Finite Group Databasebeta

← #27182597

Subgroups — #27182597

Order 12928.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265798C211yesmaximal, characteristic, complemented
2#20265467C211yesmaximal, characteristic, complemented
3#20265464C211yesmaximal, characteristic, complemented
4#20265462C211yesmaximal, characteristic, complemented
5#20265459C211yesmaximal, characteristic, complemented
6#20265433C211yesmaximal, characteristic, complemented
7#20265428C211yesmaximal, characteristic
8#20265408C211yesmaximal, characteristic, complemented
9#20265404C211yesmaximal, characteristic, complemented
10#20265399C211yesmaximal, characteristic, complemented
11#20265396C211yesmaximal, characteristic, complemented
12#20265392C211yesmaximal, characteristic, complemented
13#20265389C211yesmaximal, characteristic, complemented
14#20265204C211yesmaximal, characteristic, complemented
15#542×C101C211yesmaximal, characteristic, complemented
16C2×#13108064C2211yescharacteristic
17C2×#13108062C2211yescharacteristic
18C2×#13108061C2211yescharacteristic
19C2×#13108059C2211yescharacteristic
20C2×#13108058C2211yescharacteristic
21C22×#11717579C2211yescharacteristic
22#15692255C2211yescharacteristic
23#15692250—21no—
24#15692243C2211yescharacteristic
25#15692241—21no—
26#15692237—21no—
27#15692234C2211yescharacteristic
28#15692233—21no—
29#15692229C2211yescharacteristic
30C4×Dic202C2211yescharacteristic
31C4×D404C2211yescharacteristic
32#15692186C2211yescharacteristic
33#15692186C2211yescharacteristic
34#15692186C2211yescharacteristic
35#15692186C2211yescharacteristic
36C4⋊#11717579C2211yescharacteristic
37C4⋊#11717579C2211yescharacteristic
38C4⋊#11717579C2211yescharacteristic
39C4⋊#11717579C2211yescharacteristic
40#15692179C2211yescharacteristic
41#15692179C2211yescharacteristic
42#15692178C2211yescharacteristic
43#15692178C2211yescharacteristic
44#15692176C2211yescharacteristic
45#15692176C2211yescharacteristic
46#15692175C2211yescharacteristic
47#15692175C2211yescharacteristic
48C4⋊Q8×C101C2211yescharacteristic
49#126×C101—21no—
50#124×C101—21no—
51#124×C101C2211yescharacteristic
52#123×C101—21no—
53C22⋊Q8×C101—21no—
54C22⋊Q8×C101C2211yescharacteristic
55#121×C101C2211yescharacteristic
56C4×Q8×C101C2211yescharacteristic
57C4×D4×C101C2211yescharacteristic
58#117×C101C2211yescharacteristic
59C2×Dic202—21no—
60C2×Dic202C2311yescharacteristic
61C2×D404C2311yescharacteristic
62C2×C22⋊D101—21no—
63C2×C4×D101—21no—
64#13108064—41no—
65#13108062—41no—
66#13108061—41no—
67#13108059—41no—
68#13108058—41no—
69C2×#11717579—41no—
70C2×#11717579C2311yescharacteristic
71C2×#11717579C2311yescharacteristic
72C2×#11717579C2311yescharacteristic
73C2×#11717579C2311yescharacteristic
74C4⋊Dic101—21no—
75C4⋊Dic101C2311yescharacteristic
76C4⋊Dic101C2311yescharacteristic
77#13108053—21no—
78#13108053—21no—
79C22⋊Dic101—21no—
80#13108051—21no—
81#13108051—21no—
82#13108051—21no—
83#13108051—21no—
84C4×Dic101—21no—
85C2×Q8×C101—21no—
86C2×Q8×C101C2311yescharacteristic
87C2×D4×C101—21no—
88C2×D4×C101C2311yescharacteristic
89C22×C4×C101—21no—
90C22×C4×C101C2311yescharacteristic
91C4⋊C4×C101—21no—
92C4⋊C4×C101—21no—
93C4⋊C4×C101—21no—
94C4⋊C4×C101D411yescharacteristic
95C4⋊C4×C101D411yescharacteristic
96C4⋊C4×C101C2311yescharacteristic
97C4⋊C4×C101C2311yescharacteristic
98C22⋊C4×C101—21no—
99C22⋊C4×C101—21no—
100C22⋊C4×C101—21no—
101C22⋊C4×C101—21no—
102C22⋊C4×C101—21no—
103C22⋊C4×C101D411yescharacteristic
104C22⋊C4×C101D411yescharacteristic
105C42×C101—21no—
106C42×C101C2311yescharacteristic
107C42×C101C2311yescharacteristic
108C23×C101—21no—
109C23×C101C2×D411yescharacteristic
110C22×D101—21no—
111Q8×C101—21no—
112Q8×C101—21no—
113Q8×C101—81no—
114D4×C101—21no—
115D4×C101—21no—
116D4×C101—81no—
117C2×C4×C101—21no—
118C2×C4×C101—21no—
119C2×C4×C101—21no—
120C2×C4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—81no—
123C2×C4×C101C2×D411yescharacteristic
124C2×C4×C101C2×D411yescharacteristic
125C2×C4×C101C2×D411yescharacteristic
126C2×C4×C101C2×D411yescharacteristic
127C2×C4×C101C2×D411yescharacteristic
128C2×C4×C101C2411yescharacteristic
129C22⋊D101—81no—
130C2×Dic101—21no—
131C2×Dic101—21no—
132C2×Dic101—21no—
133D404—21no—
134D404—21no—
135C4×D101—81no—
136Dic202—21no—
137Dic202—21no—
138Dic202—81no—
139#11717579—21no—
140#11717579—21no—
141#11717579—21no—
142#11717579—21no—
143C22×C101—21no—
144C22×C101—21no—
145C22×C101—21no—
146C22×C101—41no—
147C22×C101—41no—
148C22×C101—41no—
149C22×C101C22×D411yescharacteristic
150C2×D101—41no—
151C2×D101—41no—
152C2×D101—41no—
153C4×C101—21no—
154C4×C101—21no—
155C4×C101—21no—
156C4×C101—21no—
157C4×C101—21no—
158C4×C101—41no—
159C4×C101—41no—
160C4×C101—41no—
161C4×C101H(5,𝔽2)11yescharacteristic
162C4×C1012−1+411yescharacteristic
163Dic101—41no—
164Dic101—41no—
165Dic101—41no—
166C2×C101—41no—
167C2×C101—81no—
168C2×C101#53711yescharacteristic
169C2×C101#57711yescharacteristic
170C2×C101#57711yescharacteristic
171D101—81no—
172#3263—1011nomaximal, Sylow
173C101#326311yescharacteristic, complemented, Sylow
174#542—1011no—
175#542D10111yescharacteristic, complemented
176#476—1011no—
177#476—1011no—
178#475—1011no—
179#475—1011no—
180#470—1011no—
181#469—1011no—
182#465—1011no—
183#465—1011no—
184#464—1011no—
185#464—1011no—
186#463—1011no—
187#463—1011no—
188#424—1011no—
189C2×Q16—1011no—
190C2×SD16—1011no—
191C2×SD16—1011no—
192C2×D8—1011no—
193C2×M16—1011no—
194C22×C8—1011no—
195C4⋊Q8—1011no—
196C4⋊Q8C2×D10111yescharacteristic
197#126—21no—
198#126—2021no—
199#124—21no—
200#124—1011no—
201#124—2021no—
202#124C2×D10111yescharacteristic
203#123—21no—
204#123—2021no—
205C22⋊Q8—21no—
206C22⋊Q8—1011no—
207C22⋊Q8—2021no—
208C22⋊Q8C2×D10111yescharacteristic
209#121—1011no—
210#121C2×D10111yescharacteristic
211C4×Q8—1011no—
212C4×Q8C2×D10111yescharacteristic
213C4×D4—1011no—
214C4×D4C2×D10111yescharacteristic
215#117C2×D10111yescharacteristic
216#106—1011no—
217#106—1011no—
218#106—1011no—
219#106—1011no—
220C4⋊C8—1011no—
221C4⋊C8—1011no—
222C4⋊C8—1011no—
223C4⋊C8—1011no—
224#103—1011no—
225#103—1011no—
226#103—1011no—
227#103—1011no—
228#102—1011no—
229#102—1011no—
230#102—1011no—
231#102—1011no—
232C2×Q8—21no—
233C2×Q8—1011no—
234C2×Q8—2021no—
235C2×Q8C22×D10111yescharacteristic
236C2×D4—21no—
237C2×D4—1011no—
238C2×D4—2021no—
239C2×D4C22×D10111yescharacteristic
240C22×C4—21no—
241C22×C4—2021no—
242C22×C4C22×D10111yescharacteristic
243Q16—4041no—
244SD16—4041no—
245SD16—4041no—
246D8—4041no—
247M16—4041no—
248C2×C8—1011no—
249C2×C8—1011no—
250C2×C8—1011no—
251C2×C8—1011no—
252C2×C8—4041no—
253C4⋊C4—21no—
254C4⋊C4—21no—
255C4⋊C4—21no—
256C4⋊C4—1011no—
257C4⋊C4—1011no—
258C4⋊C4—2021no—
259C4⋊C4—2021no—
260C4⋊C4—2021no—
261C4⋊C4C22⋊D10111yescharacteristic
262C4⋊C4C22⋊D10111yescharacteristic
263C4⋊C4C22×D10111yescharacteristic
264C4⋊C4C22×D10111yescharacteristic
265C22⋊C4—21no—
266C22⋊C4—21no—
267C22⋊C4—21no—
268C22⋊C4—21no—
269C22⋊C4—21no—
270C22⋊C4—2021no—
271C22⋊C4—2021no—
272C22⋊C4—2021no—
273C22⋊C4—2021no—
274C22⋊C4—2021no—
275C22⋊C4C22⋊D10111yescharacteristic
276C22⋊C4C22⋊D10111yescharacteristic
277C42—21no—
278C42—2021no—
279C42C22×D10111yescharacteristic
280C42C22×D10111yescharacteristic
281C23—21no—
282C23—2021no—
283C23C2×C22⋊D10111yescharacteristic
284Q8—21no—
285Q8—21no—
286Q8—81no—
287Q8—8081no—
288Q8—2021no—
289Q8—2021no—
290D4—21no—
291D4—21no—
292D4—81no—
293D4—8081no—
294D4—2021no—
295D4—2021no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—81no—
302C2×C4—8081no—
303C2×C4—2021no—
304C2×C4—2021no—
305C2×C4—2021no—
306C2×C4C2×C22⋊D10111yescharacteristic
307C2×C4C2×C22⋊D10111yescharacteristic
308C2×C4C2×C22⋊D10111yescharacteristic
309C2×C4C2×C22⋊D10111yescharacteristic
310C2×C4C2×C22⋊D10111yescharacteristic
311C2×C4C23×D10111yescharacteristic
312C8—2021no—
313C8—2021no—
314C8—2021no—
315C8—2021no—
316C22—21no—
317C22—21no—
318C22—21no—
319C22—41no—
320C22—41no—
321C22—41no—
322C22—4041no—
323C22—4041no—
324C22—4041no—
325C22C22×C22⋊D10111yescharacteristic
326C4—21no—
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—41no—
332C4—41no—
333C4—41no—
334C4—4041no—
335C4—4041no—
336C4—4041no—
337C4#1569230111yescharacteristic
338C4#1569230211yescharacteristic
339C2—41no—
340C2—81no—
341C2—8081no—
342C2#2026575511yescharacteristic
343C2#2026598511yescharacteristic
344C2#2026598811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#542×C101C2
socleC22×C101C22×D4
prime radical#20265428C2
hypercenter#542D101
hypocenterC101#3263
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #542

Complements

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

NH ≅ G/Ntype
1. #20265798C2semidirect
2. #20265467C2semidirect
3. #20265464C2semidirect
4. #20265462C2semidirect
5. #20265459C2semidirect
6. #20265433C2semidirect
8. #20265408C2semidirect
9. #20265404C2semidirect
10. #20265399C2semidirect
11. #20265396C2semidirect
12. #20265392C2semidirect
13. #20265389C2semidirect
14. #20265204C2semidirect
15. #542×C101C2semidirect
173. C101#3263semidirect
175. #542D101semidirect

Maximal subgroups

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

subgroupindex
#202657982
#202654672
#202654642
#202654622
#202654592
#202654332
#202654282
#202654082
#202654042
#202653992
#202653962
#202653922
#202653892
#202652042
#542×C1012
#3263101