Finite Group Databasebeta

← #27174285

Subgroups — #27174285

Order 12928.

576 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692228C211yesmaximal, characteristic
2#20265079C211yesmaximal, characteristic, complemented
3#20265077C211yesmaximal, characteristic, complemented
4#20265069C211yesmaximal, characteristic, complemented
5#20265064C211yesmaximal, characteristic, complemented
6#20265061C211yesmaximal, characteristic, complemented
7#20265055C211yesmaximal, characteristic, complemented
8#20265054C211yesmaximal, characteristic, complemented
9#20265054C211yesmaximal, characteristic, complemented
10#20265053C211yesmaximal, characteristic, complemented
11#20265052C211yesmaximal, characteristic, complemented
12#20265048C211yesmaximal, characteristic, complemented
13#20265039C211yesmaximal, characteristic, complemented
14#20265034C211yesmaximal, characteristic, complemented
15#388×C101C211yesmaximal, characteristic, complemented
16C22×D404C2211yescharacteristic
17C22×C22⋊D101C2211yescharacteristic
18C22×C22⋊D101C2211yescharacteristic
19C23×Dic101C2211yescharacteristic
20C22×C4×D101C2211yescharacteristic
21C22×C4×D101C2211yescharacteristic
22#15692228C418yescomplemented
23C2×C4⋊Dic101C2211yescharacteristic
24C2×C4⋊Dic101C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×C22⋊Dic101C2211yescharacteristic
28C2×#13108051C2211yescharacteristic
29C2×#13108051C2211yescharacteristic
30C2×#13108051C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×C4×Dic101C2211yescharacteristic
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
52C23×D101C2311yescharacteristic
53C23×D101C2311yescharacteristic
54C2×D404—41no—
55C2×D404—41no—
56C2×D404C2×C414yescomplemented
57C2×C22⋊D101—41no—
58C2×C22⋊D101—41no—
59C2×C22⋊D101—41no—
60C2×C22⋊D101—41no—
61C2×C22⋊D101C2×C414yes—
62C2×C22⋊D101C2×C414yescomplemented
63C22×Dic101—41no—
64C22×Dic101—41no—
65C22×Dic101—41no—
66C22×Dic101C2311yescharacteristic
67C22×Dic101C2311yescharacteristic
68C22×Dic101C2311yescharacteristic
69C22×Dic101C2311yescharacteristic
70C22×Dic101C2311yescharacteristic
71C22×Dic101C2311yescharacteristic
72C2×C4×D101—41no—
73C2×C4×D101—41no—
74C2×C4×D101—41no—
75C2×C4×D101—41no—
76C2×C4×D101—41no—
77C2×C4×D101C2×C414yescomplemented
78C4⋊Dic101—41no—
79C4⋊Dic101—41no—
80#13108053—41no—
81#13108053C2×C414yescomplemented
82C22⋊Dic101—41no—
83#13108051—41no—
84#13108051—41no—
85#13108051—41no—
86#13108051—41no—
87#13108051—41no—
88#13108051—41no—
89#13108051—41no—
90#13108051C2×C414yescomplemented
91C4×Dic101—41no—
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—41no—
101C22⋊C4×C101—41no—
102C22⋊C4×C101C2×C414yescomplemented
103C42×C101—41no—
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×D101—21no—
119C22×D101—21no—
120C22×D101—21no—
121C22×D101—21no—
122C22×D101—21no—
123C22×D101—21no—
124C22×D101C22×C412yes—
125C22×D101C22×C412yes—
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×C101—21no—
135C2×C4×C101—21no—
136C2×C4×C101—21no—
137C2×C4×C101—21no—
138C2×C4×C101—21no—
139C2×C4×C101—21no—
140C2×C4×C101—21no—
141C2×C4×C101—21no—
142C2×C4×C101C22×C412yes—
143C2×C4×C101C22×C412yes—
144C22⋊D101—44no—
145C22⋊D101—44no—
146C2×Dic101—21no—
147C2×Dic101—21no—
148C2×Dic101—21no—
149C2×Dic101—21no—
150C2×Dic101—21no—
151C2×Dic101—21no—
152C2×Dic101—21no—
153C2×Dic101—21no—
154C2×Dic101—21no—
155C2×Dic101—21no—
156C2×Dic101—21no—
157C2×Dic101—21no—
158C2×Dic101—21no—
159C2×Dic101—21no—
160C2×Dic101—21no—
161C2×Dic101—21no—
162C2×Dic101—44no—
163C2×Dic101C22×C412yes—
164C2×Dic101C22×C412yes—
165D404—44no—
166C4×D101—44no—
167C4×D101—44no—
168C22×C101—22no—
169C22×C101—22no—
170C22×C101—22no—
171C22×C101—41no—
172C22×C101—41no—
173C22×C101—41no—
174C22×C101—41no—
175C22×C101C23×C411yescharacteristic
176C22×C101H(5,𝔽2)11yescharacteristic
177C22×C101H(5,𝔽2)11yescharacteristic
178C22×C101H(5,𝔽2)11yescharacteristic
179C22×C101H(5,𝔽2)11yescharacteristic
180C22×C101H(5,𝔽2)11yescharacteristic
181C22×C1012−1+411yescharacteristic
182C2×D101—22no—
183C2×D101—22no—
184C2×D101—22no—
185C2×D101—22no—
186C2×D101—22no—
187C2×D101—22no—
188C2×D101—41no—
189C2×D101—41no—
190C2×D101—41no—
191C2×D101—41no—
192C2×D101—41no—
193C2×D101—41no—
194C2×D101—41no—
195C2×D101—41no—
196C4×C101—22no—
197C4×C101—22no—
198C4×C101—41no—
199C4×C101—41no—
200C4×C101—41no—
201C4×C101—41no—
202Dic101—22no—
203Dic101—22no—
204Dic101—41no—
205Dic101—41no—
206Dic101—41no—
207Dic101—41no—
208C2×C101—42no—
209C2×C101#51811yescharacteristic
210C2×C101#51811yescharacteristic
211C2×C101#52011yescharacteristic
212C2×C101#56011yescharacteristic
213C2×C101#56011yescharacteristic
214C2×C101#56211yescharacteristic
215C2×C101#56211yescharacteristic
216D101—42no—
217D101—42no—
218#2377—1011nomaximal, Sylow
219C101#237711yescharacteristic, complemented, Sylow
220C2×#121—1011no—
221#390—1011no—
222#390—1011no—
223#388—1011no—
224#388D10111yescharacteristic, complemented
225#387—1011no—
226#387—1011no—
227#386—1011no—
228#386—1011no—
229#386—1011no—
230#386—1011no—
231#385—1011no—
232#385—1011no—
233#381—1011no—
234#380—1011no—
235C22×D4—1011no—
236C22×D4—1011no—
237C22×D4—1011no—
238C23×C4—1011no—
239C23×C4—1011no—
240C23×C4—1011no—
241#121—1018no—
242C2×C4⋊C4—1011no—
243C2×C4⋊C4—1011no—
244C2×C4⋊C4—1011no—
245C2×C4⋊C4—1011no—
246C2×C4⋊C4C2×D10111yescharacteristic
247C2×C22⋊C4—1011no—
248C2×C22⋊C4—1011no—
249C2×C22⋊C4—1011no—
250C2×C22⋊C4—1011no—
251C2×C22⋊C4—1011no—
252C2×C22⋊C4—1011no—
253C2×C22⋊C4—1011no—
254C2×C22⋊C4—1011no—
255C2×C22⋊C4—1011no—
256C2×C22⋊C4C2×D10111yescharacteristic
257C2×C22⋊C4C2×D10111yescharacteristic
258C2×C22⋊C4C2×D10111yescharacteristic
259C2×C42—1011no—
260C2×C42C2×D10111yescharacteristic
261#95—1011no—
262#95—1011no—
263#95—1011no—
264#95—1011no—
265#95—1011no—
266#95—1011no—
267#95—1011no—
268#95—1011no—
269#95C2×D10111yescharacteristic
270#95C2×D10111yescharacteristic
271C24—1011no—
272C24—1011no—
273C24C22×D10111yescharacteristic
274C2×D4—1014no—
275C2×D4—1014no—
276C2×D4—1014no—
277C2×D4—4041no—
278C2×D4—4041no—
279C2×D4—4041no—
280C2×D4—4041no—
281C2×D4—4041no—
282C2×D4—4041no—
283C22×C4—1011no—
284C22×C4—1011no—
285C22×C4—1014no—
286C22×C4—1011no—
287C22×C4—1011no—
288C22×C4—1011no—
289C22×C4—1011no—
290C22×C4—4041no—
291C22×C4—4041no—
292C22×C4—4041no—
293C22×C4—4041no—
294C22×C4—4041no—
295C22×C4—4041no—
296C22×C4—4041no—
297C22×C4—4041no—
298C22×C4C22×D10111yescharacteristic
299C22×C4C22×D10111yescharacteristic
300C22×C4C22×D10111yescharacteristic
301C22×C4C22×D10111yescharacteristic
302C22×C4C22×D10111yescharacteristic
303C22×C4C22×D10111yescharacteristic
304C4⋊C4—41no—
305C4⋊C4—1014no—
306C4⋊C4—4041no—
307C4⋊C4—4041no—
308C4⋊C4—4041no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—1014no—
312C22⋊C4—4041no—
313C22⋊C4—4041no—
314C22⋊C4—4041no—
315C22⋊C4—4041no—
316C22⋊C4—4041no—
317C22⋊C4—4041no—
318C22⋊C4—4041no—
319C22⋊C4—4041no—
320C22⋊C4C4×D10114yescomplemented
321C42—41no—
322C42—4041no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—1012no—
330C23—1012no—
331C23—2021no—
332C23—2021no—
333C23—2021no—
334C23—2021no—
335C23—2021no—
336C23—2021no—
337C23—2021no—
338C23—2021no—
339C23—2021no—
340C23—2021no—
341C23—2021no—
342C23—2021no—
343C23C2×C4×D10112yes—
344C23C23×D10111yescharacteristic
345D4—4044no—
346D4—4044no—
347D4—4044no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—1012no—
365C2×C4—1012no—
366C2×C4—4044no—
367C2×C4—4044no—
368C2×C4—4044no—
369C2×C4—2021no—
370C2×C4—2021no—
371C2×C4—2021no—
372C2×C4—2021no—
373C2×C4—2021no—
374C2×C4—2021no—
375C2×C4—2021no—
376C2×C4—2021no—
377C2×C4—2021no—
378C2×C4—2021no—
379C2×C4—2021no—
380C2×C4—2021no—
381C2×C4—2021no—
382C2×C4—2021no—
383C2×C4—2021no—
384C2×C4—2021no—
385C2×C4C2×C4×D10112yes—
386C2×C4C2×C4×D10112yes—
387C22—22no—
388C22—22no—
389C22—22no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—4041no—
395C22—4041no—
396C22—4041no—
397C22—4041no—
398C22—4041no—
399C22—4041no—
400C22—4041no—
401C22—4041no—
402C22—2022no—
403C22—2022no—
404C22—2022no—
405C22—2022no—
406C22—2022no—
407C22—2022no—
408C22C22×C4×D10111yescharacteristic
409C22#1569230011yescharacteristic
410C22#1569230011yescharacteristic
411C22#1569230111yescharacteristic
412C22#1569230111yescharacteristic
413C22#1569230111yescharacteristic
414C22#1569230211yescharacteristic
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—4041no—
422C4—4041no—
423C4—4041no—
424C4—4041no—
425C4—2022no—
426C4—2022no—
427C2—42no—
428C2—4042no—
429C2—4042no—
430C2#2026564911yescharacteristic
431C2#2026564911yescharacteristic
432C2#2026565711yescharacteristic
433C2#2026591511yescharacteristic
434C2#2026591811yescharacteristic
435C2#2026592111yescharacteristic
436C2#2026592411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#388×C101C2
socle, cosocleC23×C101C24
prime radicalC2×#15692228C2
hypercenter#388D101
hypocenterC101#2377
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. #20265079C2semidirect
3. #20265077C2semidirect
4. #20265069C2semidirect
5. #20265064C2semidirect
6. #20265061C2semidirect
7. #20265055C2semidirect
8. #20265054C2semidirect
9. #20265054C2semidirect
10. #20265053C2semidirect
11. #20265052C2semidirect
12. #20265048C2semidirect
13. #20265039C2semidirect
14. #20265034C2semidirect
15. #388×C101C2semidirect
22. #15692228C4semidirect
56. C2×D404C2×C4semidirect
62. C2×C22⋊D101C2×C4semidirect
77. C2×C4×D101C2×C4semidirect
81. #13108053C2×C4semidirect
90. #13108051C2×C4semidirect
102. C22⋊C4×C101C2×C4semidirect
219. C101#2377semidirect
224. #388D101semidirect
320. C22⋊C4C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922282
#202650792
#202650772
#202650692
#202650642
#202650612
#202650552
#202650542
#202650532
#202650522
#202650482
#202650392
#202650342
#388×C1012
#2377101