Finite Group Databasebeta

← #27174038

Subgroups — #27174038

Order 12928.

704 conjugacy classes of subgroups.

The subgroup lattice has 426 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
4#20265074C211yesmaximal, characteristic, complemented
5#20265072C211yesmaximal, characteristic, complemented
6#20265052C211yesmaximal, characteristic, complemented
7#20265051C211yesmaximal, characteristic, complemented
8#20265048C211yesmaximal, characteristic, complemented
9#20265041C211yesmaximal, characteristic, complemented
10#20265035C211yesmaximal, characteristic, complemented
11#20265034C211yesmaximal, characteristic, complemented
12C4⋊C4×Dic101C211yesmaximal, characteristic, complemented
13C2×#15692162C211yesmaximal, characteristic, complemented
14#95×D101C211yesmaximal, 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×#13108053C2211yescharacteristic
28C2×#13108053C2211yescharacteristic
29C2×#13108053C2211yescharacteristic
30C2×#13108053C2211yescharacteristic
31C2×#13108053C2211yescharacteristic
32C2×C22⋊Dic101C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×C4×Dic101C2211yescharacteristic
37C2×C4×Dic101C2211yescharacteristic
38C2×C4×Dic101C2211yescharacteristic
39#15692162C2211yescharacteristic
40#15692162C2211yescharacteristic
41#15692162C2211yescharacteristic
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45#15692162C2211yescharacteristic
46#15692162C2211yescharacteristic
47C23×C4×C101C2211yescharacteristic
48C2×C4⋊C4×C101C2211yescharacteristic
49C2×C4⋊C4×C101C2211yescharacteristic
50C2×C22⋊C4×C101C2211yescharacteristic
51C2×C22⋊C4×C101C2211yescharacteristic
52#95×C101C2211yescharacteristic
53#95×C101C2211yescharacteristic
54C23×D101C2311yescharacteristic
55C2×C22⋊D101C2×C414yes—
56C2×C22⋊D101D414yes—
57C2×C22⋊D101Q814yes—
58C22×Dic101—22no—
59C22×Dic101—22no—
60C22×Dic101—41no—
61C22×Dic101—41no—
62C22×Dic101—41no—
63C22×Dic101C2×C414yescomplemented
64C22×Dic101C2311yescharacteristic
65C22×Dic101C2311yescharacteristic
66C22×Dic101C2311yescharacteristic
67C22×Dic101C2311yescharacteristic
68C22×Dic101C2311yescharacteristic
69C22×Dic101C2311yescharacteristic
70C22×Dic101C2311yescharacteristic
71C2×C4×D101—22no—
72C2×C4×D101—22no—
73C2×C4×D101—22no—
74C2×C4×D101—22no—
75C2×C4×D101—41no—
76C2×C4×D101—41no—
77C2×C4×D101—41no—
78C2×C4×D101—41no—
79C2×C4×D101C2×C414yescomplemented
80C4⋊Dic101—22no—
81C4⋊Dic101—41no—
82#13108053—22no—
83#13108053—22no—
84#13108053—41no—
85#13108053—41no—
86#13108053C2×C414yescomplemented
87C22⋊Dic101—22no—
88#13108051—22no—
89#13108051—22no—
90#13108051C2×C414yescomplemented
91C4×Dic101—22no—
92C4×Dic101—22no—
93C4×Dic101C2×C414yescomplemented
94C24×C101C2311yescharacteristic
95C22×C4×C101—22no—
96C22×C4×C101—22no—
97C22×C4×C101—41no—
98C22×C4×C101C2311yescharacteristic
99C22×C4×C101C2311yescharacteristic
100C22×C4×C101C2311yescharacteristic
101C22×C4×C101C2311yescharacteristic
102C22×C4×C101C2311yescharacteristic
103C22×C4×C101C2311yescharacteristic
104C4⋊C4×C101—22no—
105C4⋊C4×C101—41no—
106C22⋊C4×C101—22no—
107C22⋊C4×C101C2×C414yescomplemented
108C23×C101—21no—
109C23×C101—21no—
110C23×C101—21no—
111C23×C101—21no—
112C23×C101C22×C412yes—
113C23×C101C2×D412yes—
114C23×C101C2×Q812yes—
115C23×C101C2411yescharacteristic
116C22×D101—21no—
117C22×D101—21no—
118C22×D101—21no—
119C22×D101—21no—
120C22×D101C22×C412yes—
121C22×D101C2×D412yes—
122C22×D101C2×Q812yes—
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×C101—21no—
135C2×C4×C101—21no—
136C2×C4×C101—21no—
137C2×C4×C101—21no—
138C2×C4×C101—44no—
139C2×C4×C101C22×C412yes—
140C2×C4×C101C22×C412yes—
141C2×C4×C101#4112yes—
142C22⋊D101C4⋊C4116yescomplemented
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×Dic101—21no—
153C2×Dic101—21no—
154C2×Dic101—21no—
155C2×Dic101—21no—
156C2×Dic101—21no—
157C2×Dic101—21no—
158C2×Dic101—44no—
159C2×Dic101—44no—
160C2×Dic101C22×C412yes—
161C2×Dic101C22×C412yes—
162C2×Dic101C22×C412yes—
163C2×Dic101C2×D412yes—
164C2×Dic101C2×Q812yes—
165C2×Dic101#4112yes—
166C4×D101—44no—
167C4×D101—44no—
168C4×D101—44no—
169C22×C101—22no—
170C22×C101—22no—
171C22×C101—22no—
172C22×C101—22no—
173C22×C101—22no—
174C22×C101—22no—
175C22×C101C2×C4⋊C414yes—
176C22×C101C23×C411yescharacteristic
177C22×C101C22×D411yescharacteristic
178C22×C101C22×Q811yescharacteristic
179C22×C101C2×#4111yescharacteristic
180C22×C101C2×#4111yescharacteristic
181C22×C101H(5,𝔽2)11yescharacteristic
182C22×C101H(5,𝔽2)11yescharacteristic
183C2×D101—22no—
184C2×D101—22no—
185C2×D101—22no—
186C2×D101—22no—
187C2×D101—22no—
188C2×D101—22no—
189C2×D101C2×C4⋊C414yes—
190C4×C101—22no—
191C4×C101—22no—
192C4×C101—22no—
193C4×C101—41no—
194C4×C101—41no—
195C4×C101—41no—
196Dic101—22no—
197Dic101—22no—
198Dic101—22no—
199Dic101—41no—
200Dic101—41no—
201Dic101—41no—
202Dic101C2×C4⋊C414yes—
203C2×C101—24no—
204C2×C101C22×C4⋊C411yescharacteristic
205C2×C101C4×#4111yescharacteristic
206C2×C101#51811yescharacteristic
207C2×C101#54611yescharacteristic
208C2×C101#54611yescharacteristic
209C2×C101#55411yescharacteristic
210C2×C101#55411yescharacteristic
211D101—24no—
212#2349—1011nomaximal, Sylow
213C101#234911yescharacteristic, complemented, Sylow
214C2×C4×D4—1011no—
215C2×C4×D4—1011no—
216C2×C4×D4—1011no—
217#389—1011no—
218#389—1011no—
219#385—1011no—
220#385—1011no—
221#385—1011no—
222#385D10111yescharacteristic, complemented
223#382—1011no—
224#380—1011no—
225#380—1011no—
226C4×C4⋊C4—1011no—
227C2×#95—1011no—
228C2×#95—1011no—
229C22×D4—1011no—
230C23×C4—1011no—
231C23×C4—1011no—
232C23×C4—1011no—
233C23×C4—1011no—
234C23×C4—1011no—
235C23×C4C2×D10111yescharacteristic
236C4×D4—1018no—
237C4×D4—2024no—
238C4×D4—2024no—
239C2×C4⋊C4—1011no—
240C2×C4⋊C4—1011no—
241C2×C4⋊C4—1011no—
242C2×C4⋊C4—1011no—
243C2×C4⋊C4—1011no—
244C2×C4⋊C4—1011no—
245C2×C4⋊C4—1011no—
246C2×C4⋊C4C2×D10111yescharacteristic
247C2×C4⋊C4C2×D10111yescharacteristic
248C2×C22⋊C4—1011no—
249C2×C22⋊C4—1011no—
250C2×C22⋊C4—1011no—
251C2×C22⋊C4—1011no—
252C2×C22⋊C4C2×D10111yescharacteristic
253C2×C22⋊C4C2×D10111yescharacteristic
254C2×C42—1011no—
255C2×C42—1011no—
256C2×C42—1011no—
257#95—1011no—
258#95—1011no—
259#95—1011no—
260#95—1011no—
261#95—1011no—
262#95—1011no—
263#95—1011no—
264#95—1011no—
265#95C2×D10111yescharacteristic
266#95C2×D10111yescharacteristic
267C24—1011no—
268C24C22×D10111yescharacteristic
269C2×D4—1014no—
270C2×D4—1014no—
271C2×D4—1014no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—41no—
275C22×C4—1011no—
276C22×C4—1014no—
277C22×C4—1011no—
278C22×C4—1011no—
279C22×C4—1011no—
280C22×C4—1014no—
281C22×C4—1011no—
282C22×C4—1011no—
283C22×C4—1011no—
284C22×C4—4041no—
285C22×C4—4041no—
286C22×C4—4041no—
287C22×C4—4041no—
288C22×C4—4041no—
289C22×C4—4041no—
290C22×C4—4041no—
291C22×C4—2022no—
292C22×C4—2022no—
293C22×C4—2022no—
294C22×C4—2022no—
295C22×C4—2022no—
296C22×C4—2022no—
297C22×C4C22×D10111yescharacteristic
298C22×C4C22×D10111yescharacteristic
299C22×C4C22×D10111yescharacteristic
300C22×C4C22×D10111yescharacteristic
301C22×C4C22×D10111yescharacteristic
302C22×C4C22×D10111yescharacteristic
303C4⋊C4—22no—
304C4⋊C4—41no—
305C4⋊C4—1014no—
306C4⋊C4—4041no—
307C4⋊C4—4041no—
308C4⋊C4—4041no—
309C4⋊C4—2022no—
310C4⋊C4—2022no—
311C4⋊C4—2022no—
312C22⋊C4—22no—
313C22⋊C4—1014no—
314C22⋊C4—2022no—
315C22⋊C4—2022no—
316C22⋊C4—2022no—
317C22⋊C4C4×D10114yescomplemented
318C42—1014no—
319C42—2022no—
320C42—2022no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—1012no—
326C23—1012no—
327C23—1012no—
328C23—2021no—
329C23—2021no—
330C23—2021no—
331C23—2021no—
332C23C2×C4×D10112yes—
333C23D4×D10112yes—
334C23Q8×D10112yes—
335C23C23×D10111yescharacteristic
336D4—10116no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—44no—
353C2×C4—1012no—
354C2×C4—1012no—
355C2×C4—1012no—
356C2×C4—1012no—
357C2×C4—1012no—
358C2×C4—1012no—
359C2×C4—4044no—
360C2×C4—4044no—
361C2×C4—4044no—
362C2×C4—4044no—
363C2×C4—4044no—
364C2×C4—2021no—
365C2×C4—2021no—
366C2×C4—2021no—
367C2×C4—2021no—
368C2×C4—2021no—
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×C4C2×C4×D10112yes—
380C2×C4C2×C4×D10112yes—
381C2×C4#1310807312yes—
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—1014no—
389C22—2022no—
390C22—2022no—
391C22—2022no—
392C22—2022no—
393C22—2022no—
394C22—2022no—
395C22C4⋊C4×D10114yes—
396C22C22×C4×D10111yescharacteristic
397C22C2×D4×D10111yescharacteristic
398C22C2×Q8×D10111yescharacteristic
399C22#41×D10111yescharacteristic
400C22C2×#1310807311yescharacteristic
401C22#1569230011yescharacteristic
402C22#1569230111yescharacteristic
403C4—22no—
404C4—22no—
405C4—22no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—1014no—
410C4—4041no—
411C4—4041no—
412C4—4041no—
413C4—2022no—
414C4—2022no—
415C4—2022no—
416C2—24no—
417C2—2024no—
418C2C2×C4⋊C4×D10111yescharacteristic
419C2C4×#1310807311yescharacteristic
420C2#2026564911yescharacteristic
421C2#2026581411yescharacteristic
422C2#2026581911yescharacteristic
423C2#2026588111yescharacteristic
424C2#2026588511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#385×C101C2
socle, cosocleC23×C101C24
prime radicalC22×C22⋊D101C22
hypercenter#385D101
hypocenterC101#2349
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
4. #20265074C2semidirect
5. #20265072C2semidirect
6. #20265052C2semidirect
7. #20265051C2semidirect
8. #20265048C2semidirect
9. #20265041C2semidirect
10. #20265035C2semidirect
11. #20265034C2semidirect
12. C4⋊C4×Dic101C2semidirect
13. C2×#15692162C2semidirect
14. #95×D101C2semidirect
15. #385×C101C2semidirect
24. #15692215C4semidirect
63. C22×Dic101C2×C4semidirect
79. C2×C4×D101C2×C4semidirect
86. #13108053C2×C4semidirect
90. #13108051C2×C4semidirect
93. C4×Dic101C2×C4semidirect
107. C22⋊C4×C101C2×C4semidirect
142. C22⋊D101C4⋊C4semidirect
213. C101#2349semidirect
222. #385D101semidirect
317. C22⋊C4C4×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
#202650742
#202650722
#202650522
#202650512
#202650482
#202650412
#202650352
#202650342
C4⋊C4×Dic1012
C2×#156921622
#95×D1012
#385×C1012
#2349101