Finite Group Databasebeta

← #27174279

Subgroups — #27174279

Order 12928.

512 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692240C211yesmaximal, characteristic
2#20265079C211yesmaximal, characteristic
3#20265076C211yesmaximal, characteristic, complemented
4#20265074C211yesmaximal, characteristic, complemented
5#20265069C211yesmaximal, characteristic, complemented
6#20265067C211yesmaximal, characteristic, complemented
7#20265066C211yesmaximal, characteristic, complemented
8#20265066C211yesmaximal, characteristic, complemented
9#20265061C211yesmaximal, characteristic, complemented
10#20265055C211yesmaximal, characteristic
11#20265051C211yesmaximal, characteristic, complemented
12#20265051C211yesmaximal, characteristic, complemented
13#20265036C211yesmaximal, characteristic, complemented
14#20265034C211yesmaximal, characteristic, complemented
15#387×C101C211yesmaximal, characteristic, complemented
16C22×D404C2211yescharacteristic
17C22×C4×D101C2211yescharacteristic
18C22×C4×D101C2211yescharacteristic
19#15692240C418yescomplemented
20C2×C4⋊Dic101C2211yescharacteristic
21C2×C4⋊Dic101C2211yescharacteristic
22C2×C4⋊Dic101C2211yescharacteristic
23C2×#13108053C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×#13108051C2211yescharacteristic
28C2×#13108051C2211yescharacteristic
29C2×#13108051C2211yescharacteristic
30C2×#13108051C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×#13108051C2211yescharacteristic
37C2×C4×Dic101C2211yescharacteristic
38C2×C4×Dic101C2211yescharacteristic
39#15692162C2211yescharacteristic
40#15692162C2211yescharacteristic
41#15692162C2211yescharacteristic
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45C2×C4⋊C4×C101C2211yescharacteristic
46C2×C4⋊C4×C101C2211yescharacteristic
47C2×C42×C101C2211yescharacteristic
48#95×C101C2211yescharacteristic
49#95×C101C2211yescharacteristic
50#95×C101C2211yescharacteristic
51#95×C101C2211yescharacteristic
52C23×D101C2311yescharacteristic
53C23×D101C2311yescharacteristic
54C2×D404—41no—
55C2×D404—41no—
56C2×D404C2×C414yes—
57C22×Dic101C2311yescharacteristic
58C22×Dic101C2311yescharacteristic
59C22×Dic101C2311yescharacteristic
60C22×Dic101C2311yescharacteristic
61C22×Dic101C2311yescharacteristic
62C22×Dic101C2311yescharacteristic
63C2×C4×D101—41no—
64C2×C4×D101—41no—
65C2×C4×D101—41no—
66C2×C4×D101—41no—
67C2×C4×D101—41no—
68C2×C4×D101C2×C414yescomplemented
69C4⋊Dic101—41no—
70C4⋊Dic101—41no—
71C4⋊Dic101—41no—
72#13108053—41no—
73#13108053—41no—
74#13108053—41no—
75#13108053C2×C414yescomplemented
76#13108051—41no—
77#13108051—41no—
78#13108051—41no—
79#13108051—41no—
80#13108051—41no—
81#13108051—41no—
82#13108051—41no—
83#13108051C2×C414yes—
84#13108051C2×C414yes—
85#13108051C2×C414yescomplemented
86C4×Dic101—41no—
87C4×Dic101—41no—
88C22×C4×C101C2311yescharacteristic
89C22×C4×C101C2311yescharacteristic
90C22×C4×C101C2311yescharacteristic
91C22×C4×C101C2311yescharacteristic
92C22×C4×C101C2311yescharacteristic
93C22×C4×C101C2311yescharacteristic
94C22×C4×C101C2311yescharacteristic
95C4⋊C4×C101—41no—
96C4⋊C4×C101C2×C414yescomplemented
97C42×C101—41no—
98C23×C101C2411yescharacteristic
99C22×D101—21no—
100C22×D101—21no—
101C22×D101—21no—
102C22×D101—21no—
103C22×D101—21no—
104C22×D101—21no—
105C22×D101—21no—
106C22×D101—21no—
107C22×D101—21no—
108C22×D101—21no—
109C22×D101—21no—
110C22×D101—21no—
111C22×D101C22×C412yes—
112C22×D101C22×C412yes—
113C2×C4×C101—21no—
114C2×C4×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—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×C101C22×C412yes—
132C2×C4×C101C22×C412yes—
133C2×C4×C101C22×C412yes—
134C2×Dic101—21no—
135C2×Dic101—21no—
136C2×Dic101—21no—
137C2×Dic101—21no—
138C2×Dic101—21no—
139C2×Dic101—21no—
140C2×Dic101—21no—
141C2×Dic101—21no—
142C2×Dic101—21no—
143C2×Dic101—21no—
144C2×Dic101—21no—
145C2×Dic101—21no—
146C2×Dic101—21no—
147C2×Dic101—21no—
148C2×Dic101—21no—
149C2×Dic101—21no—
150C2×Dic101C22×C412yes—
151C2×Dic101C22×C412yes—
152D404—44no—
153C4×D101—44no—
154C4×D101—44no—
155C22×C101C23×C411yescharacteristic
156C22×C101H(5,𝔽2)11yescharacteristic
157C22×C101H(5,𝔽2)11yescharacteristic
158C22×C101H(5,𝔽2)11yescharacteristic
159C22×C101H(5,𝔽2)11yescharacteristic
160C22×C1012−1+411yescharacteristic
161C22×C1012−1+411yescharacteristic
162C2×D101—22no—
163C2×D101—22no—
164C2×D101—22no—
165C2×D101—22no—
166C2×D101—22no—
167C2×D101—22no—
168C2×D101—41no—
169C2×D101—41no—
170C2×D101—41no—
171C2×D101—41no—
172C2×D101—41no—
173C2×D101—41no—
174C2×D101—41no—
175C2×D101—41no—
176C4×C101—22no—
177C4×C101—22no—
178C4×C101—22no—
179C4×C101—41no—
180C4×C101—41no—
181C4×C101—41no—
182C4×C101—41no—
183Dic101—22no—
184Dic101—22no—
185Dic101—41no—
186Dic101—41no—
187Dic101—41no—
188Dic101—41no—
189C2×C101#51811yescharacteristic
190C2×C101#52011yescharacteristic
191C2×C101#52011yescharacteristic
192C2×C101#56011yescharacteristic
193C2×C101#56211yescharacteristic
194C2×C101#56211yescharacteristic
195C2×C101#56311yescharacteristic
196D101—42no—
197D101—42no—
198#2376—1011nomaximal, Sylow
199C101#237611yescharacteristic, complemented, Sylow
200C2×#123—1011no—
201#390—1011no—
202#389—1011no—
203#389—1011no—
204#388—1011no—
205#388—1011no—
206#388—1011no—
207#388—1011no—
208#387—1011no—
209#387D10111yescharacteristic, complemented
210#386—1011no—
211#385—1011no—
212#385—1011no—
213#380—1011no—
214#380—1011no—
215C22×D4—1011no—
216C23×C4—1011no—
217C23×C4—1011no—
218#123—1018no—
219C2×C4⋊C4—1011no—
220C2×C4⋊C4—1011no—
221C2×C4⋊C4—1011no—
222C2×C4⋊C4—1011no—
223C2×C4⋊C4—1011no—
224C2×C4⋊C4—1011no—
225C2×C4⋊C4—1011no—
226C2×C4⋊C4C2×D10111yescharacteristic
227C2×C4⋊C4C2×D10111yescharacteristic
228C2×C22⋊C4—1011no—
229C2×C22⋊C4—1011no—
230C2×C22⋊C4—1011no—
231C2×C22⋊C4—1011no—
232C2×C22⋊C4—1011no—
233C2×C22⋊C4—1011no—
234C2×C22⋊C4—1011no—
235C2×C22⋊C4—1011no—
236C2×C22⋊C4—1011no—
237C2×C22⋊C4—1011no—
238C2×C42—1011no—
239C2×C42—1011no—
240C2×C42C2×D10111yescharacteristic
241#95—1011no—
242#95—1011no—
243#95—1011no—
244#95—1011no—
245#95—1011no—
246#95—1011no—
247#95C2×D10111yescharacteristic
248#95C2×D10111yescharacteristic
249#95C2×D10111yescharacteristic
250#95C2×D10111yescharacteristic
251C24—1011no—
252C24—1011no—
253C2×D4—1014no—
254C2×D4—4041no—
255C2×D4—4041no—
256C22×C4—1011no—
257C22×C4—1011no—
258C22×C4—1014no—
259C22×C4—1011no—
260C22×C4—1011no—
261C22×C4—1011no—
262C22×C4—1011no—
263C22×C4—4041no—
264C22×C4—4041no—
265C22×C4—4041no—
266C22×C4—4041no—
267C22×C4—4041no—
268C22×C4C22×D10111yescharacteristic
269C22×C4C22×D10111yescharacteristic
270C22×C4C22×D10111yescharacteristic
271C22×C4C22×D10111yescharacteristic
272C22×C4C22×D10111yescharacteristic
273C22×C4C22×D10111yescharacteristic
274C22×C4C22×D10111yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—1014no—
277C4⋊C4—4041no—
278C4⋊C4—4041no—
279C4⋊C4—4041no—
280C4⋊C4—4041no—
281C4⋊C4—4041no—
282C4⋊C4—4041no—
283C4⋊C4C4×D10114yescomplemented
284C22⋊C4—1014no—
285C22⋊C4—1014no—
286C22⋊C4—1014no—
287C22⋊C4—4041no—
288C22⋊C4—4041no—
289C22⋊C4—4041no—
290C22⋊C4—4041no—
291C22⋊C4—4041no—
292C22⋊C4—4041no—
293C22⋊C4—4041no—
294C42—41no—
295C42—4041no—
296C42—4041no—
297C23—1012no—
298C23—1012no—
299C23—2021no—
300C23—2021no—
301C23—2021no—
302C23—2021no—
303C23—2021no—
304C23—2021no—
305C23—2021no—
306C23—2021no—
307C23—2021no—
308C23—2021no—
309C23—2021no—
310C23—2021no—
311C23C23×D10111yescharacteristic
312D4—4044no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—1012no—
332C2×C4—1012no—
333C2×C4—4044no—
334C2×C4—4044no—
335C2×C4—2021no—
336C2×C4—2021no—
337C2×C4—2021no—
338C2×C4—2021no—
339C2×C4—2021no—
340C2×C4—2021no—
341C2×C4—2021no—
342C2×C4—2021no—
343C2×C4—2021no—
344C2×C4—2021no—
345C2×C4—2021no—
346C2×C4—2021no—
347C2×C4—2021no—
348C2×C4—2021no—
349C2×C4—2021no—
350C2×C4—2021no—
351C2×C4C2×C4×D10112yes—
352C2×C4C2×C4×D10112yes—
353C2×C4C2×C4×D10112yes—
354C22—4041no—
355C22—4041no—
356C22—4041no—
357C22—4041no—
358C22—4041no—
359C22—4041no—
360C22—4041no—
361C22—4041no—
362C22—2022no—
363C22—2022no—
364C22—2022no—
365C22—2022no—
366C22—2022no—
367C22—2022no—
368C22C22×C4×D10111yescharacteristic
369C22#1569230011yescharacteristic
370C22#1569230111yescharacteristic
371C22#1569230111yescharacteristic
372C22#1569230111yescharacteristic
373C22#1569230211yescharacteristic
374C22#1569230311yescharacteristic
375C4—22no—
376C4—22no—
377C4—22no—
378C4—41no—
379C4—41no—
380C4—41no—
381C4—41no—
382C4—4041no—
383C4—4041no—
384C4—4041no—
385C4—4041no—
386C4—2022no—
387C4—2022no—
388C2—4042no—
389C2—4042no—
390C2#2026564911yescharacteristic
391C2#2026565711yescharacteristic
392C2#2026565911yescharacteristic
393C2#2026591811yescharacteristic
394C2#2026592311yescharacteristic
395C2#2026592411yescharacteristic
396C2#2026593111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#387×C101C2
socle, cosocleC23×C101C24
prime radicalC22×D404C22
hypercenter#387D101
hypocenterC101#2376
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
3. #20265076C2semidirect
4. #20265074C2semidirect
5. #20265069C2semidirect
6. #20265067C2semidirect
7. #20265066C2semidirect
8. #20265066C2semidirect
9. #20265061C2semidirect
11. #20265051C2semidirect
12. #20265051C2semidirect
13. #20265036C2semidirect
14. #20265034C2semidirect
15. #387×C101C2semidirect
19. #15692240C4semidirect
68. C2×C4×D101C2×C4semidirect
75. #13108053C2×C4semidirect
85. #13108051C2×C4semidirect
96. C4⋊C4×C101C2×C4semidirect
199. C101#2376semidirect
209. #387D101semidirect
283. C4⋊C4C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922402
#202650792
#202650762
#202650742
#202650692
#202650672
#202650662
#202650612
#202650552
#202650512
#202650362
#202650342
#387×C1012
#2376101