Finite Group Databasebeta

← #27173813

Subgroups — #27173813

Order 12928.

556 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#15692233C211yesmaximal, characteristic
2C2×C4×Dic202C211yesmaximal, characteristic, complemented
3C2×#15692213C211yesmaximal, characteristic
4#20265081C211yesmaximal, characteristic, complemented
5#20265076C211yesmaximal, characteristic, complemented
6#20265074C211yesmaximal, characteristic, complemented
7#20265074C211yesmaximal, characteristic, complemented
8#20265069C211yesmaximal, characteristic
9#20265069C211yesmaximal, characteristic
10#20265068C211yesmaximal, characteristic
11#20265066C211yesmaximal, characteristic
12#20265058C211yesmaximal, characteristic, complemented
13#20265036C211yesmaximal, characteristic
14#20265028C211yesmaximal, characteristic, complemented
15#387×C101C211yesmaximal, characteristic, complemented
16C22×Dic202C2211yescharacteristic
17C22×C4×D101C2211yescharacteristic
18#15692233C418yescomplemented
19C4×Dic202—24no—
20#15692213—24no—
21C2×C4⋊Dic101C2211yescharacteristic
22C2×C4⋊Dic101C2211yescharacteristic
23C2×C4⋊Dic101C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×#13108053C2211yescharacteristic
28C2×#13108053C2211yescharacteristic
29C2×#13108053C2211yescharacteristic
30C2×#13108051C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
35C2×#13108051C2211yescharacteristic
36C2×C4×Dic101C2211yescharacteristic
37C2×C4×Dic101C2211yescharacteristic
38C2×C4×Dic101C2211yescharacteristic
39C2×C4×Dic101C2211yescharacteristic
40C2×C4×Dic101C2211yescharacteristic
41#15692162C2211yescharacteristic
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45#15692162C2211yescharacteristic
46#15692162C2211yescharacteristic
47C2×C4⋊C4×C101C2211yescharacteristic
48C2×C4⋊C4×C101C2211yescharacteristic
49C2×C42×C101C2211yescharacteristic
50#95×C101C2211yescharacteristic
51#95×C101C2211yescharacteristic
52#95×C101C2211yescharacteristic
53#95×C101C2211yescharacteristic
54C23×D101C2311yescharacteristic
55C2×Dic202—22no—
56C2×Dic202—22no—
57C2×Dic202C2×C414yes—
58C22×Dic101C2311yescharacteristic
59C22×Dic101C2311yescharacteristic
60C22×Dic101C2311yescharacteristic
61C22×Dic101C2311yescharacteristic
62C22×Dic101C2311yescharacteristic
63C22×Dic101C2311yescharacteristic
64C22×Dic101C2311yescharacteristic
65C2×C4×D101—22no—
66C2×C4×D101—22no—
67C2×C4×D101C2×C414yes—
68C4⋊Dic101—22no—
69C4⋊Dic101—22no—
70C4⋊Dic101—41no—
71#13108053—22no—
72#13108053—22no—
73#13108053—22no—
74#13108053—41no—
75#13108053C2×C414yes—
76#13108053C2×C414yes—
77#13108051—22no—
78#13108051—22no—
79#13108051—41no—
80#13108051—41no—
81#13108051C2×C414yes—
82#13108051C2×C414yes—
83C4×Dic101—22no—
84C4×Dic101—22no—
85C4×Dic101—41no—
86C4×Dic101—41no—
87C4×Dic101D414yes—
88C22×C4×C101C2311yescharacteristic
89C22×C4×C101C2311yescharacteristic
90C22×C4×C101C2311yescharacteristic
91C22×C4×C101C2311yescharacteristic
92C22×C4×C101C2311yescharacteristic
93C22×C4×C101C2311yescharacteristic
94C22×C4×C101C2311yescharacteristic
95C4⋊C4×C101—22no—
96C4⋊C4×C101C2×C414yes—
97C42×C101—22no—
98C23×C101C2411yescharacteristic
99C22×D101—21no—
100C22×D101—21no—
101C22×D101—21no—
102C22×D101—21no—
103C22×D101—21no—
104C22×D101—21no—
105C22×D101C22×C412yes—
106C2×C4×C101—21no—
107C2×C4×C101—21no—
108C2×C4×C101—21no—
109C2×C4×C101—21no—
110C2×C4×C101—21no—
111C2×C4×C101—21no—
112C2×C4×C101—21no—
113C2×C4×C101—21no—
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×C101C22×C412yes—
124C2×C4×C101C22×C412yes—
125C2×C4×C101C22×C412yes—
126C2×C4×C101C2×D412yes—
127C2×Dic101—21no—
128C2×Dic101—21no—
129C2×Dic101—21no—
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×Dic101C22×C412yes—
143C2×Dic101C22×C412yes—
144C2×Dic101C22×C412yes—
145C2×Dic101C2×D412yes—
146C2×Dic101C2×D412yes—
147C2×Dic101#4112yes—
148C4×D101—28no—
149Dic202—28no—
150C22×C101C23×C411yescharacteristic
151C22×C101C22×D411yescharacteristic
152C22×C101C2×#4111yescharacteristic
153C22×C101H(5,𝔽2)11yescharacteristic
154C22×C1012−1+411yescharacteristic
155C22×C1012−1+411yescharacteristic
156C22×C1012−1+411yescharacteristic
157C2×D101—22no—
158C2×D101—22no—
159C2×D101—22no—
160C2×D101—41no—
161C2×D101—41no—
162C2×D101—41no—
163C2×D101—41no—
164C4×C101—22no—
165C4×C101—22no—
166C4×C101—22no—
167C4×C101—22no—
168C4×C101—41no—
169C4×C101—41no—
170C4×C101—41no—
171Dic101—22no—
172Dic101—22no—
173Dic101—22no—
174Dic101—41no—
175Dic101—41no—
176Dic101—41no—
177Dic101C4×D414yes—
178C2×C101C2×C4×D411yescharacteristic
179C2×C101#51911yescharacteristic
180C2×C101#52011yescharacteristic
181C2×C101#53611yescharacteristic
182C2×C101#53711yescharacteristic
183C2×C101#54111yescharacteristic
184C2×C101#54211yescharacteristic
185D101—42no—
186#2323—1011nomaximal, Sylow
187C101#232311yescharacteristic, complemented, Sylow
188C2×C22⋊Q8—1011no—
189C2×C4×Q8—1011no—
190C2×#117—1011no—
191#391—1011no—
192#389—1011no—
193#389—1011no—
194#389—1011no—
195#388—1011no—
196#388—1011no—
197#388—1011no—
198#388—1011no—
199#387—1011no—
200#387D10111yescharacteristic, complemented
201#380—1011no—
202C4×C4⋊C4—1011no—
203C22×Q8—1011no—
204C23×C4—1011no—
205C22⋊Q8—1018no—
206C4×Q8—2024no—
207#117—2024no—
208C2×C4⋊C4—1011no—
209C2×C4⋊C4—1011no—
210C2×C4⋊C4—1011no—
211C2×C4⋊C4—1011no—
212C2×C4⋊C4—1011no—
213C2×C4⋊C4—1011no—
214C2×C4⋊C4—1011no—
215C2×C4⋊C4—1011no—
216C2×C4⋊C4—1011no—
217C2×C4⋊C4C2×D10111yescharacteristic
218C2×C4⋊C4C2×D10111yescharacteristic
219C2×C22⋊C4—1011no—
220C2×C22⋊C4—1011no—
221C2×C22⋊C4—1011no—
222C2×C22⋊C4—1011no—
223C2×C22⋊C4—1011no—
224C2×C22⋊C4—1011no—
225C2×C42—1011no—
226C2×C42—1011no—
227C2×C42—1011no—
228C2×C42—1011no—
229C2×C42—1011no—
230C2×C42C2×D10111yescharacteristic
231#95—1011no—
232#95—1011no—
233#95—1011no—
234#95—1011no—
235#95—1011no—
236#95—1011no—
237#95C2×D10111yescharacteristic
238#95C2×D10111yescharacteristic
239#95C2×D10111yescharacteristic
240#95C2×D10111yescharacteristic
241C24—1011no—
242C2×Q8—1014no—
243C2×Q8—2022no—
244C2×Q8—2022no—
245C22×C4—1011no—
246C22×C4—1011no—
247C22×C4—1011no—
248C22×C4—1011no—
249C22×C4—1014no—
250C22×C4—1011no—
251C22×C4—1011no—
252C22×C4—1011no—
253C22×C4—2022no—
254C22×C4—2022no—
255C22×C4C22×D10111yescharacteristic
256C22×C4C22×D10111yescharacteristic
257C22×C4C22×D10111yescharacteristic
258C22×C4C22×D10111yescharacteristic
259C22×C4C22×D10111yescharacteristic
260C22×C4C22×D10111yescharacteristic
261C22×C4C22×D10111yescharacteristic
262C4⋊C4—22no—
263C4⋊C4—1014no—
264C4⋊C4—1014no—
265C4⋊C4—4041no—
266C4⋊C4—4041no—
267C4⋊C4—2022no—
268C4⋊C4—2022no—
269C4⋊C4—2022no—
270C4⋊C4—2022no—
271C4⋊C4—2022no—
272C4⋊C4C4×D10114yes—
273C22⋊C4—1014no—
274C22⋊C4—1014no—
275C22⋊C4—4041no—
276C22⋊C4—4041no—
277C22⋊C4—2022no—
278C22⋊C4—2022no—
279C42—22no—
280C42—1014no—
281C42—4041no—
282C42—4041no—
283C42—2022no—
284C42—2022no—
285C23—1012no—
286C23—2021no—
287C23—2021no—
288C23—2021no—
289C23—2021no—
290C23—2021no—
291C23—2021no—
292C23C23×D10111yescharacteristic
293Q8—2028no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—1012no—
312C2×C4—1012no—
313C2×C4—1012no—
314C2×C4—1012no—
315C2×C4—1012no—
316C2×C4—1012no—
317C2×C4—2028no—
318C2×C4—2021no—
319C2×C4—2021no—
320C2×C4—2021no—
321C2×C4—2021no—
322C2×C4—2021no—
323C2×C4—2021no—
324C2×C4—2021no—
325C2×C4—2021no—
326C2×C4—2021no—
327C2×C4—2021no—
328C2×C4—2021no—
329C2×C4—2021no—
330C2×C4—2021no—
331C2×C4—2021no—
332C2×C4—2021no—
333C2×C4C2×C4×D10112yes—
334C2×C4C2×C4×D10112yes—
335C2×C4C2×C4×D10112yes—
336C2×C4D4×D10112yes—
337C22—4041no—
338C22—4041no—
339C22—4041no—
340C22—4041no—
341C22—2022no—
342C22—2022no—
343C22—2022no—
344C22C22×C4×D10111yescharacteristic
345C22C2×D4×D10111yescharacteristic
346C22#41×D10111yescharacteristic
347C22#1569230111yescharacteristic
348C22#1569230211yescharacteristic
349C22#1569230211yescharacteristic
350C22#1569230311yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—22no—
355C4—41no—
356C4—41no—
357C4—41no—
358C4—1014no—
359C4—4041no—
360C4—4041no—
361C4—4041no—
362C4—2022no—
363C4—2022no—
364C4—2022no—
365C2—4042no—
366C2C4×D4×D10111yescharacteristic
367C2#2026565311yescharacteristic
368C2#2026565711yescharacteristic
369C2#2026574911yescharacteristic
370C2#2026575711yescharacteristic
371C2#2026578611yescharacteristic
372C2#2026579311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D101
commutatorC22×C101C23×C4
Fitting#387×C101C2
socle, cosocleC23×C101C24
prime radicalC23×D101C23
hypercenter#387D101
hypocenterC101#2323
LCSG ▷ C22×C101 ▷ C101
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C2×C4×Dic202C2semidirect
4. #20265081C2semidirect
5. #20265076C2semidirect
6. #20265074C2semidirect
7. #20265074C2semidirect
12. #20265058C2semidirect
14. #20265028C2semidirect
15. #387×C101C2semidirect
18. #15692233C4semidirect
187. C101#2323semidirect
200. #387D101semidirect

Maximal subgroups

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

subgroupindex
C2×#156922332
C2×C4×Dic2022
C2×#156922132
#202650812
#202650762
#202650742
#202650692
#202650682
#202650662
#202650582
#202650362
#202650282
#387×C1012
#2323101