Finite Group Databasebeta

← #27174292

Subgroups — #27174292

Order 12928.

464 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692232C211yesmaximal, characteristic
2#20265082C211yesmaximal, characteristic, complemented
3#20265081C211yesmaximal, characteristic, complemented
4#20265075C211yesmaximal, characteristic, complemented
5#20265072C211yesmaximal, characteristic, complemented
6#20265070C211yesmaximal, characteristic
7#20265065C211yesmaximal, characteristic
8#20265065C211yesmaximal, characteristic
9#20265064C211yesmaximal, characteristic, complemented
10#20265061C211yesmaximal, characteristic, complemented
11#20265060C211yesmaximal, characteristic, complemented
12#20265059C211yesmaximal, characteristic, complemented
13#20265038C211yesmaximal, characteristic
14#20265035C211yesmaximal, characteristic
15#388×C101C211yesmaximal, characteristic
16C22×Dic202C2211yescharacteristic
17C23×Dic101C2211yescharacteristic
18#15692232C418yescomplemented
19C2×C4⋊Dic101C2211yescharacteristic
20C2×C4⋊Dic101C2211yescharacteristic
21C2×#13108053C2211yescharacteristic
22C2×#13108053C2211yescharacteristic
23C2×#13108053C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×C22⋊Dic101C2211yescharacteristic
28C2×C22⋊Dic101C2211yescharacteristic
29C2×C22⋊Dic101C2211yescharacteristic
30C2×C4×Dic101C2211yescharacteristic
31C2×C4×Dic101C2211yescharacteristic
32C2×C4×Dic101C2211yescharacteristic
33#15692162C2211yescharacteristic
34#15692162C2211yescharacteristic
35#15692162C2211yescharacteristic
36#15692162C2211yescharacteristic
37#15692162C2211yescharacteristic
38#15692162C2211yescharacteristic
39#15692162C2211yescharacteristic
40#15692162C2211yescharacteristic
41#15692162C2211yescharacteristic
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45C2×C4⋊C4×C101C2211yescharacteristic
46C2×C22⋊C4×C101C2211yescharacteristic
47C2×C22⋊C4×C101C2211yescharacteristic
48C2×C22⋊C4×C101C2211yescharacteristic
49C2×C42×C101C2211yescharacteristic
50#95×C101C2211yescharacteristic
51#95×C101C2211yescharacteristic
52C2×Dic202—41no—
53C2×Dic202—41no—
54C2×Dic202C2×C414yes—
55C22×Dic101—41no—
56C22×Dic101—41no—
57C22×Dic101C2×C414yes—
58C22×Dic101C2311yescharacteristic
59C22×Dic101C2311yescharacteristic
60C22×Dic101C2311yescharacteristic
61C22×Dic101C2311yescharacteristic
62C22×Dic101C2311yescharacteristic
63C22×Dic101C2311yescharacteristic
64C22×Dic101C2311yescharacteristic
65C22×Dic101C2311yescharacteristic
66C4⋊Dic101—41no—
67C4⋊Dic101C2×C414yes—
68#13108053—41no—
69#13108053—41no—
70#13108053—41no—
71#13108053—41no—
72#13108053C2×C414yes—
73#13108053C2×C414yes—
74C22⋊Dic101—41no—
75C22⋊Dic101—41no—
76C22⋊Dic101C2×C414yes—
77C4×Dic101—41no—
78C4×Dic101—41no—
79C4×Dic101—41no—
80C24×C101C2311yescharacteristic
81C22×C4×C101C2311yescharacteristic
82C22×C4×C101C2311yescharacteristic
83C22×C4×C101C2311yescharacteristic
84C22×C4×C101C2311yescharacteristic
85C22×C4×C101C2311yescharacteristic
86C22×C4×C101C2311yescharacteristic
87C4⋊C4×C101—41no—
88C22⋊C4×C101—41no—
89C22⋊C4×C101—41no—
90C22⋊C4×C101C2×C414yes—
91C42×C101—41no—
92C23×C101—21no—
93C23×C101—21no—
94C23×C101—21no—
95C23×C101—21no—
96C23×C101—21no—
97C23×C101—21no—
98C23×C101C22×C412yes—
99C23×C101C2411yescharacteristic
100C2×C4×C101—21no—
101C2×C4×C101—21no—
102C2×C4×C101—21no—
103C2×C4×C101—21no—
104C2×C4×C101—21no—
105C2×C4×C101—21no—
106C2×C4×C101—21no—
107C2×C4×C101—21no—
108C2×C4×C101—21no—
109C2×C4×C101—21no—
110C2×C4×C101—21no—
111C2×C4×C101—21no—
112C2×C4×C101—21no—
113C2×C4×C101—21no—
114C2×C4×C101—21no—
115C2×C4×C101—21no—
116C2×C4×C101C22×C412yes—
117C2×C4×C101C22×C412yes—
118C2×Dic101—21no—
119C2×Dic101—21no—
120C2×Dic101—21no—
121C2×Dic101—21no—
122C2×Dic101—21no—
123C2×Dic101—21no—
124C2×Dic101—21no—
125C2×Dic101—21no—
126C2×Dic101—21no—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—21no—
130C2×Dic101—21no—
131C2×Dic101—21no—
132C2×Dic101—21no—
133C2×Dic101—21no—
134C2×Dic101—21no—
135C2×Dic101—21no—
136C2×Dic101—21no—
137C2×Dic101—21no—
138C2×Dic101—44no—
139C2×Dic101C22×C412yes—
140C2×Dic101C22×C412yes—
141C2×Dic101C22×C412yes—
142C2×Dic101C22×C412yes—
143Dic202—44no—
144C22×C101—22no—
145C22×C101—22no—
146C22×C101—22no—
147C22×C101—41no—
148C22×C101—41no—
149C22×C101—41no—
150C22×C101—41no—
151C22×C101C23×C411yescharacteristic
152C22×C101H(5,𝔽2)11yescharacteristic
153C22×C101H(5,𝔽2)11yescharacteristic
154C22×C101H(5,𝔽2)11yescharacteristic
155C22×C1012−1+411yescharacteristic
156C22×C1012−1+411yescharacteristic
157C22×C1012−1+411yescharacteristic
158C4×C101—22no—
159C4×C101—22no—
160C4×C101—41no—
161C4×C101—41no—
162C4×C101—41no—
163C4×C101—41no—
164Dic101—22no—
165Dic101—22no—
166Dic101—22no—
167Dic101—22no—
168Dic101—41no—
169Dic101—41no—
170Dic101—41no—
171Dic101—41no—
172C2×C101—42no—
173C2×C101#51811yescharacteristic
174C2×C101#51911yescharacteristic
175C2×C101#52011yescharacteristic
176C2×C101#56211yescharacteristic
177C2×C101#56211yescharacteristic
178C2×C101#56311yescharacteristic
179C2×C101#56311yescharacteristic
180#2378—1011nomaximal, Sylow
181C101#237811yescharacteristic, complemented, Sylow
182C2×C22⋊Q8—1011no—
183#391—1011no—
184#391—1011no—
185#389—1011no—
186#389—1011no—
187#388—1011no—
188#388—1011no—
189#388—1011no—
190#388D10111yescharacteristic
191#387—1011no—
192#387—1011no—
193#387—1011no—
194#387—1011no—
195#381—1011no—
196#380—1011no—
197C22×Q8—1011no—
198C23×C4—1011no—
199C22⋊Q8—1018no—
200C2×C4⋊C4—1011no—
201C2×C4⋊C4—1011no—
202C2×C4⋊C4—1011no—
203C2×C4⋊C4—1011no—
204C2×C4⋊C4—1011no—
205C2×C4⋊C4—1011no—
206C2×C4⋊C4—1011no—
207C2×C4⋊C4—1011no—
208C2×C4⋊C4C2×D10111yescharacteristic
209C2×C22⋊C4—1011no—
210C2×C22⋊C4—1011no—
211C2×C22⋊C4—1011no—
212C2×C22⋊C4C2×D10111yescharacteristic
213C2×C22⋊C4C2×D10111yescharacteristic
214C2×C22⋊C4C2×D10111yescharacteristic
215C2×C42—1011no—
216C2×C42—1011no—
217C2×C42—1011no—
218C2×C42C2×D10111yescharacteristic
219#95—1011no—
220#95—1011no—
221#95—1011no—
222#95—1011no—
223#95—1011no—
224#95—1011no—
225#95—1011no—
226#95—1011no—
227#95—1011no—
228#95—1011no—
229#95—1011no—
230#95—1011no—
231#95C2×D10111yescharacteristic
232#95C2×D10111yescharacteristic
233C24C22×D10111yescharacteristic
234C2×Q8—1014no—
235C2×Q8—4041no—
236C2×Q8—4041no—
237C22×C4—1014no—
238C22×C4—1011no—
239C22×C4—1011no—
240C22×C4—1011no—
241C22×C4—1011no—
242C22×C4—1011no—
243C22×C4—1011no—
244C22×C4—1011no—
245C22×C4—1011no—
246C22×C4—4041no—
247C22×C4—4041no—
248C22×C4C22×D10111yescharacteristic
249C22×C4C22×D10111yescharacteristic
250C22×C4C22×D10111yescharacteristic
251C22×C4C22×D10111yescharacteristic
252C22×C4C22×D10111yescharacteristic
253C22×C4C22×D10111yescharacteristic
254C4⋊C4—41no—
255C4⋊C4—1014no—
256C4⋊C4—1014no—
257C4⋊C4—1014no—
258C4⋊C4—4041no—
259C4⋊C4—4041no—
260C4⋊C4—4041no—
261C4⋊C4—4041no—
262C4⋊C4—4041no—
263C22⋊C4—41no—
264C22⋊C4—41no—
265C22⋊C4—1014no—
266C22⋊C4—4041no—
267C22⋊C4—4041no—
268C22⋊C4C4×D10114yes—
269C42—41no—
270C42—4041no—
271C42—4041no—
272C42—4041no—
273C23—21no—
274C23—21no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23C2×C4×D10112yes—
280C23C23×D10111yescharacteristic
281Q8—4044no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—1012no—
299C2×C4—1012no—
300C2×C4—1012no—
301C2×C4—1012no—
302C2×C4—4044no—
303C2×C4—2021no—
304C2×C4—2021no—
305C2×C4—2021no—
306C2×C4—2021no—
307C2×C4—2021no—
308C2×C4—2021no—
309C2×C4—2021no—
310C2×C4—2021no—
311C2×C4—2021no—
312C2×C4—2021no—
313C2×C4—2021no—
314C2×C4—2021no—
315C2×C4—2021no—
316C2×C4—2021no—
317C2×C4—2021no—
318C2×C4—2021no—
319C2×C4—2021no—
320C2×C4—2021no—
321C2×C4—2021no—
322C2×C4—2021no—
323C2×C4C2×C4×D10112yes—
324C2×C4C2×C4×D10112yes—
325C22—22no—
326C22—22no—
327C22—22no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22C22×C4×D10111yescharacteristic
333C22#1569230011yescharacteristic
334C22#1569230011yescharacteristic
335C22#1569230111yescharacteristic
336C22#1569230211yescharacteristic
337C22#1569230211yescharacteristic
338C22#1569230211yescharacteristic
339C4—22no—
340C4—22no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—4041no—
346C4—4041no—
347C4—4041no—
348C4—4041no—
349C4—2022no—
350C4—2022no—
351C4—2022no—
352C4—2022no—
353C2—42no—
354C2#2026564911yescharacteristic
355C2#2026565411yescharacteristic
356C2#2026565611yescharacteristic
357C2#2026592211yescharacteristic
358C2#2026592411yescharacteristic
359C2#2026592711yescharacteristic
360C2#2026592911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#388×C101C2
socle, cosocleC23×C101C24
prime radicalC24×C101C23
hypercenter#388D101
hypocenterC101#2378
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #20265082C2semidirect
3. #20265081C2semidirect
4. #20265075C2semidirect
5. #20265072C2semidirect
9. #20265064C2semidirect
10. #20265061C2semidirect
11. #20265060C2semidirect
12. #20265059C2semidirect
18. #15692232C4semidirect
181. C101#2378semidirect

Maximal subgroups

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

subgroupindex
C2×#156922322
#202650822
#202650812
#202650752
#202650722
#202650702
#202650652
#202650642
#202650612
#202650602
#202650592
#202650382
#202650352
#388×C1012
#2378101