Finite Group Databasebeta

← #27174408

Subgroups — #27174408

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×#15692212C211yesmaximal, characteristic
3#20265118C211yesmaximal, characteristic, complemented
4#20265112C211yesmaximal, characteristic
5#20265111C211yesmaximal, characteristic, complemented
6#20265108C211yesmaximal, characteristic, complemented
7#20265101C211yesmaximal, characteristic
8#20265099C211yesmaximal, characteristic
9#20265097C211yesmaximal, characteristic, complemented
10#20265090C211yesmaximal, characteristic
11C4×#13108053C211yesmaximal, characteristic, complemented
12C4×#13108053C211yesmaximal, characteristic, complemented
13C4×#13108051C211yesmaximal, characteristic
14C4×#13108051C211yesmaximal, characteristic
15C4×C4⋊C4×C101C211yesmaximal, characteristic, complemented
16C22×Dic202C2211yescharacteristic
17C22×C4×D101C2211yescharacteristic
18C4×Dic202—24no—
19#15692212—24no—
20C2×C4⋊Dic101C2211yescharacteristic
21C2×#13108053C2211yescharacteristic
22C2×#13108053C2211yescharacteristic
23C2×#13108053C2211yescharacteristic
24C2×#13108053C2211yescharacteristic
25C2×#13108053C2211yescharacteristic
26C2×#13108053C2211yescharacteristic
27C2×#13108053C2211yescharacteristic
28C2×#13108053C2211yescharacteristic
29C2×#13108051C2211yescharacteristic
30C2×#13108051C2211yescharacteristic
31C2×#13108051C2211yescharacteristic
32C2×#13108051C2211yescharacteristic
33C2×#13108051C2211yescharacteristic
34C2×#13108051C2211yescharacteristic
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
46C2×C4⋊C4×C101C2211yescharacteristic
47C2×C4⋊C4×C101C2211yescharacteristic
48C2×C42×C101C2211yescharacteristic
49C2×C42×C101C2211yescharacteristic
50C2×C42×C101C2211yescharacteristic
51#95×C101C2211yescharacteristic
52#95×C101C2211yescharacteristic
53C23×D101C2311yescharacteristic
54C2×Dic202—22no—
55C2×Dic202—22no—
56C2×Dic202—22no—
57C22×Dic101C2311yescharacteristic
58C22×Dic101C2311yescharacteristic
59C22×Dic101C2311yescharacteristic
60C22×Dic101C2311yescharacteristic
61C22×Dic101C2311yescharacteristic
62C22×Dic101C2311yescharacteristic
63C22×Dic101C2311yescharacteristic
64C2×C4×D101—22no—
65C2×C4×D101—22no—
66C2×C4×D101Q814yescomplemented
67C4⋊Dic101—22no—
68#13108053—22no—
69#13108053—22no—
70#13108053—22no—
71#13108053—22no—
72#13108053—22no—
73#13108053—22no—
74#13108053—22no—
75#13108053—22no—
76#13108051—22no—
77#13108051—22no—
78#13108051—22no—
79#13108051—22no—
80#13108051—22no—
81#13108051—22no—
82C4×Dic101—22no—
83C4×Dic101—22no—
84C4×Dic101D414yes—
85C22×C4×C101C2311yescharacteristic
86C22×C4×C101C2311yescharacteristic
87C22×C4×C101C2311yescharacteristic
88C22×C4×C101C2311yescharacteristic
89C22×C4×C101C2311yescharacteristic
90C22×C4×C101C2311yescharacteristic
91C22×C4×C101C2311yescharacteristic
92C4⋊C4×C101—22no—
93C4⋊C4×C101—22no—
94C42×C101—22no—
95C42×C101—22no—
96C42×C101—22no—
97C23×C101C2411yescharacteristic
98C22×D101—21no—
99C22×D101—21no—
100C22×D101—21no—
101C22×D101—21no—
102C22×D101—21no—
103C22×D101—21no—
104C22×D101C2×Q812yes—
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—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×C101C2×D412yes—
122C2×C4×C101C2×Q812yes—
123C2×C4×C101#4112yes—
124C2×C4×C101#4112yes—
125C2×C4×C101#4112yes—
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×Dic101C2×D412yes—
143C2×Dic101C2×D412yes—
144C2×Dic101C2×Q812yes—
145C2×Dic101#4112yes—
146C2×Dic101#4112yes—
147C4×D101—28no—
148Dic202—28no—
149C22×C101C22×D411yescharacteristic
150C22×C101C22×Q811yescharacteristic
151C22×C101C2×#4111yescharacteristic
152C22×C101C2×#4111yescharacteristic
153C22×C101C2×#4111yescharacteristic
154C22×C101C2×#4111yescharacteristic
155C22×C101C2×#4111yescharacteristic
156C2×D101—22no—
157C2×D101—22no—
158C2×D101—22no—
159C2×D101—41no—
160C2×D101—41no—
161C2×D101—41no—
162C2×D101—41no—
163C4×C101—22no—
164C4×C101—22no—
165C4×C101—41no—
166C4×C101—41no—
167C4×C101—41no—
168C4×C101—41no—
169C4×C101C22⋊Q814yes—
170Dic101—22no—
171Dic101—22no—
172Dic101—22no—
173Dic101—22no—
174Dic101—22no—
175Dic101—41no—
176Dic101—41no—
177C2×C101C2×C22⋊Q811yescharacteristic
178C2×C101#52511yescharacteristic
179C2×C101#52911yescharacteristic
180C2×C101#52911yescharacteristic
181C2×C101#53211yescharacteristic
182C2×C101#53311yescharacteristic
183C2×C101#53311yescharacteristic
184D101—42no—
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⋊C4D10111yescharacteristic, complemented
200C4×C22⋊C4—1011no—
201C4×C22⋊C4—1011no—
202C22×Q8—1011no—
203C23×C4—1011no—
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⋊C4C2×D10111yescharacteristic
216C2×C4⋊C4C2×D10111yescharacteristic
217C2×C22⋊C4—1011no—
218C2×C22⋊C4—1011no—
219C2×C22⋊C4—1011no—
220C2×C22⋊C4—1011no—
221C2×C22⋊C4—1011no—
222C2×C22⋊C4—1011no—
223C2×C42—1011no—
224C2×C42—1011no—
225C2×C42—1011no—
226C2×C42C2×D10111yescharacteristic
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
239C24—1011no—
240C2×Q8—2022no—
241C2×Q8—2022no—
242C2×Q8—2022no—
243C22×C4—1011no—
244C22×C4—1011no—
245C22×C4—1011no—
246C22×C4—1011no—
247C22×C4—1011no—
248C22×C4—1014no—
249C22×C4—1011no—
250C22×C4—1011no—
251C22×C4—2022no—
252C22×C4—2022no—
253C22×C4C22×D10111yescharacteristic
254C22×C4C22×D10111yescharacteristic
255C22×C4C22×D10111yescharacteristic
256C22×C4C22×D10111yescharacteristic
257C22×C4C22×D10111yescharacteristic
258C22×C4C22×D10111yescharacteristic
259C22×C4C22×D10111yescharacteristic
260C4⋊C4—22no—
261C4⋊C4—22no—
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—2022no—
272C22⋊C4—2022no—
273C22⋊C4—2022no—
274C22⋊C4—2022no—
275C22⋊C4—2022no—
276C22⋊C4—2022no—
277C42—22no—
278C42—22no—
279C42—22no—
280C42—1014no—
281C42—2022no—
282C42—2022no—
283C23—1012no—
284C23—2021no—
285C23—2021no—
286C23—2021no—
287C23—2021no—
288C23—2021no—
289C23—2021no—
290C23C23×D10111yescharacteristic
291Q8—2028no—
292C2×C4—21no—
293C2×C4—21no—
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—1012no—
309C2×C4—1012no—
310C2×C4—1012no—
311C2×C4—1012no—
312C2×C4—1012no—
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×C4—2028no—
330C2×C4D4×D10112yes—
331C2×C4Q8×D10112yes—
332C2×C4#1310807412yes—
333C2×C4#1310807412yes—
334C2×C4#1310807412yes—
335C22—4041no—
336C22—4041no—
337C22—4041no—
338C22—4041no—
339C22—2022no—
340C22—2022no—
341C22—2022no—
342C22C2×D4×D10111yescharacteristic
343C22C2×Q8×D10111yescharacteristic
344C22#41×D10111yescharacteristic
345C22#41×D10111yescharacteristic
346C22C2×#1310807411yescharacteristic
347C22C2×#1310807411yescharacteristic
348C22C2×#1310807411yescharacteristic
349C4—22no—
350C4—22no—
351C4—41no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—4041no—
356C4—4041no—
357C4—2022no—
358C4—2022no—
359C4—2022no—
360C4—2022no—
361C4—2022no—
362C4#1569223314yes—
363C2—4042no—
364C2C2×#1569223311yescharacteristic
365C2#2026568611yescharacteristic
366C2#2026570511yescharacteristic
367C2#2026571111yescharacteristic
368C2#2026572411yescharacteristic
369C2#2026573111yescharacteristic
370C2#2026573111yescharacteristic
subgroupG/H
centerC22×C4C22×D101
commutator, socle, cosocleC23×C101C24
FrattiniC23C23×D101
FittingC4×C4⋊C4×C101C2
prime radicalC23×D101C23
hypercenterC4×C4⋊C4D101
hypocenterC101#2396
LCSG ▷ C23×C101 ▷ C101
UCSC1 ◁ C22×C4 ◁ C4×C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×C4×Dic202C2semidirect
3. #20265118C2semidirect
5. #20265111C2semidirect
6. #20265108C2semidirect
9. #20265097C2semidirect
11. C4×#13108053C2semidirect
12. C4×#13108053C2semidirect
15. C4×C4⋊C4×C101C2semidirect
66. C2×C4×D101Q8semidirect
186. C101#2396semidirect
199. C4×C4⋊C4D101semidirect

Maximal subgroups

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

subgroupindex
C2×C4×Dic2022
C2×#156922122
#202651182
#202651122
#202651112
#202651082
#202651012
#202650992
#202650972
#202650902
C4×#131080532
C4×#131080512
C4×C4⋊C4×C1012
#2396101