Finite Group Databasebeta

← #27173758

Subgroups — #27173758

Order 12928.

568 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×Dic101C211yesmaximal, characteristic
2#20265071C211yesmaximal, characteristic, complemented
3#20265065C211yesmaximal, characteristic, complemented
4#20265062C211yesmaximal, characteristic, complemented
5#20265060C211yesmaximal, characteristic, complemented
6#20265057C211yesmaximal, characteristic
7#20265050C211yesmaximal, characteristic, complemented
8#20265049C211yesmaximal, characteristic, complemented
9#20265044C211yesmaximal, characteristic, complemented
10#20265041C211yesmaximal, characteristic, complemented
11#20265038C211yesmaximal, characteristic, complemented
12#20265035C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic101C211yesmaximal, characteristic, complemented
14C2×#15692162C211yesmaximal, characteristic, complemented
15#386×C101C211yesmaximal, characteristic
16C23×Dic101C2211yescharacteristic
17C23×Dic101C2211yescharacteristic
18C23×Dic101C2211yescharacteristic
19D4×Dic101C418yescomplemented
20C2×C4⋊Dic101C2211yescharacteristic
21C2×#13108053C2211yescharacteristic
22C2×#13108053C2211yescharacteristic
23C2×#13108053C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×C22⋊Dic101C2211yescharacteristic
26C2×C22⋊Dic101C2211yescharacteristic
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
45C22×D4×C101C2211yescharacteristic
46C23×C4×C101C2211yescharacteristic
47C2×C22⋊C4×C101C2211yescharacteristic
48C2×C22⋊C4×C101C2211yescharacteristic
49C2×C22⋊C4×C101C2211yescharacteristic
50#95×C101C2211yescharacteristic
51#95×C101C2211yescharacteristic
52C22×Dic101—22no—
53C22×Dic101—41no—
54C22×Dic101—41no—
55C22×Dic101—41no—
56C22×Dic101—41no—
57C22×Dic101—41no—
58C22×Dic101—41no—
59C22×Dic101C2×C414yes—
60C22×Dic101C2×C414yescomplemented
61C22×Dic101C2311yescharacteristic
62C22×Dic101C2311yescharacteristic
63C22×Dic101C2311yescharacteristic
64C22×Dic101C2311yescharacteristic
65C22×Dic101C2311yescharacteristic
66C22×Dic101C2311yescharacteristic
67C22×Dic101C2311yescharacteristic
68C22×Dic101C2311yescharacteristic
69C4⋊Dic101C2×C414yescomplemented
70#13108053—41no—
71#13108053—41no—
72#13108053—41no—
73#13108053—41no—
74C22⋊Dic101—22no—
75C22⋊Dic101—41no—
76C22⋊Dic101—41no—
77C22⋊Dic101C2×C414yes—
78C22⋊Dic101C2×C414yescomplemented
79C4×Dic101—41no—
80C4×Dic101—41no—
81C4×Dic101C2×C414yescomplemented
82C24×C101C2311yescharacteristic
83C24×C101C2311yescharacteristic
84C2×D4×C101—22no—
85C2×D4×C101—22no—
86C2×D4×C101C2×C414yes—
87C22×C4×C101—22no—
88C22×C4×C101—41no—
89C22×C4×C101—41no—
90C22×C4×C101C2311yescharacteristic
91C22×C4×C101C2311yescharacteristic
92C22×C4×C101C2311yescharacteristic
93C22×C4×C101C2311yescharacteristic
94C22×C4×C101C2311yescharacteristic
95C22⋊C4×C101—22no—
96C22⋊C4×C101—41no—
97C22⋊C4×C101—41no—
98C23×C101—21no—
99C23×C101—21no—
100C23×C101—21no—
101C23×C101—21no—
102C23×C101—21no—
103C23×C101—21no—
104C23×C101—21no—
105C23×C101—21no—
106C23×C101—21no—
107C23×C101—21no—
108C23×C101C22×C412yes—
109C23×C101C22×C412yes—
110C23×C101#4112yes—
111C23×C101#4112yes—
112C23×C101C2411yescharacteristic
113D4×C101—28no—
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—44no—
129C2×C4×C101C22×C412yes—
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—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×Dic101—44no—
151C2×Dic101—44no—
152C2×Dic101—44no—
153C2×Dic101C22×C412yes—
154C2×Dic101C22×C412yes—
155C2×Dic101C22×C412yes—
156C2×Dic101C22×C412yes—
157C22×C101—22no—
158C22×C101—22no—
159C22×C101—22no—
160C22×C101—22no—
161C22×C101—22no—
162C22×C101—22no—
163C22×C101—22no—
164C22×C101—22no—
165C22×C101—22no—
166C22×C101—41no—
167C22×C101—41no—
168C22×C101—41no—
169C22×C101—41no—
170C22×C101#11714yes—
171C22×C101C23×C411yescharacteristic
172C22×C101C2×#4111yescharacteristic
173C22×C101C2×#4111yescharacteristic
174C22×C101H(5,𝔽2)11yescharacteristic
175C22×C101H(5,𝔽2)11yescharacteristic
176C22×C101H(5,𝔽2)11yescharacteristic
177C22×C1012−1+411yescharacteristic
178C4×C101—22no—
179C4×C101—41no—
180C4×C101—41no—
181C4×C101—41no—
182C4×C101—41no—
183Dic101—22no—
184Dic101—22no—
185Dic101—22no—
186Dic101—22no—
187Dic101—41no—
188Dic101—41no—
189Dic101—41no—
190Dic101—41no—
191C2×C101—24no—
192C2×C101—42no—
193C2×C101C2×#11711yescharacteristic
194C2×C101#51811yescharacteristic
195C2×C101#52011yescharacteristic
196C2×C101#53811yescharacteristic
197C2×C101#53811yescharacteristic
198C2×C101#53911yescharacteristic
199C2×C101#53911yescharacteristic
200#2318—1011nomaximal, Sylow
201C101#231811yescharacteristic, complemented, Sylow
202C2×C4×D4—1011no—
203#388—1011no—
204#388—1011no—
205#387—1011no—
206#387—1011no—
207#386—1011no—
208#386D10111yescharacteristic
209#385—1011no—
210#385—1011no—
211#383—1011no—
212#382—1011no—
213#381—1011no—
214#380—1011no—
215C4×C22⋊C4—1011no—
216C2×#95—1011no—
217C22×D4C2×D10111yescharacteristic
218C23×C4—1011no—
219C23×C4—1011no—
220C23×C4—1011no—
221C23×C4C2×D10111yescharacteristic
222C4×D4—1018no—
223C2×C4⋊C4—1011no—
224C2×C4⋊C4—1011no—
225C2×C4⋊C4—1011no—
226C2×C4⋊C4—1011no—
227C2×C4⋊C4—1011no—
228C2×C22⋊C4—1011no—
229C2×C22⋊C4—1011no—
230C2×C22⋊C4—1011no—
231C2×C22⋊C4—1011no—
232C2×C22⋊C4—1011no—
233C2×C22⋊C4C2×D10111yescharacteristic
234C2×C22⋊C4C2×D10111yescharacteristic
235C2×C22⋊C4C2×D10111yescharacteristic
236C2×C42—1011no—
237C2×C42—1011no—
238C2×C42—1011no—
239#95—1011no—
240#95—1011no—
241#95—1011no—
242#95—1011no—
243#95—1011no—
244#95—1011no—
245#95—1011no—
246#95—1011no—
247#95—1011no—
248#95—1011no—
249#95—1011no—
250#95—1011no—
251#95C2×D10111yescharacteristic
252#95C2×D10111yescharacteristic
253C24C22×D10111yescharacteristic
254C24C22×D10111yescharacteristic
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C4×D10114yes—
258C22×C4—22no—
259C22×C4—41no—
260C22×C4—41no—
261C22×C4—1014no—
262C22×C4—1014no—
263C22×C4—1011no—
264C22×C4—1011no—
265C22×C4—1011no—
266C22×C4—1011no—
267C22×C4—1011no—
268C22×C4—1011no—
269C22×C4—1011no—
270C22×C4—1011no—
271C22×C4—4041no—
272C22×C4—4041no—
273C22×C4—4041no—
274C22×C4—4041no—
275C22×C4—4041no—
276C22×C4—4041no—
277C22×C4—2022no—
278C22×C4C22×D10111yescharacteristic
279C22×C4C22×D10111yescharacteristic
280C22×C4C22×D10111yescharacteristic
281C22×C4C22×D10111yescharacteristic
282C22×C4C22×D10111yescharacteristic
283C4⋊C4—1014no—
284C4⋊C4—4041no—
285C4⋊C4—4041no—
286C4⋊C4—4041no—
287C4⋊C4—4041no—
288C22⋊C4—22no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—1014no—
292C22⋊C4—1014no—
293C22⋊C4—4041no—
294C22⋊C4—4041no—
295C22⋊C4—2022no—
296C42—1014no—
297C42—4041no—
298C42—4041no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—21no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23C2×C4×D10112yes—
310C23C2×C4×D10112yes—
311C23#1310807312yes—
312C23#1310807312yes—
313C23C23×D10111yescharacteristic
314D4—28no—
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—44no—
330C2×C4—1012no—
331C2×C4—1012no—
332C2×C4—1012no—
333C2×C4—1012no—
334C2×C4—4044no—
335C2×C4—4044no—
336C2×C4—4044no—
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×C4—2021no—
352C2×C4—2021no—
353C2×C4—2021no—
354C2×C4—2021no—
355C2×C4—2021no—
356C2×C4—2021no—
357C2×C4C2×C4×D10112yes—
358C22—22no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22#1569221114yes—
372C22C22×C4×D10111yescharacteristic
373C22C2×#1310807311yescharacteristic
374C22C2×#1310807311yescharacteristic
375C22#1569230011yescharacteristic
376C22#1569230011yescharacteristic
377C22#1569230111yescharacteristic
378C22#1569230211yescharacteristic
379C4—22no—
380C4—41no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—4041no—
385C4—4041no—
386C4—4041no—
387C4—4041no—
388C4—2022no—
389C4—2022no—
390C4—2022no—
391C4—2022no—
392C2—24no—
393C2—42no—
394C2C2×#1569221111yescharacteristic
395C2#2026564911yescharacteristic
396C2#2026565611yescharacteristic
397C2#2026576211yescharacteristic
398C2#2026576411yescharacteristic
399C2#2026577111yescharacteristic
400C2#2026577211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#386×C101C2
socle, cosocleC23×C101C24
prime radicalC22×D4×C101C22
hypercenter#386D101
hypocenterC101#2318
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #20265071C2semidirect
3. #20265065C2semidirect
4. #20265062C2semidirect
5. #20265060C2semidirect
7. #20265050C2semidirect
8. #20265049C2semidirect
9. #20265044C2semidirect
10. #20265041C2semidirect
11. #20265038C2semidirect
12. #20265035C2semidirect
13. C22⋊C4×Dic101C2semidirect
14. C2×#15692162C2semidirect
19. D4×Dic101C4semidirect
60. C22×Dic101C2×C4semidirect
69. C4⋊Dic101C2×C4semidirect
78. C22⋊Dic101C2×C4semidirect
81. C4×Dic101C2×C4semidirect
201. C101#2318semidirect

Maximal subgroups

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

subgroupindex
C2×D4×Dic1012
#202650712
#202650652
#202650622
#202650602
#202650572
#202650502
#202650492
#202650442
#202650412
#202650382
#202650352
C22⋊C4×Dic1012
C2×#156921622
#386×C1012
#2318101