Finite Group Databasebeta

← #27174406

Subgroups — #27174406

Order 12928.

528 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×Dic202C211yesmaximal, characteristic, complemented
2C2×#15692214C211yesmaximal, characteristic
3#20265119C211yesmaximal, characteristic, complemented
4#20265113C211yesmaximal, characteristic
5#20265111C211yesmaximal, characteristic, complemented
6#20265109C211yesmaximal, characteristic, complemented
7#20265100C211yesmaximal, characteristic
8#20265100C211yesmaximal, characteristic
9#20265098C211yesmaximal, characteristic, complemented
10#20265089C211yesmaximal, characteristic
11C4×#13108053C211yesmaximal, characteristic, complemented
12C4×#13108053C211yesmaximal, characteristic, complemented
13C4×C4⋊Dic101C211yesmaximal, characteristic, complemented
14C4×C22⋊Dic101C211yesmaximal, characteristic
15C4×C22⋊C4×C101C211yesmaximal, characteristic
16C22×Dic202C2211yescharacteristic
17C4×Dic202—24no—
18#15692214—24no—
19C2×C4⋊Dic101C2211yescharacteristic
20C2×C4⋊Dic101C2211yescharacteristic
21C2×C4⋊Dic101C2211yescharacteristic
22C2×C4⋊Dic101C2211yescharacteristic
23C2×C4⋊Dic101C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×#13108053C2211yescharacteristic
28C2×#13108053C2211yescharacteristic
29C2×#13108053C2211yescharacteristic
30C2×C22⋊Dic101C2211yescharacteristic
31C2×C22⋊Dic101C2211yescharacteristic
32C2×C22⋊Dic101C2211yescharacteristic
33C2×C22⋊Dic101C2211yescharacteristic
34C2×C4×Dic101C2211yescharacteristic
35C2×C4×Dic101C2211yescharacteristic
36C2×C4×Dic101C2211yescharacteristic
37C2×C4×Dic101C2211yescharacteristic
38#15692162C2211yescharacteristic
39#15692162C2211yescharacteristic
40#15692162C2211yescharacteristic
41#15692162C2211yescharacteristic
42#15692162C2211yescharacteristic
43#15692162C2211yescharacteristic
44#15692162C2211yescharacteristic
45#15692162C2211yescharacteristic
46C23×C4×C101C2211yescharacteristic
47C2×C22⋊C4×C101C2211yescharacteristic
48C2×C22⋊C4×C101C2211yescharacteristic
49C2×C42×C101C2211yescharacteristic
50C2×C42×C101C2211yescharacteristic
51#95×C101C2211yescharacteristic
52#95×C101C2211yescharacteristic
53C2×Dic202—22no—
54C2×Dic202—22no—
55C2×Dic202—22no—
56C22×Dic101C2311yescharacteristic
57C22×Dic101C2311yescharacteristic
58C22×Dic101C2311yescharacteristic
59C22×Dic101C2311yescharacteristic
60C22×Dic101C2311yescharacteristic
61C22×Dic101C2311yescharacteristic
62C22×Dic101C2311yescharacteristic
63C22×Dic101C2311yescharacteristic
64C4⋊Dic101—22no—
65C4⋊Dic101—22no—
66C4⋊Dic101—22no—
67C4⋊Dic101—22no—
68C4⋊Dic101—22no—
69#13108053—22no—
70#13108053—22no—
71#13108053—22no—
72#13108053—22no—
73#13108053—22no—
74#13108053—22no—
75C22⋊Dic101—22no—
76C22⋊Dic101—22no—
77C22⋊Dic101—22no—
78C22⋊Dic101—22no—
79C4×Dic101—22no—
80C4×Dic101—22no—
81C4×Dic101—22no—
82C4×Dic101D414yes—
83C24×C101C2311yescharacteristic
84C22×C4×C101—22no—
85C22×C4×C101—22no—
86C22×C4×C101Q814yescomplemented
87C22×C4×C101C2311yescharacteristic
88C22×C4×C101C2311yescharacteristic
89C22×C4×C101C2311yescharacteristic
90C22×C4×C101C2311yescharacteristic
91C22×C4×C101C2311yescharacteristic
92C22×C4×C101C2311yescharacteristic
93C22⋊C4×C101—22no—
94C22⋊C4×C101—22no—
95C42×C101—22no—
96C42×C101—22no—
97C23×C101—21no—
98C23×C101—21no—
99C23×C101—21no—
100C23×C101—21no—
101C23×C101—21no—
102C23×C101—21no—
103C23×C101C2×Q812yes—
104C23×C101C2411yescharacteristic
105C2×C4×C101—21no—
106C2×C4×C101—21no—
107C2×C4×C101—21no—
108C2×C4×C101—21no—
109C2×C4×C101—21no—
110C2×C4×C101—21no—
111C2×C4×C101—28no—
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×C101C2×D412yes—
119C2×C4×C101C2×Q812yes—
120C2×C4×C101C2×Q812yes—
121C2×C4×C101#4112yes—
122C2×C4×C101#4112yes—
123C2×C4×C101#4112yes—
124C2×Dic101—21no—
125C2×Dic101—21no—
126C2×Dic101—21no—
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×Dic101—21no—
143C2×Dic101—21no—
144C2×Dic101C2×D412yes—
145C2×Dic101C2×D412yes—
146C2×Dic101#4112yes—
147C2×Dic101#4112yes—
148Dic202—28no—
149C22×C101—22no—
150C22×C101—22no—
151C22×C101—22no—
152C22×C101—41no—
153C22×C101—41no—
154C22×C101—41no—
155C22×C101—41no—
156C22×C101C22×D411yescharacteristic
157C22×C101C22×Q811yescharacteristic
158C22×C101C2×#4111yescharacteristic
159C22×C101C2×#4111yescharacteristic
160C22×C101C2×#4111yescharacteristic
161C22×C101C2×#4111yescharacteristic
162C22×C101C2×#4111yescharacteristic
163C4×C101—22no—
164C4×C101—22no—
165C4×C101—22no—
166C4×C101—41no—
167C4×C101—41no—
168C4×C101C22⋊Q814yes—
169Dic101—22no—
170Dic101—22no—
171Dic101—22no—
172Dic101—22no—
173Dic101—41no—
174Dic101—41no—
175Dic101—41no—
176Dic101—41no—
177C2×C101—42no—
178C2×C101C2×C22⋊Q811yescharacteristic
179C2×C101#52511yescharacteristic
180C2×C101#52911yescharacteristic
181C2×C101#52911yescharacteristic
182C2×C101#53211yescharacteristic
183C2×C101#53311yescharacteristic
184C2×C101#53311yescharacteristic
185#2396—1011nomaximal, Sylow
186C101#239611yescharacteristic, complemented, Sylow
187C2×C4×Q8—1011no—
188C2×#117—1011no—
189#400—1011no—
190#399—1011no—
191#398—1011no—
192#398—1011no—
193#396—1011no—
194#396—1011no—
195#395—1011no—
196#393—1011no—
197C4×C4⋊C4—1011no—
198C4×C4⋊C4—1011no—
199C4×C4⋊C4—1011no—
200C4×C22⋊C4—1011no—
201C4×C22⋊C4D10111yescharacteristic
202C22×Q8—1011no—
203C23×C4C2×D10111yescharacteristic
204C4×Q8—2024no—
205#117—2024no—
206C2×C4⋊C4—1011no—
207C2×C4⋊C4—1011no—
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×C22⋊C4—1011no—
218C2×C22⋊C4—1011no—
219C2×C22⋊C4—1011no—
220C2×C22⋊C4—1011no—
221C2×C22⋊C4C2×D10111yescharacteristic
222C2×C22⋊C4C2×D10111yescharacteristic
223C2×C42—1011no—
224C2×C42—1011no—
225C2×C42—1011no—
226C2×C42—1011no—
227C2×C42C2×D10111yescharacteristic
228C2×C42C2×D10111yescharacteristic
229#95—1011no—
230#95—1011no—
231#95—1011no—
232#95—1011no—
233#95—1011no—
234#95—1011no—
235#95—1011no—
236#95—1011no—
237#95C2×D10111yescharacteristic
238#95C2×D10111yescharacteristic
239C24C22×D10111yescharacteristic
240C2×Q8—2022no—
241C2×Q8—2022no—
242C2×Q8—2022no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—1011no—
246C22×C4—1011no—
247C22×C4—1011no—
248C22×C4—1011no—
249C22×C4—1011no—
250C22×C4—1011no—
251C22×C4—1011no—
252C22×C4—1011no—
253C22×C4Dic20214yescomplemented
254C22×C4C22×D10111yescharacteristic
255C22×C4C22×D10111yescharacteristic
256C22×C4C22×D10111yescharacteristic
257C22×C4C22×D10111yescharacteristic
258C22×C4C22×D10111yescharacteristic
259C22×C4C22×D10111yescharacteristic
260C4⋊C4—2022no—
261C4⋊C4—2022no—
262C4⋊C4—2022no—
263C4⋊C4—2022no—
264C4⋊C4—2022no—
265C4⋊C4—2022no—
266C4⋊C4—2022no—
267C4⋊C4—2022no—
268C4⋊C4—2022no—
269C4⋊C4—2022no—
270C4⋊C4—2022no—
271C22⋊C4—22no—
272C22⋊C4—22no—
273C22⋊C4—2022no—
274C22⋊C4—2022no—
275C22⋊C4—2022no—
276C22⋊C4—2022no—
277C42—22no—
278C42—22no—
279C42—1014no—
280C42—2022no—
281C42—2022no—
282C42—2022no—
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23C2×Dic20212yes—
290C23C23×D10111yescharacteristic
291Q8—2028no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—28no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—1012no—
306C2×C4—1012no—
307C2×C4—1012no—
308C2×C4—1012no—
309C2×C4—2021no—
310C2×C4—2021no—
311C2×C4—2021no—
312C2×C4—2021no—
313C2×C4—2021no—
314C2×C4—2021no—
315C2×C4—2021no—
316C2×C4—2021no—
317C2×C4—2021no—
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×C4D4×D10112yes—
330C2×C4C2×Dic20212yes—
331C2×C4C2×Dic20212yes—
332C2×C4#1310807312yes—
333C2×C4#1310807412yes—
334C2×C4#1310807412yes—
335C22—22no—
336C22—22no—
337C22—22no—
338C22—41no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22C2×D4×D10111yescharacteristic
343C22C22×Dic20211yescharacteristic
344C22#41×D10111yescharacteristic
345C22#41×D10111yescharacteristic
346C22C2×#1310807311yescharacteristic
347C22C2×#1310807411yescharacteristic
348C22C2×#1310807411yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—41no—
353C4—41no—
354C4—4041no—
355C4—4041no—
356C4—4041no—
357C4—4041no—
358C4—2022no—
359C4—2022no—
360C4—2022no—
361C4—2022no—
362C4#1569223214yes—
363C2—42no—
364C2C2×#1569223211yescharacteristic
365C2#2026568611yescharacteristic
366C2#2026570711yescharacteristic
367C2#2026570911yescharacteristic
368C2#2026572411yescharacteristic
369C2#2026573011yescharacteristic
370C2#2026573411yescharacteristic
subgroupG/H
centerC22×C4C22×D101
commutator, socle, cosocleC23×C101C24
FrattiniC23C23×D101
FittingC4×C22⋊C4×C101C2
prime radicalC24×C101C23
hypercenterC4×C22⋊C4D101
hypocenterC101#2396
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C22×C4 ◁ C4×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic202C2semidirect
3. #20265119C2semidirect
5. #20265111C2semidirect
6. #20265109C2semidirect
9. #20265098C2semidirect
11. C4×#13108053C2semidirect
12. C4×#13108053C2semidirect
13. C4×C4⋊Dic101C2semidirect
86. C22×C4×C101Q8semidirect
186. C101#2396semidirect
253. C22×C4Dic202semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic2022
C2×#156922142
#202651192
#202651132
#202651112
#202651092
#202651002
#202650982
#202650892
C4×#131080532
C4×C4⋊Dic1012
C4×C22⋊Dic1012
C4×C22⋊C4×C1012
#2396101