Finite Group Databasebeta

← #27174030

Subgroups — #27174030

Order 12928.

776 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×C4×C22⋊D101C211yesmaximal, characteristic
2C2×#15692215C211yesmaximal, characteristic
3C2×#15692215C211yesmaximal, characteristic
4C22×#13108053C211yesmaximal, characteristic, complemented
5C2×C4⋊C4×D101C211yesmaximal, characteristic, complemented
6#20265074C211yesmaximal, characteristic, complemented
7#20265072C211yesmaximal, characteristic, complemented
8#20265052C211yesmaximal, characteristic, complemented
9#20265051C211yesmaximal, characteristic, complemented
10#20265048C211yesmaximal, characteristic, complemented
11#20265046C211yesmaximal, characteristic, complemented
12#20265036C211yesmaximal, characteristic, complemented
13#20265035C211yesmaximal, characteristic, complemented
14#20265028C211yesmaximal, characteristic, complemented
15#385×C101C211yesmaximal, characteristic, complemented
16C22×C22⋊D101C2211yescharacteristic
17C23×Dic101C2211yescharacteristic
18C23×Dic101C2211yescharacteristic
19C22×C4×D101C2211yescharacteristic
20C22×C4×D101C2211yescharacteristic
21C22×C4×D101C2211yescharacteristic
22C4×C22⋊D101—24no—
23#15692215—24no—
24#15692215C418yescomplemented
25C2×C4⋊Dic101C2211yescharacteristic
26C2×C4⋊Dic101C2211yescharacteristic
27C2×#13108053—24no—
28C2×#13108053C2211yescharacteristic
29C2×#13108053C2211yescharacteristic
30C2×#13108053C2211yescharacteristic
31C2×#13108053C2211yescharacteristic
32C2×#13108053C2211yescharacteristic
33C2×#13108053C2211yescharacteristic
34C2×#13108053C2211yescharacteristic
35C2×#13108053C2211yescharacteristic
36C2×#13108053C2211yescharacteristic
37C4⋊C4×D101—24no—
38C2×C22⋊Dic101C2211yescharacteristic
39C2×#13108051C2211yescharacteristic
40C2×#13108051C2211yescharacteristic
41C2×#13108051C2211yescharacteristic
42C2×C4×Dic101C2211yescharacteristic
43C2×C4×Dic101C2211yescharacteristic
44C2×C4×Dic101C2211yescharacteristic
45#15692162C2211yescharacteristic
46#15692162C2211yescharacteristic
47#15692162C2211yescharacteristic
48#15692162C2211yescharacteristic
49C23×C4×C101C2211yescharacteristic
50C2×C4⋊C4×C101C2211yescharacteristic
51C2×C4⋊C4×C101C2211yescharacteristic
52C2×C22⋊C4×C101C2211yescharacteristic
53C2×C22⋊C4×C101C2211yescharacteristic
54#95×C101C2211yescharacteristic
55#95×C101C2211yescharacteristic
56C23×D101C2311yescharacteristic
57C2×C22⋊D101C2×C414yes—
58C2×C22⋊D101D414yes—
59C2×C22⋊D101Q814yes—
60C22×Dic101—22no—
61C22×Dic101—22no—
62C22×Dic101—22no—
63C22×Dic101—22no—
64C22×Dic101—41no—
65C22×Dic101C2×C414yescomplemented
66C22×Dic101C2311yescharacteristic
67C22×Dic101C2311yescharacteristic
68C22×Dic101C2311yescharacteristic
69C22×Dic101C2311yescharacteristic
70C22×Dic101C2311yescharacteristic
71C22×Dic101C2311yescharacteristic
72C22×Dic101C2311yescharacteristic
73C2×C4×D101—22no—
74C2×C4×D101—22no—
75C2×C4×D101—22no—
76C2×C4×D101—22no—
77C2×C4×D101—22no—
78C2×C4×D101—22no—
79C2×C4×D101—41no—
80C2×C4×D101—41no—
81C2×C4×D101C2×C414yescomplemented
82C4⋊Dic101—22no—
83C4⋊Dic101—41no—
84#13108053—22no—
85#13108053—22no—
86#13108053—22no—
87#13108053—22no—
88#13108053—22no—
89#13108053—22no—
90#13108053—22no—
91#13108053C2×C414yescomplemented
92#13108053D414yescomplemented
93C22⋊Dic101—22no—
94#13108051—22no—
95#13108051—22no—
96#13108051C2×C414yescomplemented
97C4×Dic101—22no—
98C4×Dic101—22no—
99C4×Dic101C2×C414yescomplemented
100C24×C101C2311yescharacteristic
101C22×C4×C101—22no—
102C22×C4×C101—22no—
103C22×C4×C101—41no—
104C22×C4×C101C2311yescharacteristic
105C22×C4×C101C2311yescharacteristic
106C22×C4×C101C2311yescharacteristic
107C22×C4×C101C2311yescharacteristic
108C22×C4×C101C2311yescharacteristic
109C22×C4×C101C2311yescharacteristic
110C4⋊C4×C101—22no—
111C4⋊C4×C101—41no—
112C22⋊C4×C101—22no—
113C22⋊C4×C101C2×C414yescomplemented
114C23×C101—21no—
115C23×C101—21no—
116C23×C101—21no—
117C23×C101—21no—
118C23×C101C22×C412yes—
119C23×C101C2×D412yes—
120C23×C101C2×Q812yes—
121C23×C101C2411yescharacteristic
122C22×D101—21no—
123C22×D101—21no—
124C22×D101—21no—
125C22×D101—21no—
126C22×D101C22×C412yes—
127C22×D101C2×D412yes—
128C22×D101C2×Q812yes—
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×C101—21no—
143C2×C4×C101—21no—
144C2×C4×C101—44no—
145C2×C4×C101C22×C412yes—
146C2×C4×C101C22×C412yes—
147C2×C4×C101C2×D412yes—
148C22⋊D101C4⋊C4116yescomplemented
149C2×Dic101—21no—
150C2×Dic101—21no—
151C2×Dic101—21no—
152C2×Dic101—21no—
153C2×Dic101—28no—
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—21no—
163C2×Dic101—44no—
164C2×Dic101C22×C412yes—
165C2×Dic101C22×C412yes—
166C2×Dic101C22×C412yes—
167C2×Dic101C2×D412yes—
168C2×Dic101C2×D412yes—
169C2×Dic101C2×D412yes—
170C2×Dic101C2×Q812yes—
171C2×Dic101#4112yes—
172C4×D101—28no—
173C4×D101—44no—
174C4×D101—44no—
175C22×C101—22no—
176C22×C101—22no—
177C22×C101—22no—
178C22×C101—22no—
179C22×C101—22no—
180C22×C101—22no—
181C22×C101C2×C4⋊C414yescomplemented
182C22×C101C23×C411yescharacteristic
183C22×C101C22×D411yescharacteristic
184C22×C101C22×D411yescharacteristic
185C22×C101C22×Q811yescharacteristic
186C22×C101C2×#4111yescharacteristic
187C22×C101H(5,𝔽2)11yescharacteristic
188C22×C1012−1+411yescharacteristic
189C2×D101—22no—
190C2×D101—22no—
191C2×D101—22no—
192C2×D101—22no—
193C2×D101—22no—
194C2×D101—22no—
195C2×D101C2×C4⋊C414yescomplemented
196C4×C101—22no—
197C4×C101—22no—
198C4×C101—22no—
199C4×C101—41no—
200C4×C101—41no—
201C4×C101—41no—
202Dic101—22no—
203Dic101—22no—
204Dic101—41no—
205Dic101—41no—
206Dic101—41no—
207Dic101C2×C4⋊C414yescomplemented
208Dic101C4×D414yescomplemented
209C2×C101—24no—
210C2×C101C22×C4⋊C411yescharacteristic
211C2×C101C2×C4×D411yescharacteristic
212C2×C101#52011yescharacteristic
213C2×C101D4211yescharacteristic
214C2×C101#54711yescharacteristic
215C2×C101D4×Q811yescharacteristic
216C2×C101#55411yescharacteristic
217D101—24no—
218D4×C4⋊C4—1011nomaximal, Sylow
219C101D4×C4⋊C411yescharacteristic, complemented, Sylow
220C2×C4×D4—1011no—
221C2×C4×D4—1011no—
222C2×C4×D4—1011no—
223C22×C4⋊C4—1011no—
224C22×C4⋊C4—1011no—
225#389—1011no—
226#389—1011no—
227#385—1011no—
228#385—1011no—
229#385—1011no—
230#385D10111yescharacteristic, complemented
231#384—1011no—
232#380—1011no—
233#380—1011no—
234C4×C4⋊C4—1011no—
235C22×D4—1011no—
236C23×C4—1011no—
237C23×C4—1011no—
238C23×C4—1011no—
239C23×C4—1011no—
240C23×C4—1011no—
241C23×C4C2×D10111yescharacteristic
242C4×D4—1018no—
243C4×D4—2024no—
244C4×D4—2024no—
245C2×C4⋊C4—1011no—
246C2×C4⋊C4—1011no—
247C2×C4⋊C4—1011no—
248C2×C4⋊C4—1011no—
249C2×C4⋊C4—1011no—
250C2×C4⋊C4—1011no—
251C2×C4⋊C4—1011no—
252C2×C4⋊C4—1011no—
253C2×C4⋊C4—1011no—
254C2×C4⋊C4—1011no—
255C2×C4⋊C4—1011no—
256C2×C4⋊C4—2024no—
257C2×C4⋊C4—2024no—
258C2×C4⋊C4C2×D10111yescharacteristic
259C2×C4⋊C4C2×D10111yescharacteristic
260C2×C22⋊C4—1011no—
261C2×C22⋊C4—1011no—
262C2×C22⋊C4—1011no—
263C2×C22⋊C4—1011no—
264C2×C22⋊C4C2×D10111yescharacteristic
265C2×C22⋊C4C2×D10111yescharacteristic
266C2×C42—1011no—
267C2×C42—1011no—
268C2×C42—1011no—
269#95—1011no—
270#95—1011no—
271#95—1011no—
272#95—1011no—
273#95C2×D10111yescharacteristic
274#95C2×D10111yescharacteristic
275C24—1011no—
276C24C22×D10111yescharacteristic
277C2×D4—1014no—
278C2×D4—1014no—
279C2×D4—1014no—
280C22×C4—22no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—1011no—
284C22×C4—1011no—
285C22×C4—1014no—
286C22×C4—1011no—
287C22×C4—1011no—
288C22×C4—1014no—
289C22×C4—1011no—
290C22×C4—1011no—
291C22×C4—1011no—
292C22×C4—4041no—
293C22×C4—4041no—
294C22×C4—4041no—
295C22×C4—2022no—
296C22×C4—2022no—
297C22×C4—2022no—
298C22×C4—2022no—
299C22×C4—2022no—
300C22×C4—2022no—
301C22×C4—2022no—
302C22×C4—2022no—
303C22×C4—2022no—
304C22×C4—2022no—
305C22×C4C22×D10111yescharacteristic
306C22×C4C22×D10111yescharacteristic
307C22×C4C22×D10111yescharacteristic
308C22×C4C22×D10111yescharacteristic
309C22×C4C22×D10111yescharacteristic
310C22×C4C22×D10111yescharacteristic
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—1014no—
314C4⋊C4—1014no—
315C4⋊C4—4041no—
316C4⋊C4—2022no—
317C4⋊C4—2022no—
318C4⋊C4—2022no—
319C4⋊C4—2022no—
320C4⋊C4—2022no—
321C4⋊C4—2022no—
322C4⋊C4—2022no—
323C4⋊C4—2022no—
324C22⋊C4—22no—
325C22⋊C4—1014no—
326C22⋊C4—2022no—
327C22⋊C4—2022no—
328C22⋊C4—2022no—
329C22⋊C4C4×D10114yescomplemented
330C42—1014no—
331C42—2022no—
332C42—2022no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—1012no—
338C23—1012no—
339C23—1012no—
340C23—2021no—
341C23—2021no—
342C23—2021no—
343C23—2021no—
344C23C2×C4×D10112yes—
345C23D4×D10112yes—
346C23Q8×D10112yes—
347C23C23×D10111yescharacteristic
348D4—10116no—
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—44no—
365C2×C4—1012no—
366C2×C4—1012no—
367C2×C4—1012no—
368C2×C4—1012no—
369C2×C4—1012no—
370C2×C4—1012no—
371C2×C4—1012no—
372C2×C4—1012no—
373C2×C4—4044no—
374C2×C4—4044no—
375C2×C4—4044no—
376C2×C4—2028no—
377C2×C4—2028no—
378C2×C4—2021no—
379C2×C4—2021no—
380C2×C4—2021no—
381C2×C4—2021no—
382C2×C4—2021no—
383C2×C4—2021no—
384C2×C4—2021no—
385C2×C4—2021no—
386C2×C4—2021no—
387C2×C4—2021no—
388C2×C4—2021no—
389C2×C4—2021no—
390C2×C4—2021no—
391C2×C4C2×C4×D10112yes—
392C2×C4C2×C4×D10112yes—
393C2×C4D4×D10112yes—
394C22—22no—
395C22—22no—
396C22—22no—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—1014no—
401C22—2022no—
402C22—2022no—
403C22—2022no—
404C22—2022no—
405C22—2022no—
406C22—2022no—
407C22C4⋊C4×D10114yescomplemented
408C22C22×C4×D10111yescharacteristic
409C22C2×D4×D10111yescharacteristic
410C22C2×D4×D10111yescharacteristic
411C22C2×Q8×D10111yescharacteristic
412C22#41×D10111yescharacteristic
413C22#1569230011yescharacteristic
414C22#1569230211yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—1014no—
422C4—1014no—
423C4—4041no—
424C4—4041no—
425C4—4041no—
426C4—2022no—
427C4—2022no—
428C2—24no—
429C2—2024no—
430C2C2×C4⋊C4×D10111yescharacteristic
431C2C4×D4×D10111yescharacteristic
432C2#2026565611yescharacteristic
433C2#2026580911yescharacteristic
434C2#2026582511yescharacteristic
435C2#2026584011yescharacteristic
436C2#2026588111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#385×C101C2
socle, cosocleC23×C101C24
prime radicalC22×C22⋊D101C22
hypercenter#385D101
hypocenterC101D4×C4⋊C4
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
4. C22×#13108053C2semidirect
5. C2×C4⋊C4×D101C2semidirect
6. #20265074C2semidirect
7. #20265072C2semidirect
8. #20265052C2semidirect
9. #20265051C2semidirect
10. #20265048C2semidirect
11. #20265046C2semidirect
12. #20265036C2semidirect
13. #20265035C2semidirect
14. #20265028C2semidirect
15. #385×C101C2semidirect
24. #15692215C4semidirect
65. C22×Dic101C2×C4semidirect
81. C2×C4×D101C2×C4semidirect
91. #13108053C2×C4semidirect
92. #13108053D4semidirect
96. #13108051C2×C4semidirect
99. C4×Dic101C2×C4semidirect
113. C22⋊C4×C101C2×C4semidirect
148. C22⋊D101C4⋊C4semidirect
181. C22×C101C2×C4⋊C4semidirect
195. C2×D101C2×C4⋊C4semidirect
207. Dic101C2×C4⋊C4semidirect
208. Dic101C4×D4semidirect
219. C101D4×C4⋊C4semidirect
230. #385D101semidirect
329. C22⋊C4C4×D101semidirect
407. C22C4⋊C4×D101semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D1012
C2×#156922152
C22×#131080532
C2×C4⋊C4×D1012
#202650742
#202650722
#202650522
#202650512
#202650482
#202650462
#202650362
#202650352
#202650282
#385×C1012
D4×C4⋊C4101