Finite Group Databasebeta

← #27181332

Subgroups — #27181332

Order 12928.

396 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265780C211yesmaximal, characteristic, complemented
2#20265660C211yesmaximal, characteristic, complemented
3#20265496C211yesmaximal, characteristic, complemented
4#20265488C211yesmaximal, characteristic, complemented
5#20265482C211yesmaximal, characteristic, complemented
6#20265479C211yesmaximal, characteristic, complemented
7#20265375C211yesmaximal, characteristic, complemented
8#20265370C211yesmaximal, characteristic, complemented
9#20265322C211yesmaximal, characteristic, complemented
10#20265289C211yesmaximal, characteristic, complemented
11#20265280C211yesmaximal, characteristic, complemented
12SD16×Dic101C211yesmaximal, characteristic, complemented
13#20265257C211yesmaximal, characteristic, complemented
14#20265235C211yesmaximal, characteristic, complemented
15#449×C101C211yesmaximal, characteristic, complemented
16C2×#13108061C2211yescharacteristic
17C2×#13108059C2211yescharacteristic
18#15692251C2211yescharacteristic
19#15692246C2211yescharacteristic
20#15692239—21no—
21#15692237—21no—
22#15692229C2211yescharacteristic
23#15692227—21no—
24#15692226—21no—
25Q8×Dic101C2211yescharacteristic
26Q8×Dic101C2211yescharacteristic
27D4×Dic101—21no—
28D4×Dic101—21no—
29D4×Dic101C2211yescharacteristic
30D4×Dic101C2211yescharacteristic
31D4×Dic101C2211yescharacteristic
32#15692217C2211yescharacteristic
33#15692214—21no—
34#15692214C2211yescharacteristic
35C2×C4⋊Dic101—21no—
36C2×C4⋊Dic101C2211yescharacteristic
37#15692188C2211yescharacteristic
38#15692188C2211yescharacteristic
39#15692187C2211yescharacteristic
40#15692187C2211yescharacteristic
41#15692184C2211yescharacteristic
42#15692184C2211yescharacteristic
43#15692179C2211yescharacteristic
44#15692177C2211yescharacteristic
45#15692177C2211yescharacteristic
46#15692176C2211yescharacteristic
47#15692176C2211yescharacteristic
48#15692176C2211yescharacteristic
49C22⋊#11717579C2211yescharacteristic
50#15692165C2211yescharacteristic
51C8×Dic101C2211yescharacteristic
52C2×#41×C101C2211yescharacteristic
53C2×SD16×C101C2211yescharacteristic
54C2×D8×C101C2211yescharacteristic
55#121×C101C2211yescharacteristic
56#103×C101C2211yescharacteristic
57#102×C101C2211yescharacteristic
58C22⋊C8×C101C2211yescharacteristic
59C2×D404C2311yescharacteristic
60C2×C22⋊D101—21no—
61C2×C22⋊D101—21no—
62C2×C22⋊D101—21no—
63C22×Dic101—21no—
64C22×Dic101—21no—
65C22×Dic101—21no—
66C22×Dic101—21no—
67C2×C4×D101—21no—
68#13108061—22no—
69#13108059—22no—
70C2×#11717579C2311yescharacteristic
71C2×#11717579C2311yescharacteristic
72C4⋊Dic101—21no—
73C4⋊Dic101—21no—
74C4⋊Dic101—21no—
75C4⋊Dic101—21no—
76C4⋊Dic101D411yescharacteristic
77C4⋊Dic101D411yescharacteristic
78C4⋊Dic101C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80C4⋊Dic101C2311yescharacteristic
81#13108053—21no—
82C22⋊Dic101—21no—
83C22⋊Dic101—21no—
84C22⋊Dic101—21no—
85C22⋊Dic101D411yescharacteristic
86C22⋊Dic101D411yescharacteristic
87#13108051—21no—
88#13108051—21no—
89C4×Dic101—21no—
90C4×Dic101—21no—
91C4×Dic101C2311yescharacteristic
92C4×Dic101C2311yescharacteristic
93#41×C101—22no—
94#41×C101—22no—
95C2×Q8×C101C2311yescharacteristic
96C2×D4×C101—21no—
97C2×D4×C101—21no—
98C2×D4×C101C2311yescharacteristic
99C2×D4×C101C2311yescharacteristic
100C22×C4×C101—21no—
101C22×C4×C101C2311yescharacteristic
102SD16×C101—22no—
103D8×C101—22no—
104C2×C8×C101C2311yescharacteristic
105C2×C8×C101C2311yescharacteristic
106C4⋊C4×C101C2311yescharacteristic
107C22⋊C4×C101—21no—
108C23×C101—21no—
109C23×C101—21no—
110C23×C101C2×D411yescharacteristic
111C22×D101—21no—
112Q8×C101—21no—
113Q8×C101#4112yes—
114D4×C101—24no—
115D4×C101—21no—
116D4×C101—21no—
117D4×C101—21no—
118D4×C101—81no—
119D4×C101#4112yes—
120C2×C4×C101—24no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—21no—
125C2×C4×C101C2×D411yescharacteristic
126C2×C4×C101C2411yescharacteristic
127C22⋊D101—81no—
128C22⋊D101—81no—
129C22⋊D101—81no—
130C2×Dic101—21no—
131C2×Dic101—21no—
132C2×Dic101—21no—
133C2×Dic101—81no—
134C2×Dic101—81no—
135C2×Dic101—81no—
136C2×Dic101—81no—
137C2×Dic101C2×D411yescharacteristic
138C2×Dic101C2×D411yescharacteristic
139C2×Dic101C2×D411yescharacteristic
140C2×Dic101C2×D411yescharacteristic
141D404—21no—
142D404—21no—
143C4×D101—81no—
144C8×C101—21no—
145C8×C101—21no—
146#11717579—21no—
147#11717579—21no—
148C22×C101—21no—
149C22×C101—22no—
150C22×C101—21no—
151C22×C101—21no—
152C22×C101—41no—
153C22×C101—41no—
154C22×C101—41no—
155C22×C101—41no—
156C22×C101—41no—
157C22×C101C22×D411yescharacteristic
158C2×D101—41no—
159C2×D101—41no—
160C2×D101—41no—
161C4×C101—21no—
162C4×C101—22no—
163C4×C101—41no—
164C4×C101C2×#4111yescharacteristic
165C4×C101C2×#4111yescharacteristic
166Dic101—21no—
167Dic101—21no—
168Dic101—21no—
169Dic101—21no—
170Dic101—41no—
171Dic101—41no—
172Dic101—41no—
173C2×C101—41no—
174C2×C101—42no—
175C2×C101—81no—
176C2×C101#52511yescharacteristic
177C2×C101#57611yescharacteristic
178C2×C101#57711yescharacteristic
179D101—81no—
180#3119—1011nomaximal, Sylow
181C101#311911yescharacteristic, complemented, Sylow
182#540—1011no—
183#520—1011no—
184#481—1011no—
185#480—1011no—
186#479—1011no—
187#478—1011no—
188#460—1011no—
189#459—1011no—
190#449—1011no—
191#449D10111yescharacteristic, complemented
192#442—1011no—
193#440—1011no—
194C4×SD16—1011no—
195C4×D8—1011no—
196#433—1011no—
197C2×#41C2×D10111yescharacteristic
198C2×SD16—1011no—
199C2×SD16C2×D10111yescharacteristic
200C2×D8—1011no—
201C2×D8C2×D10111yescharacteristic
202#127—1011no—
203#125—1011no—
204#123—2021no—
205#123—2021no—
206#121—1011no—
207#121—2021no—
208#121—2021no—
209#121C2×D10111yescharacteristic
210C4×Q8—1011no—
211C4×Q8—1011no—
212C4×D4—1011no—
213C4×D4—1011no—
214C4×D4—1011no—
215C4×D4—1011no—
216C4×D4—2021no—
217C4×D4—2021no—
218#117—1011no—
219#117—2021no—
220C2×C4⋊C4—1011no—
221C2×C4⋊C4—2021no—
222#107—1011no—
223#107—1011no—
224#106—1011no—
225#106—1011no—
226C4⋊C8—1011no—
227C4⋊C8—1011no—
228#103—1011no—
229#103C2×D10111yescharacteristic
230#102—1011no—
231#102—1011no—
232#102—1011no—
233#102—1011no—
234#102—1011no—
235#102C2×D10111yescharacteristic
236C22⋊C8—1011no—
237C22⋊C8C2×D10111yescharacteristic
238#97—1011no—
239C4×C8—1011no—
240#41—22no—
241#41—22no—
242C2×Q8C22×D10111yescharacteristic
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—1011no—
246C2×D4—2021no—
247C2×D4—2021no—
248C2×D4—2021no—
249C2×D4C22×D10111yescharacteristic
250C2×D4C22×D10111yescharacteristic
251C22×C4—21no—
252C22×C4—2021no—
253C22×C4—2021no—
254C22×C4—2021no—
255C22×C4—2021no—
256C22×C4—2021no—
257C22×C4C22×D10111yescharacteristic
258SD16—22no—
259SD16—2022no—
260D8—22no—
261D8—2022no—
262C2×C8—1011no—
263C2×C8—1011no—
264C2×C8C22×D10111yescharacteristic
265C2×C8C22×D10111yescharacteristic
266C4⋊C4—1011no—
267C4⋊C4—1011no—
268C4⋊C4—1011no—
269C4⋊C4—1011no—
270C4⋊C4—1011no—
271C4⋊C4—2021no—
272C4⋊C4—2021no—
273C4⋊C4—2021no—
274C4⋊C4—2021no—
275C4⋊C4—2021no—
276C4⋊C4C22×D10111yescharacteristic
277C22⋊C4—21no—
278C22⋊C4—1011no—
279C22⋊C4—1011no—
280C22⋊C4—2021no—
281C22⋊C4—2021no—
282C22⋊C4—2021no—
283C22⋊C4—2021no—
284C22⋊C4—2021no—
285C42—1011no—
286C42—1011no—
287C42—2021no—
288C42—2021no—
289C23—21no—
290C23—21no—
291C23—2021no—
292C23D4×D10111yescharacteristic
293Q8—21no—
294Q8#1310807312yes—
295D4—24no—
296D4—21no—
297D4—21no—
298D4—21no—
299D4—81no—
300D4—8081no—
301D4—8081no—
302D4—8081no—
303D4—2021no—
304D4—2021no—
305D4#1310807312yes—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—8081no—
312C2×C4—8081no—
313C2×C4—8081no—
314C2×C4—8081no—
315C2×C4—8081no—
316C2×C4—1011no—
317C2×C4—1011no—
318C2×C4—1011no—
319C2×C4—1011no—
320C2×C4—2021no—
321C2×C4—2021no—
322C2×C4—2021no—
323C2×C4D4×D10111yescharacteristic
324C2×C4C23×D10111yescharacteristic
325C8—21no—
326C8—21no—
327C8—2021no—
328C8—2021no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—21no—
333C22—41no—
334C22—41no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—4041no—
339C22—4041no—
340C22—4041no—
341C22C2×D4×D10111yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—4041no—
346C4—4041no—
347C4—4041no—
348C4—2021no—
349C4—2021no—
350C4—2021no—
351C4—2021no—
352C4C2×#1310807311yescharacteristic
353C4C2×#1310807311yescharacteristic
354C2—41no—
355C2—42no—
356C2—81no—
357C2—8081no—
358C2#2026568311yescharacteristic
359C2#2026598111yescharacteristic
360C2#2026598411yescharacteristic
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#449×C101C2
socleC22×C101C22×D4
hypercenter#449D101
hypocenterC101#3119
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #20265780C2semidirect
2. #20265660C2semidirect
3. #20265496C2semidirect
4. #20265488C2semidirect
5. #20265482C2semidirect
6. #20265479C2semidirect
7. #20265375C2semidirect
8. #20265370C2semidirect
9. #20265322C2semidirect
10. #20265289C2semidirect
11. #20265280C2semidirect
12. SD16×Dic101C2semidirect
13. #20265257C2semidirect
14. #20265235C2semidirect
15. #449×C101C2semidirect
181. C101#3119semidirect
191. #449D101semidirect

Maximal subgroups

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

subgroupindex
#202657802
#202656602
#202654962
#202654882
#202654822
#202654792
#202653752
#202653702
#202653222
#202652892
#202652802
SD16×Dic1012
#202652572
#202652352
#449×C1012
#3119101