Finite Group Databasebeta

← #35324765

Subgroups — #35324765

Order 21248.

312 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 59 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#24917842C211yesmaximal, characteristic
2#356×D83C212yesmaximal
3#24905636C211yesmaximal, characteristic, complemented
4#24905635C212yesmaximal, complemented
5C83×#1413C211yesmaximal, characteristic, complemented
6#19128257C411yescharacteristic
7C4⋊Q8×D83C2211yescharacteristic
8#19128071C411yescharacteristic
9#19128070C411yescharacteristic
10#19128069C2211yescharacteristic
11#19128050—22no—
12#19127708—22no—
13#101×D83—22no—
14#19127653C411yescharacteristic
15#19127500C2212yes—
16#19127499—22no—
17#491×C83C2211yescharacteristic
18#356×C83C2212yes—
19C2×Q8×D83D412yes—
20C2×#13050573—21no—
21M16×D83—42no—
22#14581792C2×C411yescharacteristic
23#14581791—21no—
24#14581791—41no—
25#14581790D412yes—
26#14581784C2×C411yescharacteristic
27#14581770—41no—
28#14581757—41no—
29#14581750—21no—
30C4⋊C4×D83—41no—
31C42×D83C2×C411yescharacteristic
32#14581717—22no—
33#14581716—22no—
34#14581715—22no—
35#14581711—22no—
36#14581710—42no—
37#14581703C2×C411yescharacteristic
38C8×Dic83—21no—
39C4⋊Q8×C83C2311yescharacteristic
40#125×C83C2×C411yescharacteristic
41#103×C83—22no—
42#101×C83—22no—
43#97×C83C2×C411yescharacteristic
44C2×Dic166—22no—
45C2×Dic166—41no—
46C2×Dic166C22⋊C412yes—
47Q8×D83—42no—
48Q8×D83—42no—
49C2×C4×D83—21no—
50C2×C4×D83—22no—
51C2×C4×D83C22⋊C411yescharacteristic
52#13050574—42no—
53#13050573—42no—
54#13050573—82no—
55C2×C83⋊C8—21no—
56C8×D83—82no—
57C4⋊Dic83—21no—
58C4⋊Dic83—41no—
59Dic83⋊C4—42no—
60Dic83⋊C4—41no—
61Dic83⋊C4—41no—
62(C2×D83)⋊C4—41no—
63C4×Dic83—21no—
64C4×Dic83—41no—
65C4×Dic83C22⋊C411yescharacteristic
66C2×Q8×C83C2×D412yes—
67M16×C83—42no—
68C2×C8×C83—21no—
69C4⋊C4×C83—21no—
70C4⋊C4×C83—41no—
71C4⋊C4×C83—41no—
72C42×C83C22×C411yescharacteristic
73C22×D83#9911yescharacteristic
74Q8×C83—42no—
75C2×C4×C83—21no—
76C2×C4×C83—22no—
77C2×C4×C83—41no—
78C2×C4×C83C2×C22⋊C411yescharacteristic
79C2×Dic83—21no—
80C2×Dic83—22no—
81C2×Dic83—41no—
82C2×Dic83#9911yescharacteristic
83C4×D83—22no—
84C4×D83—42no—
85C4×D83—42no—
86C4×D83—42no—
87Dic166—42no—
88Dic166—82no—
89Dic166—82no—
90C8×C83—41no—
91C8×C83—82no—
92C83⋊C8—41no—
93C83⋊C8—82no—
94C22×C83C2×#9911yescharacteristic
95C2×D83—41no—
96C2×D83#35412yes—
97C4×C83—21no—
98C4×C83—41no—
99C4×C83—42no—
100C4×C83—81no—
101Dic83—22no—
102Dic83—21no—
103Dic83—42no—
104Dic83—81no—
105#13599—831nomaximal, Sylow
106C2×C83—21no—
107C2×C83C2×#35411yescharacteristic
108D83—22no—
109#3139—831no—
110C2×#356—832no—
111#1413—831no—
112#1413—832no—
113#1413D8311yescharacteristic, complemented
114C83#1359911yescharacteristic, complemented, Sylow
115#533—831no—
116C2×C4⋊Q8—831no—
117#491—831no—
118#491—831no—
119#491—831no—
120#491C2×D8311yescharacteristic
121#487—1662no—
122#419—1662no—
123C2×#101—1662no—
124C4×M16—831no—
125#356—832no—
126#356—1662no—
127#356C2×D8312yes—
128C22×Q8—832no—
129C2×M16—3322no—
130C2×M16—1661no—
131C4⋊Q8—3321no—
132C4⋊Q8—832no—
133C4⋊Q8—831no—
134C4⋊Q8—1661no—
135C4⋊Q8C22×D8311yescharacteristic
136#125—831no—
137#125C4×D8311yescharacteristic
138C22⋊Q8—3321no—
139C4×Q8—3321no—
140#117—1661no—
141C2×C4⋊C4—3321no—
142C2×C42—831no—
143#103—22no—
144#103—1662no—
145#103—1662no—
146#103—1662no—
147#101—22no—
148#101—3322no—
149#101—1662no—
150#97—831no—
151#97C4×D8311yescharacteristic
152C4×C8—1661no—
153C2×Q8—3321no—
154C2×Q8—3322no—
155C2×Q8—3322no—
156C2×Q8—832no—
157C2×Q8—1662no—
158C2×Q8D4×D8312yes—
159C22×C4—831no—
160C22×C4—1661no—
161C22×C4—1662no—
162M16—42no—
163M16—3322no—
164M16—3322no—
165M16—6642no—
166C2×C8—21no—
167C2×C8—6642no—
168C2×C8—1661no—
169C4⋊C4—21no—
170C4⋊C4—41no—
171C4⋊C4—41no—
172C4⋊C4—3321no—
173C4⋊C4—3322no—
174C4⋊C4—3321no—
175C4⋊C4—3321no—
176C4⋊C4—1661no—
177C22⋊C4—3321no—
178C42—3321no—
179C42—831no—
180C42—1661no—
181C42C2×C4×D8311yescharacteristic
182C23—831no—
183Q8—42no—
184Q8—3322no—
185Q8—6642no—
186Q8—6642no—
187C2×C4—21no—
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—3322no—
191C2×C4—3322no—
192C2×C4—3322no—
193C2×C4—3321no—
194C2×C4—831no—
195C2×C4—1662no—
196C2×C4—1662no—
197C2×C4—1661no—
198C2×C4C22⋊C4×D8311yescharacteristic
199C8—41no—
200C8—82no—
201C8—3321no—
202C8—6642no—
203C22—3321no—
204C22—832no—
205C22#99×D8311yescharacteristic
206C4—21no—
207C4—41no—
208C4—42no—
209C4—81no—
210C4—3322no—
211C4—6641no—
212C4—1662no—
213C4—1661no—
214C2—21no—
215C2—1662no—
216C2#354×D8311yescharacteristic
subgroupG/H
centerC2#354×D83
commutatorC42×C83C22×C4
FrattiniC4⋊Q8C22×D83
FittingC83×#1413C2
socleC2×C83C2×#354
prime radicalC22×D83#99
cosocleC4⋊Q8×C83C23
hypercenter#1413D83
hypocenterC83#13599
LCSG ▷ C42×C83 ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C4⋊Q8 ◁ #1413

Complements

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

NH ≅ G/Ntype
3. #24905636C2semidirect
4. #24905635C2semidirect
5. C83×#1413C2semidirect
113. #1413D83semidirect
114. C83#13599semidirect

Maximal subgroups

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

subgroupindex
#249178422
#356×D832
#249056362
#249056352
C83×#14132
#1359983