, complemented
2#20265539C211yesmaximal, characteristic, complemented
3#20265512C211yesmaximal, characteristic, complemented
4#20265490C211yesmaximal, characteristic, complemented
5#20265469C211yesmaximal, characteristic, complemented
6#20265421C211yesmaximal, characteristic, complemented
7#20265411C211yesmaximal, characteristic, complemented
8#20265392C211yesmaximal, characteristic, complemented
9#20265320C211yesmaximal, characteristic, complemented
10#20265315C211yesmaximal, characteristic, complemented
11#20265277C211yesmaximal, characteristic, complemented
12#20265249C211yesmaximal, characteristic, complemented
13#20265176C211yesmaximal, characteristic, complemented
14C2×#15692178C211yesmaximal, characteristic, complemented
15#547×C101C211yesmaximal, characteristic, complemented
16C2×#13108074C2211yescharacteristic
17C22×Dic202C2211yescharacteristic
18C2×#13108062C2211yescharacteristic
19C2×#13108062C2211yescharacteristic
20C2×#13108059C2211yescharacteristic
21C2×#13108058C2211yescharacteristic
22C22×#11717579C2211yescharacteristic
23#15692243C2211yescharacteristic
24#15692241—21no—
25#15692234C2211yescharacteristic
26#15692233—21no—
27#15692229C2211yescharacteristic
28#15692227—21no—
29C4×Dic202C2211yescharacteristic
30#15692215—21no—
31#15692188C2211yescharacteristic
32#15692188C2211yescharacteristic
33#15692188C2211yescharacteristic
34C4⋊#11717579C2211yescharacteristic
35#15692178C2211yescharacteristic
36#15692178C2211yescharacteristic
37#15692178C2211yescharacteristic
38#15692178C2211yescharacteristic
39#15692178C2211yescharacteristic
40#15692178C2211yescharacteristic
41#15692176C2211yescharacteristic
42#15692175C2211yescharacteristic
43#15692175C2211yescharacteristic
44#15692175C2211yescharacteristic
45C22⋊#11717579C2211yescharacteristic
46C22⋊#11717579C2211yescharacteristic
47#15692166C2211yescharacteristic
48C2×#41×C101—21no—
49C4⋊Q8×C101C2211yescharacteristic
50#123×C101—21no—
51#123×C101—21no—
52C22⋊Q8×C101—21no—
53#121×C101C2211yescharacteristic
54#121×C101C2211yescharacteristic
55C4×D4×C101C2211yescharacteristic
56C4×D4×C101C2211yescharacteristic
57C2×C4⋊C4×C101C2211yescharacteristic
58C2×C4⋊C4×C101C2211yescharacteristic
59#13108074—22no—
60#13108074—41no—
61C2×Dic202—22no—
62C2×Dic202—21no—
63C2×Dic202—41no—
64C2×Dic202C2311yescharacteristic
65C2×Dic202C2311yescharacteristic
66C2×D404C2311yescharacteristic
67C2×C22⋊D101—21no—
68C2×C22⋊D101—21no—
69C22×Dic101—21no—
70C2×C4×D101—21no—
71#13108062—22no—
72#13108062—41no—
73#13108059—41no—
74#13108058—41no—
75C2×#11717579—41no—
76C2×#11717579C2311yescharacteristic
77C2×#11717579C2311yescharacteristic
78C2×#11717579C2311yescharacteristic
79C2×#11717579C2311yescharacteristic
80C4⋊Dic101C2311yescharacteristic
81#13108053—21no—
82#13108053—21no—
83C22⋊Dic101—21no—
84#13108051—21no—
85#13108051—21no—
86#13108051—21no—
87C4×Dic101—21no—
88#41×C101—44no—
89C2×Q8×C101—21no—
90C2×D4×C101—21no—
91C2×D4×C101—21no—
92C2×D4×C101D411yescharacteristic
93C2×D4×C101C2311yescharacteristic
94C22×C4×C101—21no—
95C22×C4×C101—21no—
96C22×C4×C101—21no—
97C22×C4×C101C2311yescharacteristic
98C22×C4×C101C2311yescharacteristic
99C4⋊C4×C101—21no—
100C4⋊C4×C101—21no—
101C4⋊C4×C101—21no—
102C4⋊C4×C101D411yescharacteristic
103C4⋊C4×C101C2311yescharacteristic
104C4⋊C4×C101C2311yescharacteristic
105C4⋊C4×C101C2311yescharacteristic
106C22⋊C4×C101—21no—
107C22⋊C4×C101—21no—
108C22⋊C4×C101—21no—
109C22⋊C4×C101D411yescharacteristic
110C22⋊C4×C101D411yescharacteristic
111C42×C101C2311yescharacteristic
112C23×C101—21no—
113C23×C101C2×D411yescharacteristic
114C23×C101C2×D411yescharacteristic
115C22×D101—21no—
116Q8×C101—42no—
117D4×C101—22no—
118D4×C101—21no—
119D4×C101—21no—
120D4×C101—42no—
121D4×C101—42no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101—21no—
126C2×C4×C101—21no—
127C2×C4×C101—21no—
128C2×C4×C101—21no—
129C2×C4×C101—42no—
130C2×C4×C101—42no—
131C2×C4×C101—42no—
132C2×C4×C101C2×D411yescharacteristic
133C2×C4×C101C2×D411yescharacteristic
134C2×C4×C101C2×D411yescharacteristic
135C2×C4×C101C2×D411yescharacteristic
136C2×C4×C101C2411yescharacteristic
137C22⋊D101—42no—
138C22⋊D101—42no—
139C2×Dic101—21no—
140C2×Dic101—21no—
141C2×Dic101—21no—
142C2×Dic101—42no—
143D404—21no—
144D404—21no—
145C4×D101—42no—
146Dic202—21no—
147Dic202—21no—
148Dic202—21no—
149Dic202—42no—
150Dic202#4112yescomplemented
151#11717579—21no—
152#11717579—21no—
153#11717579—21no—
154#11717579—21no—
155C22×C101—21no—
156C22×C101—21no—
157C22×C101—21no—
158C22×C101—21no—
159C22×C101—21no—
160C22×C101—41no—
161C22×C101—41no—
162C22×C101—41no—
163C22×C101AGL(1,ℤ/8)12yes—
164C22×C101C22×D411yescharacteristic
165C2×D101—41no—
166C2×D101—41no—
167C2×D101—41no—
168C4×C101—21no—
169C4×C101—21no—
170C4×C101—21no—
171C4×C101—21no—
172C4×C101—41no—
173C4×C101—41no—
174C4×C101—41no—
175C4×C101C2×#4111yescharacteristic
176C4×C101H(5,𝔽2)11yescharacteristic
177Dic101—22no—
178Dic101—41no—
179Dic101—41no—
180C2×C101—22no—
181C2×C101—41no—
182C2×C101—81no—
183C2×C101#54611yescharacteristic
184C2×C101C2×AGL(1,ℤ/8)11yescharacteristic
185C2×C101#57811yescharacteristic
186D101—81no—
187#3308—1011nomaximal, Sylow
188C101#330811yescharacteristic, complemented, Sylow
189#548—1011no—
190#547D10111yescharacteristic, complemented
191#489—1011no—
192#484—1011no—
193#480—1011no—
194#477—1011no—
195#468—1011no—
196#466—1011no—
197#463—1011no—
198#449—1011no—
199#448—1011no—
200#440—1011no—
201#435—1011no—
202#417—1011no—
203C2×#103—1011no—
204C2×#41—21no—
205C2×#41—1011no—
206C22×Q8—1011no—
207C2×SD16—1011no—
208C2×SD16—1011no—
209C2×D8—1011no—
210C2×M16—1011no—
211C22×C8—1011no—
212C4⋊Q8C2×D10111yescharacteristic
213#124—1011no—
214#124—2021no—
215#123—21no—
216#123—21no—
217C22⋊Q8—21no—
218C22⋊Q8—1011no—
219C22⋊Q8—2021no—
220#121—1011no—
221#121—2021no—
222#121C2×D10111yescharacteristic
223#121C2×D10111yescharacteristic
224C4×Q8—1011no—
225C4×D4—2021no—
226C4×D4C2×D10111yescharacteristic
227C4×D4C2×D10111yescharacteristic
228C2×C4⋊C4C2×D10111yescharacteristic
229C2×C4⋊C4C2×D10111yescharacteristic
230#107—1011no—
231#107—1011no—
232#107—1011no—
233C4⋊C8—1011no—
234#103—1011no—
235#103—1011no—
236#103—1011no—
237#103—1011no—
238#103—1011no—
239#103—1011no—
240#102—1011no—
241#102—1011no—
242#102—1011no—
243#102—1011no—
244C22⋊C8—1011no—
245C22⋊C8—1011no—
246#97—1011no—
247#41—44no—
248#41—4041no—
249#41—2022no—
250C2×Q8—21no—
251C2×Q8—1011no—
252C2×Q8—1011no—
253C2×Q8—4041no—
254C2×Q8—2022no—
255C2×Q8—2021no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—1011no—
259C2×D4—2021no—
260C2×D4—2021no—
261C2×D4C22⋊D10111yescharacteristic
262C2×D4C22×D10111yescharacteristic
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—2021no—
267C22×C4—2021no—
268C22×C4C22×D10111yescharacteristic
269C22×C4C22×D10111yescharacteristic
270SD16—4041no—
271SD16—2022no—
272D8—4041no—
273M16—4041no—
274C2×C8—1011no—
275C2×C8—1011no—
276C2×C8—1011no—
277C2×C8—1011no—
278C2×C8—4041no—
279C4⋊C4—21no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—1011no—
283C4⋊C4—2021no—
284C4⋊C4—2021no—
285C4⋊C4C22⋊D10111yescharacteristic
286C4⋊C4C22×D10111yescharacteristic
287C4⋊C4C22×D10111yescharacteristic
288C4⋊C4C22×D10111yescharacteristic
289C22⋊C4—21no—
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—2021no—
293C22⋊C4—2021no—
294C22⋊C4—2021no—
295C22⋊C4—2021no—
296C22⋊C4C22⋊D10111yescharacteristic
297C22⋊C4C22⋊D10111yescharacteristic
298C42—2021no—
299C42C22×D10111yescharacteristic
300C23—21no—
301C23—2021no—
302C23C2×C22⋊D10111yescharacteristic
303C23C2×C22⋊D10111yescharacteristic
304Q8—42no—
305Q8—1012no—
306Q8—4042no—
307Q8—2021no—
308Q8—2021no—
309Q8—2021no—
310D4—22no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—42no—
315D4—4042no—
316D4—4042no—
317D4—2021no—
318D4—2021no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—4042no—
330C2×C4—4042no—
331C2×C4—2021no—
332C2×C4—2021no—
333C2×C4—2021no—
334C2×C4C2×C22⋊D10111yescharacteristic
335C2×C4C2×C22⋊D10111yescharacteristic
336C2×C4C2×C22⋊D10111yescharacteristic
337C2×C4C2×C22⋊D10111yescharacteristic
338C2×C4C23×D10111yescharacteristic
339C8—2021no—
340C8—2021no—
341C8—2021no—
342C8—2021no—
343C22—21no—
344C22—21no—
345C22—21no—
346C22—21no—
347C22—21no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—4041no—
352C22—4041no—
353C22—4041no—
354C22#1569228112yes—
355C22C22×C22⋊D10111yescharacteristic
356C4—21no—
357C4—21no—
358C4—21no—
359C4—21no—
360C4—41no—
361C4—41no—
362C4—41no—
363C4—4041no—
364C4—4041no—
365C4—2022no—
366C4#41×D10111yescharacteristic
367C4#1569230111yescharacteristic
368C2—22no—
369C2—41no—
370C2—81no—
371C2—8081no—
372C2#2026581511yescharacteristic
373C2C2×#1569228111yescharacteristic
374C2#2026599211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#547×C101C2
socleC22×C101C22×D4
hypercenter#547D101
hypocenterC101#3308
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×D4 ◁ #547

Complements

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

NH ≅ G/Ntype
1. #20265836C2semidirect
2. #20265539C2semidirect
3. #20265512C2semidirect
4. #20265490C2semidirect
5. #20265469C2semidirect
6. #20265421C2semidirect
7. #20265411C2semidirect
8. #20265392C2semidirect
9. #20265320C2semidirect
10. #20265315C2semidirect
11. #20265277C2semidirect
12. #20265249C2semidirect
13. #20265176C2semidirect
14. C2×#15692178C2semidirect
15. #547×C101C2semidirect
150. Dic202#41semidirect
188. C101#3308semidirect
190. #547D101semidirect

Maximal subgroups

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

subgroupindex
#202658362
#202655392
#202655122
#202654902
#202654692
#202654212
#202654112
#202653922
#202653202
#202653152
#202652772
#202652492
#202651762
C2×#156921782
#547×C1012
#3308101