Finite Group Databasebeta

← #28362210

Subgroups — #28362210

Order 13952.

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

normal subgroups only

This group has 308 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
1D4×D436C211yesmaximal, characteristic
2#20295374C211yesmaximal, characteristic, complemented
3#20295228C211yesmaximal, characteristic, complemented
4#20294769C211yesmaximal, characteristic, complemented
5D4⋊#11724784C211yesmaximal, characteristic, complemented
6#20294753C211yesmaximal, characteristic, complemented
7#474×C109C211yesmaximal, characteristic, complemented
8C22×D436—21no—
9C2×D4×D109—21no—
10C2×#13182016—21no—
11#16812696C2211yescharacteristic
12#16812674—21no—
13#16812673—21no—
14#16812669—22no—
15C4×D436C2211yescharacteristic
16C4⋊#11724784C2211yescharacteristic, complemented
17#16812623—21no—
18#16812619—22no—
19C4×#11724784C2211yescharacteristic, complemented
20C4⋊Q8×C109C2211yescharacteristic, complemented
21C4×D4×C109C2211yescharacteristic
22#107×C109—21no—
23C4⋊C8×C109C2211yescharacteristic, complemented
24#102×C109—21no—
25C23×D109—22no—
26C2×D436—22no—
27C2×D436—21no—
28C2×D436—21no—
29C2×D436—42no—
30C2×D436—42no—
31C2×D436D411yescharacteristic
32C2×D436D412yes—
33C2×C22⋊D109—22no—
34D4×D109—42no—
35D4×D109—42no—
36C2×C4×D109—21no—
37#13182016—42no—
38C2×#11724784—21no—
39C2×#11724784—21no—
40C4⋊Dic109D411yescharacteristic
41#13182006—22no—
42#13182006—21no—
43C2×Q8×C109—21no—
44C2×D4×C109D411yescharacteristic
45C22×C4×C109—21no—
46C2×C8×C109—21no—
47C4⋊C4×C109—21no—
48C4⋊C4×C109—21no—
49C4⋊C4×C109D411yescharacteristic
50C22⋊C4×C109—21no—
51C42×C109C2311yescharacteristic
52C23×C109—21no—
53C22×D109—21no—
54C22×D109—22no—
55C22×D109—44no—
56C22×D109—44no—
57C22×D109—82no—
58Q8×C109—41no—
59Q8×C109—41no—
60D4×C109—21no—
61D4×C109SD1612yescomplemented
62C2×C4×C109—21no—
63C2×C4×C109—41no—
64C2×C4×C109—41no—
65C2×C4×C109—41no—
66C2×C4×C109C2×D411yescharacteristic
67C2×C4×C109C2×D411yescharacteristic
68C2×C4×C109C2×D411yescharacteristic
69C22⋊D109—44no—
70C2×Dic109—21no—
71D436—21no—
72D436—42no—
73D436—44no—
74D436—41no—
75D436—41no—
76D436—41no—
77D436—41no—
78D436D812yes—
79C4×D109—41no—
80C4×D109—41no—
81C8×C109—41no—
82#11724784—41no—
83#11724784—41no—
84C22×C109—22no—
85C22×C109—41no—
86C22×C109—41no—
87C22×C109#12011yescharacteristic
88C2×D109—22no—
89C2×D109—41no—
90C2×D109—41no—
91C2×D109—42no—
92C2×D109—42no—
93C2×D109—42no—
94C2×D109—88no—
95C4×C109—21no—
96C4×C109—41no—
97C4×C109—81no—
98C4×C109C2×D811yescharacteristic
99C4×C109C2×SD1611yescharacteristic
100C4×C109AGL(1,ℤ/8)11yescharacteristic
101C4×C109AGL(1,ℤ/8)11yescharacteristic
102Dic109—41no—
103C2×C109—42no—
104C2×C109#44711yescharacteristic
105C2×C109#45011yescharacteristic
106C2×C109#45511yescharacteristic
107D109—42no—
108D109—82no—
109#1639—1091nomaximal, Sylow
110C109#163911yescharacteristic, complemented, Sylow
111D42—1091no—
112#486—1091no—
113#474D10911yescharacteristic, complemented
114#461—1091no—
115#331—1091no—
116D4⋊C8—1091no—
117D4⋊C8—1091no—
118C22×D4—2181no—
119C22×D4—2181no—
120C2×SD16—2181no—
121C4⋊Q8C2×D10911yescharacteristic, complemented
122#127—1091no—
123#121—2181no—
124#121—2181no—
125#120—2182no—
126C4×D4—1091no—
127C4×D4C2×D10911yescharacteristic
128#107—21no—
129C4⋊C8—1091no—
130C4⋊C8C2×D10911yescharacteristic, complemented
131#103—2181no—
132#102—21no—
133#102—2182no—
134C4×C8—1091no—
135C24—2182no—
136C2×Q8—21no—
137C2×D4—1091no—
138C2×D4—1092no—
139C2×D4—4362no—
140C2×D4—4362no—
141C2×D4—4362no—
142C2×D4—4362no—
143C2×D4—2182no—
144C2×D4—2182no—
145C2×D4—2181no—
146C2×D4—2181no—
147C2×D4C22⋊D10911yescharacteristic
148C22×C4—21no—
149C22×C4—2181no—
150SD16—4362no—
151C2×C8—21no—
152C2×C8—2181no—
153C2×C8—2181no—
154C4⋊C4—21no—
155C4⋊C4—21no—
156C4⋊C4—1091no—
157C4⋊C4C22⋊D10911yescharacteristic
158C22⋊C4—21no—
159C22⋊C4—2182no—
160C22⋊C4—2181no—
161C42C22×D10911yescharacteristic
162C23—21no—
163C23—8722no—
164C23—4364no—
165C23—4364no—
166C23—2182no—
167C23—2181no—
168Q8—41no—
169Q8—41no—
170D4—21no—
171D4—1092no—
172D4—4362no—
173D4—4364no—
174D4—4364no—
175D4—4361no—
176D4—4361no—
177D4—4361no—
178D4—4361no—
179D4—2181no—
180D4#1318201612yescomplemented
181C2×C4—21no—
182C2×C4—41no—
183C2×C4—41no—
184C2×C4—41no—
185C2×C4—4361no—
186C2×C4—4361no—
187C2×C4—2181no—
188C2×C4D4×D10911yescharacteristic
189C2×C4D4×D10911yescharacteristic
190C2×C4C2×C22⋊D10911yescharacteristic
191C8—41no—
192C8—4361no—
193C8—4361no—
194C22—22no—
195C22—41no—
196C22—41no—
197C22—8728no—
198C22—4362no—
199C22—4362no—
200C22—4362no—
201C22—4361no—
202C22—4361no—
203C22—2182no—
204C22#1681266711yescharacteristic
205C4—21no—
206C4—41no—
207C4—81no—
208C4—4361no—
209C4D8×D10911yescharacteristic
210C4C2×#1318201611yescharacteristic
211C4#1681272311yescharacteristic
212C4#1681272411yescharacteristic
213C2—42no—
214C2—8722no—
215C2—4362no—
216C2#2029515811yescharacteristic
217C2#2029517611yescharacteristic
218C2#2029520211yescharacteristic
subgroupG/H
centerC22#16812667
commutator, cosocleC42×C109C23
FrattiniC42C22×D109
Fitting#474×C109C2
socleC22×C109#120
prime radicalD4×D436C2
hypercenter#474D109
hypocenterC109#1639
LCSG ▷ C42×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #474

Complements

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

NH ≅ G/Ntype
2. #20295374C2semidirect
3. #20295228C2semidirect
4. #20294769C2semidirect
5. D4⋊#11724784C2semidirect
6. #20294753C2semidirect
7. #474×C109C2semidirect
16. C4⋊#11724784C22semidirect
19. C4×#11724784C22semidirect
20. C4⋊Q8×C109C22semidirect
23. C4⋊C8×C109C22semidirect
61. D4×C109SD16semidirect
110. C109#1639semidirect
113. #474D109semidirect
121. C4⋊Q8C2×D109semidirect
130. C4⋊C8C2×D109semidirect
180. D4#13182016semidirect

Maximal subgroups

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

subgroupindex
D4×D4362
#202953742
#202952282
#202947692
D4⋊#117247842
#202947532
#474×C1092
#1639109