Finite Group Databasebeta

← #27183006

Subgroups — #27183006

Order 12928.

388 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265882C211yesmaximal, characteristic
2#20265840C211yesmaximal, characteristic
3#20265526C211yesmaximal, characteristic, complemented
4#20265514C211yesmaximal, characteristic, complemented
5#20265471C211yesmaximal, characteristic, complemented
6#20265419C211yesmaximal, characteristic, complemented
7#20265416C211yesmaximal, characteristic, complemented
8#20265386C211yesmaximal, characteristic, complemented
9#20265335C211yesmaximal, characteristic, complemented
10#20265334C211yesmaximal, characteristic, complemented
11#20265271C211yesmaximal, characteristic, complemented
12#20265240C211yesmaximal, characteristic
13C2×#15692178C211yesmaximal, characteristic, complemented
14#103×D101C211yesmaximal, characteristic, complemented
15#484×C101C211yesmaximal, characteristic, complemented
16C22×Dic202C2211yescharacteristic
17C2×Q8×D101C2211yescharacteristic
18C2×Dic404C2211yescharacteristic
19C2×Dic404C2211yescharacteristic
20C2×#13108064C2211yescharacteristic
21C22×#11717579C2211yescharacteristic
22C2×C8×D101C2211yescharacteristic
23#15692254—21no—
24#15692254C2211yescharacteristic
25#15692253C2211yescharacteristic
26#15692245—21no—
27#15692235C2211yescharacteristic
28#15692235C2211yescharacteristic
29#15692234—21no—
30#15692234C2211yescharacteristic
31#15692233—21no—
32#15692233—21no—
33#15692232—21no—
34#15692220C2211yescharacteristic
35#15692220C2211yescharacteristic
36C4×C22⋊D101—21no—
37C4×C22⋊D101C2211yescharacteristic
38#15692215—21no—
39C4⋊C4×D101C2211yescharacteristic
40#15692188C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692180C2211yescharacteristic
43#15692180C2211yescharacteristic
44#15692179C2211yescharacteristic
45#15692178C2211yescharacteristic
46#15692178C2211yescharacteristic
47#15692178C2211yescharacteristic
48#15692178C2211yescharacteristic
49#15692178C2211yescharacteristic
50#15692167C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C22⋊Q8×C101C2211yescharacteristic
53C2×C4⋊C4×C101C2211yescharacteristic
54#107×C101C2211yescharacteristic
55#107×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#103×C101C2211yescharacteristic
58C22⋊C8×C101C2211yescharacteristic
59C2×Dic202—22no—
60C2×Dic202—21no—
61C2×Dic202—21no—
62C2×Dic202—21no—
63C2×Dic202—41no—
64C2×Dic202C2311yescharacteristic
65C2×Dic202C2311yescharacteristic
66C2×Dic202C2311yescharacteristic
67Q8×D101—22no—
68Q8×D101—41no—
69C2×C22⋊D101D411yescharacteristic
70C22×Dic101—21no—
71C2×C4×D101—21no—
72C2×C4×D101—21no—
73C2×C4×D101C2311yescharacteristic
74Dic404—22no—
75Dic404—41no—
76#13108064—41no—
77C2×#11717579—41no—
78C2×#11717579C2311yescharacteristic
79C2×#11717579C2311yescharacteristic
80C8×D101—41no—
81C4⋊Dic101—21no—
82C4⋊Dic101—21no—
83C4⋊Dic101C2311yescharacteristic
84#13108053—21no—
85#13108053—21no—
86#13108053—21no—
87#13108053—21no—
88#13108053—21no—
89#13108053—21no—
90#13108053—21no—
91#13108053D411yescharacteristic
92C22⋊Dic101—21no—
93C22⋊Dic101D411yescharacteristic
94#13108051—21no—
95#13108051—21no—
96#13108051D411yescharacteristic
97C4×Dic101—21no—
98C4×Dic101—21no—
99C4×Dic101C2311yescharacteristic
100C2×Q8×C101C2311yescharacteristic
101C22×C4×C101—21no—
102C22×C4×C101C2311yescharacteristic
103C2×C8×C101C2311yescharacteristic
104C2×C8×C101C2311yescharacteristic
105C4⋊C4×C101—21no—
106C4⋊C4×C101—21no—
107C4⋊C4×C101C2311yescharacteristic
108C4⋊C4×C101C2311yescharacteristic
109C4⋊C4×C101C2311yescharacteristic
110C22⋊C4×C101—21no—
111C23×C101C2×D411yescharacteristic
112C22×D101C2×D411yescharacteristic
113Q8×C101—21no—
114Q8×C101—21no—
115C2×C4×C101—21no—
116C2×C4×C101—21no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—81no—
122C2×C4×C101C2×D411yescharacteristic
123C2×C4×C101C2411yescharacteristic
124C22⋊D101Q1614yescomplemented
125C2×Dic101—21no—
126C2×Dic101—21no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—42no—
130C2×Dic101C2×D411yescharacteristic
131C2×Dic101C2×D411yescharacteristic
132C2×Dic101C2×D411yescharacteristic
133C4×D101—21no—
134C4×D101—21no—
135C4×D101—42no—
136C4×D101—81no—
137Dic202—21no—
138Dic202—21no—
139Dic202—21no—
140Dic202—21no—
141Dic202—21no—
142Dic202—42no—
143Dic202—42no—
144Dic202—81no—
145Dic202#4112yes—
146C8×C101—21no—
147C8×C101—21no—
148#11717579—21no—
149#11717579—21no—
150C22×C101—21no—
151C22×C101—21no—
152C22×C101C2×Q1612yes—
153C22×C101C22×D411yescharacteristic
154C2×D101—21no—
155C2×D101—21no—
156C2×D101C2×Q1612yes—
157C4×C101—21no—
158C4×C101—41no—
159C4×C101—41no—
160C4×C101—41no—
161C4×C101—41no—
162C4×C101C2×#4111yescharacteristic
163C4×C101H(5,𝔽2)11yescharacteristic
164Dic101—22no—
165Dic101—21no—
166Dic101—21no—
167Dic101—41no—
168Dic101—41no—
169Dic101—41no—
170Dic101C2×Q1612yes—
171C2×C101—22no—
172C2×C101#54611yescharacteristic
173C2×C101C22×Q1611yescharacteristic
174C2×C101#57711yescharacteristic
175D101—22no—
176#3299—1011nomaximal, Sylow
177C101#329911yescharacteristic, complemented, Sylow
178#554—1011no—
179D4×Q8—1011no—
180#487—1011no—
181#484—1011no—
182#484D10111yescharacteristic, complemented
183#477—1011no—
184#467—1011no—
185#467—1011no—
186#462—1011no—
187#451—1011no—
188#451—1011no—
189C4×Q16—1011no—
190C8×D4—1011no—
191C2×#103—1011no—
192C2×#103—1011no—
193C22×Q8—1011no—
194C22×Q8—1011no—
195C2×Q16—1011no—
196C2×Q16—1011no—
197C2×Q16—1011no—
198C22×C8—1011no—
199C22×C8—1011no—
200C4⋊Q8—1011no—
201C4⋊Q8—1011no—
202C4⋊Q8—2021no—
203#125—2021no—
204C22⋊Q8—1011no—
205C22⋊Q8—1011no—
206C22⋊Q8—1011no—
207C22⋊Q8—2021no—
208C22⋊Q8—2021no—
209C22⋊Q8—2021no—
210C22⋊Q8—2021no—
211C22⋊Q8C2×D10111yescharacteristic
212C4×Q8—1011no—
213C4×Q8—1011no—
214C4×D4—1011no—
215C4×D4—2021no—
216C4×D4—2021no—
217C2×C4⋊C4—1011no—
218C2×C4⋊C4C2×D10111yescharacteristic
219#107—1011no—
220#107C2×D10111yescharacteristic
221#107C2×D10111yescharacteristic
222C4⋊C8—1011no—
223#103—1011no—
224#103—1011no—
225#103—1011no—
226#103—1011no—
227#103—1011no—
228#103—1011no—
229#103—1011no—
230#103—1011no—
231#103C2×D10111yescharacteristic
232#103C2×D10111yescharacteristic
233C22⋊C8—1011no—
234C22⋊C8C2×D10111yescharacteristic
235C4×C8—1011no—
236C2×Q8—1011no—
237C2×Q8—1011no—
238C2×Q8—1011no—
239C2×Q8—4041no—
240C2×Q8—4041no—
241C2×Q8—2021no—
242C2×Q8—2021no—
243C2×Q8—2022no—
244C2×Q8—2022no—
245C2×Q8—2021no—
246C2×Q8C22×D10111yescharacteristic
247C2×D4—1011no—
248C22×C4—21no—
249C22×C4—1011no—
250C22×C4—2021no—
251C22×C4—2021no—
252C22×C4—2021no—
253C22×C4C22×D10111yescharacteristic
254Q16—4041no—
255Q16—4041no—
256Q16—2022no—
257C2×C8—1011no—
258C2×C8—1011no—
259C2×C8—4041no—
260C2×C8—4041no—
261C2×C8C22×D10111yescharacteristic
262C2×C8C22×D10111yescharacteristic
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—1011no—
266C4⋊C4—1011no—
267C4⋊C4—2021no—
268C4⋊C4—2021no—
269C4⋊C4—2021no—
270C4⋊C4—2021no—
271C4⋊C4—2021no—
272C4⋊C4—2021no—
273C4⋊C4—2021no—
274C4⋊C4—2021no—
275C4⋊C4—2021no—
276C4⋊C4C22×D10111yescharacteristic
277C4⋊C4C22×D10111yescharacteristic
278C4⋊C4C22×D10111yescharacteristic
279C22⋊C4—21no—
280C22⋊C4—1011no—
281C22⋊C4—1011no—
282C22⋊C4—2021no—
283C22⋊C4—2021no—
284C22⋊C4—2021no—
285C42—1011no—
286C42—2021no—
287C42—2021no—
288C23—1011no—
289C23D4×D10111yescharacteristic
290Q8—21no—
291Q8—21no—
292Q8—8081no—
293Q8—1012no—
294Q8—4042no—
295Q8—4042no—
296Q8—2021no—
297Q8—2021no—
298Q8—2021no—
299Q8—2021no—
300Q8—2021no—
301D4—1014no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—81no—
309C2×C4—8081no—
310C2×C4—1011no—
311C2×C4—1011no—
312C2×C4—1011no—
313C2×C4—4042no—
314C2×C4—4042no—
315C2×C4—2021no—
316C2×C4—2021no—
317C2×C4—2021no—
318C2×C4—2021no—
319C2×C4—2021no—
320C2×C4—2021no—
321C2×C4D4×D10111yescharacteristic
322C2×C4C23×D10111yescharacteristic
323C8—21no—
324C8—21no—
325C8—2021no—
326C8—2021no—
327C22—21no—
328C22—21no—
329C22—1012no—
330C22—2021no—
331C22—2021no—
332C22Q16×D10112yes—
333C22C2×D4×D10111yescharacteristic
334C4—21no—
335C4—41no—
336C4—41no—
337C4—41no—
338C4—41no—
339C4—1012no—
340C4—4041no—
341C4—4041no—
342C4—4041no—
343C4—2021no—
344C4—2021no—
345C4—2022no—
346C4#41×D10111yescharacteristic
347C4#1569230111yescharacteristic
348C2—22no—
349C2—2022no—
350C2#2026581911yescharacteristic
351C2C2×Q16×D10111yescharacteristic
352C2#2026598711yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#484×C101C2
socleC22×C101C22×D4
prime radicalC2×C22⋊D101D4
hypercenter#484D101
hypocenterC101#3299
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
3. #20265526C2semidirect
4. #20265514C2semidirect
5. #20265471C2semidirect
6. #20265419C2semidirect
7. #20265416C2semidirect
8. #20265386C2semidirect
9. #20265335C2semidirect
10. #20265334C2semidirect
11. #20265271C2semidirect
13. C2×#15692178C2semidirect
14. #103×D101C2semidirect
15. #484×C101C2semidirect
124. C22⋊D101Q16semidirect
177. C101#3299semidirect
182. #484D101semidirect

Maximal subgroups

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

subgroupindex
#202658822
#202658402
#202655262
#202655142
#202654712
#202654192
#202654162
#202653862
#202653352
#202653342
#202652712
#202652402
C2×#156921782
#103×D1012
#484×C1012
#3299101