Finite Group Databasebeta

← #27174078

Subgroups — #27174078

Order 12928.

632 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692215C211yesmaximal, characteristic
2C2×#15692212C211yesmaximal, characteristic, complemented
3C2×#15692211C211yesmaximal, characteristic, complemented
4#20265077C211yesmaximal, characteristic
5#20265077C211yesmaximal, characteristic
6#20265070C211yesmaximal, characteristic, complemented
7#20265069C211yesmaximal, characteristic, complemented
8#20265067C211yesmaximal, characteristic, complemented
9#20265046C211yesmaximal, characteristic, complemented
10#20265036C211yesmaximal, characteristic, complemented
11#20265035C211yesmaximal, characteristic, complemented
12C4×#13108053C211yesmaximal, characteristic, complemented
13#20265028C211yesmaximal, characteristic, complemented
14#20265028C211yesmaximal, characteristic, complemented
15#388×C101C211yesmaximal, characteristic, complemented
16C22×C22⋊D101C2211yescharacteristic
17C23×Dic101C2211yescharacteristic
18C22×C4×D101C2211yescharacteristic
19#15692215C418yescomplemented
20#15692212—24no—
21#15692211—24no—
22C2×C4⋊Dic101C2211yescharacteristic
23C2×#13108053C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×#13108053C2211yescharacteristic
28C2×#13108053C2211yescharacteristic
29C2×#13108053C2211yescharacteristic
30C2×C22⋊Dic101C2211yescharacteristic
31C2×C22⋊Dic101C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×#13108051C2211yescharacteristic
37C2×C4×Dic101C2211yescharacteristic
38C2×C4×Dic101C2211yescharacteristic
39C2×C4×Dic101C2211yescharacteristic
40C2×C4×Dic101C2211yescharacteristic
41C2×C4×Dic101C2211yescharacteristic
42C2×C4×Dic101C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45#15692162C2211yescharacteristic
46#15692162C2211yescharacteristic
47C2×C4⋊C4×C101C2211yescharacteristic
48C2×C22⋊C4×C101C2211yescharacteristic
49C2×C22⋊C4×C101C2211yescharacteristic
50C2×C22⋊C4×C101C2211yescharacteristic
51C2×C42×C101C2211yescharacteristic
52#95×C101C2211yescharacteristic
53#95×C101C2211yescharacteristic
54C23×D101C2311yescharacteristic
55C2×C22⋊D101—22no—
56C2×C22⋊D101—22no—
57C2×C22⋊D101C2×C414yes—
58C22×Dic101—22no—
59C22×Dic101—22no—
60C22×Dic101C2×C414yes—
61C22×Dic101C2311yescharacteristic
62C22×Dic101C2311yescharacteristic
63C22×Dic101C2311yescharacteristic
64C22×Dic101C2311yescharacteristic
65C22×Dic101C2311yescharacteristic
66C22×Dic101C2311yescharacteristic
67C22×Dic101C2311yescharacteristic
68C2×C4×D101—22no—
69C2×C4×D101—22no—
70C2×C4×D101C2×C414yes—
71C4⋊Dic101—41no—
72#13108053—22no—
73#13108053—22no—
74#13108053—22no—
75#13108053—22no—
76#13108053—22no—
77#13108053C2×C414yes—
78#13108053D414yescomplemented
79C22⋊Dic101—22no—
80C22⋊Dic101—41no—
81#13108051—22no—
82#13108051—22no—
83#13108051—41no—
84#13108051—41no—
85#13108051C2×C414yes—
86C4×Dic101—22no—
87C4×Dic101—22no—
88C4×Dic101—22no—
89C4×Dic101—22no—
90C4×Dic101—22no—
91C4×Dic101C2×C414yes—
92C24×C101C2311yescharacteristic
93C22×C4×C101C2311yescharacteristic
94C22×C4×C101C2311yescharacteristic
95C22×C4×C101C2311yescharacteristic
96C22×C4×C101C2311yescharacteristic
97C22×C4×C101C2311yescharacteristic
98C22×C4×C101C2311yescharacteristic
99C4⋊C4×C101—41no—
100C22⋊C4×C101—22no—
101C22⋊C4×C101—41no—
102C22⋊C4×C101C2×C414yes—
103C42×C101—22no—
104C23×C101—21no—
105C23×C101—21no—
106C23×C101—21no—
107C23×C101—21no—
108C23×C101—21no—
109C23×C101—21no—
110C23×C101C22×C412yes—
111C23×C101C2411yescharacteristic
112C22×D101—21no—
113C22×D101—21no—
114C22×D101—21no—
115C22×D101—21no—
116C22×D101—21no—
117C22×D101—21no—
118C22×D101C22×C412yes—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
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—21no—
130C2×C4×C101—21no—
131C2×C4×C101—21no—
132C2×C4×C101—21no—
133C2×C4×C101—21no—
134C2×C4×C101C22×C412yes—
135C2×C4×C101C22×C412yes—
136C2×C4×C101C2×D412yes—
137C22⋊D101—28no—
138C2×Dic101—21no—
139C2×Dic101—21no—
140C2×Dic101—21no—
141C2×Dic101—21no—
142C2×Dic101—28no—
143C2×Dic101—21no—
144C2×Dic101—21no—
145C2×Dic101—21no—
146C2×Dic101—21no—
147C2×Dic101—21no—
148C2×Dic101—21no—
149C2×Dic101—21no—
150C2×Dic101—21no—
151C2×Dic101—21no—
152C2×Dic101C22×C412yes—
153C2×Dic101C22×C412yes—
154C2×Dic101C22×C412yes—
155C2×Dic101C2×D412yes—
156C2×Dic101C2×D412yes—
157C2×Dic101#4112yes—
158C2×Dic101#4112yes—
159C2×Dic101#4112yes—
160C4×D101—28no—
161C22×C101—22no—
162C22×C101—22no—
163C22×C101—22no—
164C22×C101—41no—
165C22×C101—41no—
166C22×C101—41no—
167C22×C101—41no—
168C22×C101C23×C411yescharacteristic
169C22×C101C22×D411yescharacteristic
170C22×C101C2×#4111yescharacteristic
171C22×C101C2×#4111yescharacteristic
172C22×C101C2×#4111yescharacteristic
173C22×C101H(5,𝔽2)11yescharacteristic
174C22×C1012−1+411yescharacteristic
175C2×D101—22no—
176C2×D101—22no—
177C2×D101—22no—
178C2×D101—41no—
179C2×D101—41no—
180C2×D101—41no—
181C2×D101—41no—
182C4×C101—22no—
183C4×C101—22no—
184C4×C101—22no—
185C4×C101—41no—
186C4×C101—41no—
187C4×C101—41no—
188Dic101—22no—
189Dic101—22no—
190Dic101—41no—
191Dic101—41no—
192Dic101—41no—
193Dic101#11714yescomplemented
194Dic101C4×D414yes—
195C2×C101—42no—
196C2×C101C2×#11711yescharacteristic
197C2×C101C2×C4×D411yescharacteristic
198C2×C101#52011yescharacteristic
199C2×C101#54711yescharacteristic
200C2×C101#55011yescharacteristic
201C2×C101#55511yescharacteristic
202C2×C101#55611yescharacteristic
203D101—42no—
204#2354—1011nomaximal, Sylow
205C101#235411yescharacteristic, complemented, Sylow
206C2×C4×D4—1011no—
207C2×#117—1011no—
208C2×#117—1011no—
209#390—1011no—
210#390—1011no—
211#388—1011no—
212#388—1011no—
213#388—1011no—
214#388D10111yescharacteristic, complemented
215#384—1011no—
216#380—1011no—
217#380—1011no—
218C4×C4⋊C4—1011no—
219C4×C4⋊C4—1011no—
220C4×C4⋊C4—1011no—
221C22×D4—1011no—
222C23×C4—1011no—
223C23×C4—1011no—
224C4×D4—1018no—
225#117—2024no—
226#117—2024no—
227C2×C4⋊C4—1011no—
228C2×C4⋊C4—1011no—
229C2×C4⋊C4—1011no—
230C2×C4⋊C4—1011no—
231C2×C4⋊C4—1011no—
232C2×C4⋊C4—1011no—
233C2×C4⋊C4—1011no—
234C2×C4⋊C4—1011no—
235C2×C4⋊C4C2×D10111yescharacteristic
236C2×C22⋊C4—1011no—
237C2×C22⋊C4—1011no—
238C2×C22⋊C4—1011no—
239C2×C22⋊C4—1011no—
240C2×C22⋊C4—1011no—
241C2×C22⋊C4—1011no—
242C2×C22⋊C4—1011no—
243C2×C22⋊C4C2×D10111yescharacteristic
244C2×C22⋊C4C2×D10111yescharacteristic
245C2×C22⋊C4C2×D10111yescharacteristic
246C2×C42—1011no—
247C2×C42—1011no—
248C2×C42—1011no—
249C2×C42—1011no—
250C2×C42—1011no—
251C2×C42—1011no—
252C2×C42C2×D10111yescharacteristic
253#95—1011no—
254#95—1011no—
255#95—1011no—
256#95—1011no—
257#95C2×D10111yescharacteristic
258#95C2×D10111yescharacteristic
259C24—1011no—
260C24C22×D10111yescharacteristic
261C2×D4—1014no—
262C2×D4—2022no—
263C2×D4—2022no—
264C22×C4—1011no—
265C22×C4—1011no—
266C22×C4—1014no—
267C22×C4—1011no—
268C22×C4—1011no—
269C22×C4—1014no—
270C22×C4—1011no—
271C22×C4—1011no—
272C22×C4—1011no—
273C22×C4—2022no—
274C22×C4—2022no—
275C22×C4—2022no—
276C22×C4—2022no—
277C22×C4C22×D10111yescharacteristic
278C22×C4C22×D10111yescharacteristic
279C22×C4C22×D10111yescharacteristic
280C22×C4C22×D10111yescharacteristic
281C22×C4C22×D10111yescharacteristic
282C22×C4C22×D10111yescharacteristic
283C4⋊C4—41no—
284C4⋊C4—1014no—
285C4⋊C4—1014no—
286C4⋊C4—4041no—
287C4⋊C4—2022no—
288C4⋊C4—2022no—
289C4⋊C4—2022no—
290C4⋊C4—2022no—
291C4⋊C4—2022no—
292C22⋊C4—22no—
293C22⋊C4—41no—
294C22⋊C4—1014no—
295C22⋊C4—4041no—
296C22⋊C4—4041no—
297C22⋊C4—4041no—
298C22⋊C4—2022no—
299C22⋊C4—2022no—
300C22⋊C4—2022no—
301C22⋊C4C4×D10114yes—
302C42—22no—
303C42—1014no—
304C42—2022no—
305C42—2022no—
306C42—2022no—
307C42—2022no—
308C42—2022no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—1012no—
316C23—2021no—
317C23—2021no—
318C23—2021no—
319C23—2021no—
320C23—2021no—
321C23—2021no—
322C23C2×C4×D10112yes—
323C23C23×D10111yescharacteristic
324D4—2028no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—1012no—
341C2×C4—1012no—
342C2×C4—1012no—
343C2×C4—1012no—
344C2×C4—1012no—
345C2×C4—1012no—
346C2×C4—1012no—
347C2×C4—1012no—
348C2×C4—2028no—
349C2×C4—2028no—
350C2×C4—2021no—
351C2×C4—2021no—
352C2×C4—2021no—
353C2×C4—2021no—
354C2×C4—2021no—
355C2×C4—2021no—
356C2×C4—2021no—
357C2×C4—2021no—
358C2×C4—2021no—
359C2×C4—2021no—
360C2×C4—2021no—
361C2×C4—2021no—
362C2×C4—2021no—
363C2×C4C2×C4×D10112yes—
364C2×C4C2×C4×D10112yes—
365C2×C4D4×D10112yes—
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—4041no—
374C22—4041no—
375C22—4041no—
376C22—4041no—
377C22—2022no—
378C22—2022no—
379C22—2022no—
380C22C22×C4×D10111yescharacteristic
381C22C2×D4×D10111yescharacteristic
382C22#41×D10111yescharacteristic
383C22#41×D10111yescharacteristic
384C22#41×D10111yescharacteristic
385C22#1569230011yescharacteristic
386C22#1569230211yescharacteristic
387C4—22no—
388C4—22no—
389C4—22no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—1014no—
394C4—1014no—
395C4—4041no—
396C4—4041no—
397C4—4041no—
398C4—2022no—
399C4—2022no—
400C2—42no—
401C2—4042no—
402C2#117×D10111yescharacteristic
403C2C4×D4×D10111yescharacteristic
404C2#2026565611yescharacteristic
405C2#2026582511yescharacteristic
406C2#2026584511yescharacteristic
407C2#2026588911yescharacteristic
408C2#2026589411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#388×C101C2
socle, cosocleC23×C101C24
prime radicalC22×C22⋊D101C22
hypercenter#388D101
hypocenterC101#2354
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#15692212C2semidirect
3. C2×#15692211C2semidirect
6. #20265070C2semidirect
7. #20265069C2semidirect
8. #20265067C2semidirect
9. #20265046C2semidirect
10. #20265036C2semidirect
11. #20265035C2semidirect
12. C4×#13108053C2semidirect
13. #20265028C2semidirect
14. #20265028C2semidirect
15. #388×C101C2semidirect
19. #15692215C4semidirect
78. #13108053D4semidirect
193. Dic101#117semidirect
205. C101#2354semidirect
214. #388D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922152
C2×#156922122
C2×#156922112
#202650772
#202650702
#202650692
#202650672
#202650462
#202650362
#202650352
C4×#131080532
#202650282
#388×C1012
#2354101