Finite Group Databasebeta

← #27173721

Subgroups — #27173721

Order 12928.

880 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13108074C211yesmaximal, characteristic
2C2×#15692213C211yesmaximal, characteristic, complemented
3C2×C4⋊C4×D101C211yesmaximal, characteristic, complemented
4#20265081C212yesmaximal, complemented
5#20265077C212yesmaximal, complemented
6#20265055C212yesmaximal, complemented
7#20265053C212yesmaximal, complemented
8#20265037C211yesmaximal, characteristic, complemented
9#20265034C211yesmaximal, characteristic, complemented
10#20265034C211yesmaximal, characteristic, complemented
11#380×C101C211yesmaximal, characteristic, complemented
12C2×#13108074—24no—
13C2×#13108074—24no—
14C2×#13108074C418yescomplemented
15C22×Dic202C2211yescharacteristic
16C22×D404C2211yescharacteristic
17C22×C22⋊D101C2211yescharacteristic
18C22×C22⋊D101C2211yescharacteristic
19C22×C4×D101C2211yescharacteristic
20C22×C4×D101C2212yes—
21C22×C4×D101C2211yescharacteristic
22#15692213—24no—
23C2×C4⋊Dic101C2211yescharacteristic
24C2×C4⋊Dic101C2211yescharacteristic
25C2×#13108053C2212yes—
26C4⋊C4×D101—24no—
27C2×C22⋊Dic101C2212yes—
28C2×#13108051C2212yes—
29C2×#13108051C2212yes—
30C2×#13108051C2212yes—
31C2×C4×Dic101C2212yes—
32#15692162C2212yes—
33#15692162C2212yes—
34#15692162C2212yes—
35C23×C4×C101C2211yescharacteristic
36C2×C4⋊C4×C101C2211yescharacteristic
37C2×C4⋊C4×C101C2211yescharacteristic
38C2×C22⋊C4×C101C2212yes—
39#95×C101C2212yes—
40C23×D101C2311yescharacteristic
41C23×D101C2311yescharacteristic
42#13108074—232no—
43C2×Dic202—22no—
44C2×Dic202—22no—
45C2×Dic202C2×C414yescomplemented
46C2×D404—22no—
47C2×D404—22no—
48C2×D404C2×C414yes—
49C2×C22⋊D101—22no—
50C2×C22⋊D101—22no—
51C2×C22⋊D101—22no—
52C2×C22⋊D101—22no—
53C2×C22⋊D101C2×C414yes—
54C2×C22⋊D101C2×C414yescomplemented
55C22×Dic101C2312yes—
56C22×Dic101C2312yes—
57C22×Dic101C2311yescharacteristic
58C22×Dic101C2311yescharacteristic
59C2×C4×D101—22no—
60C2×C4×D101—22no—
61C2×C4×D101—24no—
62C2×C4×D101—42no—
63C2×C4×D101—42no—
64C2×C4×D101C2×C414yes—
65C2×C4×D101C2×C414yescomplemented
66C2×C4×D101D414yes—
67C2×C4×D101D414yes—
68C4⋊Dic101—22no—
69C4⋊Dic101—22no—
70#13108053—24no—
71C22⋊Dic101—42no—
72#13108051—24no—
73#13108051—42no—
74#13108051—42no—
75C4×Dic101—24no—
76C24×C101C2311yescharacteristic
77C22×C4×C101—22no—
78C22×C4×C101—22no—
79C22×C4×C101C2×C414yescomplemented
80C22×C4×C101C2312yes—
81C22×C4×C101C2312yes—
82C22×C4×C101C2311yescharacteristic
83C22×C4×C101C2311yescharacteristic
84C4⋊C4×C101—22no—
85C4⋊C4×C101—22no—
86C22⋊C4×C101—42no—
87C23×C101—21no—
88C23×C101—21no—
89C23×C101—21no—
90C23×C101—21no—
91C23×C101—21no—
92C23×C101—21no—
93C23×C101C22×C412yes—
94C23×C101C2411yescharacteristic
95C22×D101—21no—
96C22×D101—21no—
97C22×D101—21no—
98C22×D101—21no—
99C22×D101—21no—
100C22×D101—21no—
101C22×D101—21no—
102C22×D101—21no—
103C22×D101—21no—
104C22×D101—21no—
105C22×D101C22×C412yes—
106C22×D101C22×C412yes—
107C22×D101C2×D412yes—
108C22×D101C2×D412yes—
109C2×C4×C101—21no—
110C2×C4×C101—21no—
111C2×C4×C101—28no—
112C2×C4×C101—22no—
113C2×C4×C101—22no—
114C2×C4×C101—22no—
115C2×C4×C101—22no—
116C2×C4×C101—22no—
117C2×C4×C101—22no—
118C2×C4×C101C22×C412yes—
119C2×C4×C101C22×C412yes—
120C2×C4×C101C2×D412yes—
121C2×C4×C101C2×D412yes—
122C22⋊D101—28no—
123C22⋊D101—28no—
124C2×Dic101—21no—
125C2×Dic101—21no—
126C2×Dic101—22no—
127C2×Dic101—22no—
128C2×Dic101—22no—
129C2×Dic101—22no—
130C2×Dic101—22no—
131C2×Dic101—22no—
132C2×Dic101C22×C412yes—
133C2×Dic101C22×C412yes—
134C2×Dic101C2×D412yes—
135C2×Dic101C2×D412yes—
136D404—28no—
137C4×D101—28no—
138C4×D101—48no—
139C4×D101C22⋊C4116yescomplemented
140Dic202—28no—
141C22×C101—22no—
142C22×C101—22no—
143C22×C101—22no—
144C22×C101—41no—
145C22×C101—41no—
146C22×C101—41no—
147C22×C101—41no—
148C22×C101C23×C411yescharacteristic
149C22×C101C22×D411yescharacteristic
150C22×C101C22×D411yescharacteristic
151C22×C101H(5,𝔽2)11yescharacteristic
152C22×C101H(5,𝔽2)11yescharacteristic
153C22×C1012−1+411yescharacteristic
154C22×C1012−1+411yescharacteristic
155C2×D101—22no—
156C2×D101—22no—
157C2×D101—22no—
158C2×D101—22no—
159C2×D101—22no—
160C2×D101—22no—
161C2×D101—22no—
162C2×D101—22no—
163C2×D101—22no—
164C2×D101—41no—
165C2×D101—41no—
166C2×D101—41no—
167C2×D101—41no—
168C2×D101C2×C22⋊C414yes—
169C4×C101—22no—
170C4×C101—42no—
171C4×C101—42no—
172C4×C101C2×C22⋊C414yes—
173Dic101—22no—
174Dic101—42no—
175Dic101—42no—
176Dic101C2×C22⋊C414yes—
177C2×C101—42no—
178C2×C101C22×C22⋊C411yescharacteristic
179C2×C101#52011yescharacteristic
180C2×C101#52011yescharacteristic
181C2×C101#53511yescharacteristic
182C2×C101#53611yescharacteristic
183C2×C101#53711yescharacteristic
184C2×C101#53711yescharacteristic
185D101—24no—
186D101—42no—
187#2311—1011nomaximal, Sylow
188C101#231111yescharacteristic, complemented, Sylow
189C22×#41—1011no—
190C2×#117—1011no—
191C22×C4⋊C4—1011no—
192#391—1012no—
193#390—1012no—
194#386—1012no—
195#386—1012no—
196#380—1011no—
197#380—1011no—
198#380—1011no—
199#380D10111yescharacteristic, complemented
200C2×#41—1018no—
201C2×#41—2024no—
202C2×#41—2024no—
203C22×Q8—1011no—
204C22×D4—1011no—
205C22×D4—1011no—
206C22×D4—1011no—
207C23×C4—1011no—
208C23×C4—1011no—
209C23×C4—1012no—
210C23×C4C2×D10111yescharacteristic
211#117—2024no—
212C2×C4⋊C4—1011no—
213C2×C4⋊C4—1012no—
214C2×C4⋊C4—1011no—
215C2×C4⋊C4—2024no—
216C2×C4⋊C4C2×D10111yescharacteristic
217C2×C4⋊C4C2×D10111yescharacteristic
218C2×C22⋊C4—1012no—
219C2×C22⋊C4—1012no—
220C2×C22⋊C4—1012no—
221C2×C22⋊C4—1012no—
222C2×C22⋊C4C2×D10112yes—
223C2×C42—1012no—
224#95—1012no—
225#95—1012no—
226#95—1012no—
227#95C2×D10112yes—
228C24—1011no—
229C24—1011no—
230C24C22×D10111yescharacteristic
231#41—20232no—
232C2×Q8—1014no—
233C2×Q8—2022no—
234C2×Q8—2022no—
235C2×D4—1014no—
236C2×D4—1014no—
237C2×D4—1014no—
238C2×D4—2022no—
239C2×D4—2022no—
240C2×D4—2022no—
241C2×D4—2022no—
242C2×D4—2022no—
243C2×D4—2022no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—1011no—
247C22×C4—1014no—
248C22×C4—1014no—
249C22×C4—1014no—
250C22×C4—1011no—
251C22×C4—1014no—
252C22×C4—1012no—
253C22×C4—1012no—
254C22×C4—4042no—
255C22×C4—4042no—
256C22×C4—2024no—
257C22×C4—2022no—
258C22×C4—2022no—
259C22×C4C4×D10114yescomplemented
260C22×C4C22×D10112yes—
261C22×C4C22×D10112yes—
262C22×C4C22×D10111yescharacteristic
263C22×C4C22×D10111yescharacteristic
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—2024no—
267C4⋊C4—2022no—
268C4⋊C4—2022no—
269C22⋊C4—42no—
270C22⋊C4—4042no—
271C22⋊C4—4042no—
272C22⋊C4—4042no—
273C22⋊C4—2024no—
274C42—2024no—
275C23—21no—
276C23—21no—
277C23—21no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—1012no—
282C23—1012no—
283C23—1012no—
284C23—1012no—
285C23—2021no—
286C23—2021no—
287C23—2021no—
288C23—2021no—
289C23—2021no—
290C23—2021no—
291C23—2021no—
292C23—2021no—
293C23—2021no—
294C23—2021no—
295C23C2×C4×D10112yes—
296C23C23×D10111yescharacteristic
297Q8—2028no—
298D4—2028no—
299D4—2028no—
300D4—2028no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—28no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—1012no—
311C2×C4—1012no—
312C2×C4—1012no—
313C2×C4—10116no—
314C2×C4—1012no—
315C2×C4—4048no—
316C2×C4—2022no—
317C2×C4—2022no—
318C2×C4—2022no—
319C2×C4—2022no—
320C2×C4—2022no—
321C2×C4—2022no—
322C2×C4—2021no—
323C2×C4—2021no—
324C2×C4—2028no—
325C2×C4C2×C4×D10112yes—
326C2×C4C2×C4×D10112yes—
327C2×C4D4×D10112yes—
328C2×C4D4×D10112yes—
329C22—22no—
330C22—22no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—1014no—
337C22—4041no—
338C22—4041no—
339C22—4041no—
340C22—4041no—
341C22—2022no—
342C22—2022no—
343C22—2022no—
344C22—2022no—
345C22—2022no—
346C22—2022no—
347C22—2022no—
348C22—2022no—
349C22—2022no—
350C22C22×C4×D10111yescharacteristic
351C22C2×D4×D10111yescharacteristic
352C22C2×D4×D10111yescharacteristic
353C22#1569230011yescharacteristic
354C22#1569230111yescharacteristic
355C22#1569230211yescharacteristic
356C22#1569230311yescharacteristic
357C4—22no—
358C4—42no—
359C4—42no—
360C4—1014no—
361C4—4042no—
362C4—4042no—
363C4—2022no—
364C4C22⋊C4×D10114yes—
365C2—42no—
366C2—4042no—
367C2—2024no—
368C2C2×C22⋊C4×D10111yescharacteristic
369C2#2026565711yescharacteristic
370C2#2026565811yescharacteristic
371C2#2026574311yescharacteristic
372C2#2026575011yescharacteristic
373C2#2026575611yescharacteristic
374C2#2026575711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#380×C101C2
socle, cosocleC23×C101C24
prime radicalC22×#13108074C2
hypercenter#380D101
hypocenterC101#2311
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #380

Complements

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

NH ≅ G/Ntype
2. C2×#15692213C2semidirect
3. C2×C4⋊C4×D101C2semidirect
4. #20265081C2semidirect
5. #20265077C2semidirect
6. #20265055C2semidirect
7. #20265053C2semidirect
8. #20265037C2semidirect
9. #20265034C2semidirect
10. #20265034C2semidirect
11. #380×C101C2semidirect
14. C2×#13108074C4semidirect
45. C2×Dic202C2×C4semidirect
54. C2×C22⋊D101C2×C4semidirect
65. C2×C4×D101C2×C4semidirect
79. C22×C4×C101C2×C4semidirect
139. C4×D101C22⋊C4semidirect
188. C101#2311semidirect
199. #380D101semidirect
259. C22×C4C4×D101semidirect

Maximal subgroups

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

subgroupindex
C22×#131080742
C2×#156922132
C2×C4⋊C4×D1012
#202650812
#202650772
#202650552
#202650532
#202650372
#202650342
#380×C1012
#2311101