Finite Group Databasebeta

← #27184198

Subgroups — #27184198

Order 12928.

368 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20265912C211yesmaximal, characteristic, complemented
2#20265569C211yesmaximal, characteristic, complemented
3#20265567C211yesmaximal, characteristic
4#20265559C211yesmaximal, characteristic, complemented
5#20265399C211yesmaximal, characteristic, complemented
6#20265396C211yesmaximal, characteristic, complemented
7#20265392C211yesmaximal, characteristic, complemented
8#20265389C211yesmaximal, characteristic, complemented
9#20265379C211yesmaximal, characteristic, complemented
10#20265377C211yesmaximal, characteristic, complemented
11#20265308C211yesmaximal, characteristic, complemented
12#20265287C211yesmaximal, characteristic, complemented
13#20265282C211yesmaximal, characteristic, complemented
14C4×#13108061C211yesmaximal, characteristic, complemented
15#556×C101C211yesmaximal, characteristic, complemented
16C2×#13108064C2211yescharacteristic
17C2×#13108064C2211yescharacteristic
18C2×#13108062C2211yescharacteristic
19C2×#13108061C2211yescharacteristic
20C2×#13108061C2211yescharacteristic
21C2×#13108061C2211yescharacteristic
22C2×#13108061C2211yescharacteristic
23C2×#13108059C2211yescharacteristic
24C2×#13108059C2211yescharacteristic
25#15692252C2211yescharacteristic
26#15692244—21no—
27#15692243C2211yescharacteristic
28#15692243C2211yescharacteristic
29#15692240—21no—
30#15692240—21no—
31C4×Dic202C2211yescharacteristic
32C4×D404C2211yescharacteristic
33C4×D404C2211yescharacteristic
34#15692217—21no—
35#15692186C2211yescharacteristic
36C4⋊#11717579C2211yescharacteristic
37C4⋊#11717579C2211yescharacteristic
38C4⋊#11717579C2211yescharacteristic
39#15692179C2211yescharacteristic
40#15692179C2211yescharacteristic
41#15692178C2211yescharacteristic
42#15692176C2211yescharacteristic
43#15692175C2211yescharacteristic
44#15692175C2211yescharacteristic
45#15692166C2211yescharacteristic
46#15692166C2211yescharacteristic
47C4×#11717579C2211yescharacteristic
48C4⋊Q8×C101C2211yescharacteristic
49#126×C101—21no—
50#126×C101—21no—
51#124×C101C2211yescharacteristic
52#124×C101C2211yescharacteristic
53C22⋊Q8×C101—21no—
54C4×Q8×C101C2211yescharacteristic
55C4×Q8×C101C2211yescharacteristic
56C4×Q8×C101C2211yescharacteristic
57C4×D4×C101C2211yescharacteristic
58#117×C101—21no—
59C2×Dic202C2311yescharacteristic
60C2×D404—21no—
61C2×D404—21no—
62C2×D404C2311yescharacteristic
63C2×D404C2311yescharacteristic
64C2×C4×D101—21no—
65C2×C4×D101—21no—
66#13108064—22no—
67#13108064—41no—
68#13108062—41no—
69#13108061—22no—
70#13108061—41no—
71#13108061—41no—
72#13108061—41no—
73#13108059—22no—
74#13108059—41no—
75C2×#11717579C2311yescharacteristic
76C2×#11717579C2311yescharacteristic
77C2×#11717579C2311yescharacteristic
78C2×#11717579C2311yescharacteristic
79C4⋊Dic101C2311yescharacteristic
80#13108053—21no—
81#13108051—21no—
82#13108051—21no—
83#13108051—21no—
84#13108051—21no—
85#13108051—21no—
86#13108051—21no—
87C4×Dic101—21no—
88C2×Q8×C101D411yescharacteristic
89C2×Q8×C101C2311yescharacteristic
90C2×Q8×C101C2311yescharacteristic
91C2×D4×C101C2311yescharacteristic
92C22×C4×C101—21no—
93C4⋊C4×C101—21no—
94C4⋊C4×C101—21no—
95C4⋊C4×C101—21no—
96C4⋊C4×C101—21no—
97C4⋊C4×C101D411yescharacteristic
98C4⋊C4×C101D411yescharacteristic
99C4⋊C4×C101D411yescharacteristic
100C4⋊C4×C101C2311yescharacteristic
101C22⋊C4×C101—21no—
102C22⋊C4×C101—21no—
103C22⋊C4×C101—21no—
104C22⋊C4×C101—21no—
105C22⋊C4×C101—21no—
106C42×C101—21no—
107C42×C101—21no—
108C42×C101C2311yescharacteristic
109C42×C101C2311yescharacteristic
110C42×C101C2311yescharacteristic
111C23×C101—21no—
112C22×D101—21no—
113C22×D101—21no—
114Q8×C101—21no—
115Q8×C101—21no—
116Q8×C101—21no—
117Q8×C101—21no—
118Q8×C101—41no—
119D4×C101—21no—
120D4×C101—21no—
121C2×C4×C101—21no—
122C2×C4×C101—21no—
123C2×C4×C101—21no—
124C2×C4×C101—42no—
125C2×C4×C101C2×D411yescharacteristic
126C2×C4×C101C2×D411yescharacteristic
127C2×C4×C101C2×D411yescharacteristic
128C2×C4×C101C2×D411yescharacteristic
129C2×C4×C101C2×D411yescharacteristic
130C2×C4×C101C2×D411yescharacteristic
131C2×C4×C101C2411yescharacteristic
132C2×Dic101—21no—
133C2×Dic101—21no—
134D404—21no—
135D404—21no—
136D404—21no—
137D404—21no—
138D404—42no—
139D404—81no—
140C4×D101—81no—
141C4×D101—81no—
142Dic202—21no—
143Dic202—21no—
144#11717579—21no—
145#11717579—21no—
146#11717579—21no—
147#11717579#4112yes—
148C22×C101—41no—
149C22×C101—41no—
150C22×C101—41no—
151C22×C101C22×D411yescharacteristic
152C2×D101—41no—
153C2×D101—41no—
154C2×D101—41no—
155C2×D101—41no—
156C2×D101—41no—
157C2×D101—41no—
158C4×C101—21no—
159C4×C101—21no—
160C4×C101—21no—
161C4×C101—21no—
162C4×C101—21no—
163C4×C101—21no—
164C4×C101—41no—
165C4×C101—41no—
166C4×C101—41no—
167C4×C101C2×#4111yescharacteristic
168C4×C101H(5,𝔽2)11yescharacteristic
169Dic101—41no—
170Dic101—41no—
171C2×C101—81no—
172C2×C101#55011yescharacteristic
173C2×C101#57511yescharacteristic
174C2×C101#57711yescharacteristic
175D101—81no—
176D101—81no—
177#3401—1011nomaximal, Sylow
178C101#340111yescharacteristic, complemented, Sylow
179#559—1011no—
180#556D10111yescharacteristic, complemented
181#497—1011no—
182#496—1011no—
183#495—1011no—
184#464—1011no—
185#464—1011no—
186#463—1011no—
187#463—1011no—
188#461—1011no—
189#460—1011no—
190#446—1011no—
191#442—1011no—
192#441—1011no—
193C4×SD16—1011no—
194C2×Q16—1011no—
195C2×Q16—1011no—
196C2×SD16—1011no—
197C2×SD16—1011no—
198C2×SD16—1011no—
199C2×SD16—1011no—
200C2×SD16—1011no—
201C2×D8—1011no—
202C2×D8—1011no—
203C4⋊Q8C2×D10111yescharacteristic
204#127—1011no—
205#126—21no—
206#126—21no—
207#124—1011no—
208#124—1011no—
209#124—2021no—
210#124C2×D10111yescharacteristic
211#124C2×D10111yescharacteristic
212#123—2021no—
213#123—2021no—
214C22⋊Q8—21no—
215C4×Q8—1011no—
216C4×Q8C2×D10111yescharacteristic
217C4×Q8C2×D10111yescharacteristic
218C4×Q8C2×D10111yescharacteristic
219C4×D4—1011no—
220C4×D4—1011no—
221C4×D4—2021no—
222C4×D4C2×D10111yescharacteristic
223#117—21no—
224#106—1011no—
225C4⋊C8—1011no—
226C4⋊C8—1011no—
227C4⋊C8—1011no—
228#103—1011no—
229#103—1011no—
230#103—1011no—
231#102—1011no—
232#102—1011no—
233#102—1011no—
234#97—1011no—
235#97—1011no—
236C4×C8—1011no—
237C2×Q8—1011no—
238C2×Q8C22⋊D10111yescharacteristic
239C2×Q8C22×D10111yescharacteristic
240C2×Q8C22×D10111yescharacteristic
241C2×D4—1011no—
242C2×D4—1011no—
243C2×D4—2021no—
244C2×D4—2021no—
245C2×D4C22×D10111yescharacteristic
246C22×C4—21no—
247C22×C4—2021no—
248C22×C4—2021no—
249Q16—4041no—
250Q16—2022no—
251SD16—4041no—
252SD16—4041no—
253SD16—4041no—
254SD16—4041no—
255SD16—2022no—
256D8—4041no—
257D8—2022no—
258C2×C8—1011no—
259C2×C8—1011no—
260C2×C8—1011no—
261C2×C8—1011no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—21no—
265C4⋊C4—21no—
266C4⋊C4—1011no—
267C4⋊C4—2021no—
268C4⋊C4C22⋊D10111yescharacteristic
269C4⋊C4C22⋊D10111yescharacteristic
270C4⋊C4C22⋊D10111yescharacteristic
271C4⋊C4C22×D10111yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—21no—
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—2021no—
278C22⋊C4—2021no—
279C22⋊C4—2021no—
280C22⋊C4—2021no—
281C22⋊C4—2021no—
282C22⋊C4—2021no—
283C42—21no—
284C42—21no—
285C42—2021no—
286C42C22×D10111yescharacteristic
287C42C22×D10111yescharacteristic
288C42C22×D10111yescharacteristic
289C23—21no—
290C23—2021no—
291C23—2021no—
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—41no—
297Q8—2021no—
298Q8—2021no—
299D4—21no—
300D4—21no—
301D4—8081no—
302D4—4042no—
303D4—2021no—
304D4—2021no—
305D4—2021no—
306D4—2021no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—42no—
311C2×C4—8081no—
312C2×C4—8081no—
313C2×C4—2021no—
314C2×C4—2021no—
315C2×C4C2×C22⋊D10111yescharacteristic
316C2×C4C2×C22⋊D10111yescharacteristic
317C2×C4C2×C22⋊D10111yescharacteristic
318C2×C4C2×C22⋊D10111yescharacteristic
319C2×C4C2×C22⋊D10111yescharacteristic
320C2×C4C2×C22⋊D10111yescharacteristic
321C2×C4C23×D10111yescharacteristic
322C8—1012no—
323C8—2021no—
324C8—2021no—
325C8—2021no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—4041no—
330C22—4041no—
331C22—4041no—
332C22—4041no—
333C22—4041no—
334C22—4041no—
335C22C22×C22⋊D10111yescharacteristic
336C4—21no—
337C4—21no—
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—41no—
343C4—41no—
344C4—41no—
345C4—4041no—
346C4—4041no—
347C4#41×D10111yescharacteristic
348C4#1569230111yescharacteristic
349C2—81no—
350C2—8081no—
351C2—8081no—
352C2#2026584611yescharacteristic
353C2#2026597511yescharacteristic
354C2#2026598511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#556×C101C2
socleC22×C101C22×D4
prime radical#20265567C2
hypercenter#556D101
hypocenterC101#3401
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #556

Complements

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

NH ≅ G/Ntype
1. #20265912C2semidirect
2. #20265569C2semidirect
4. #20265559C2semidirect
5. #20265399C2semidirect
6. #20265396C2semidirect
7. #20265392C2semidirect
8. #20265389C2semidirect
9. #20265379C2semidirect
10. #20265377C2semidirect
11. #20265308C2semidirect
12. #20265287C2semidirect
13. #20265282C2semidirect
14. C4×#13108061C2semidirect
15. #556×C101C2semidirect
178. C101#3401semidirect
180. #556D101semidirect

Maximal subgroups

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

subgroupindex
#202659122
#202655692
#202655672
#202655592
#202653992
#202653962
#202653922
#202653892
#202653792
#202653772
#202653082
#202652872
#202652822
C4×#131080612
#556×C1012
#3401101