Finite Group Databasebeta

← #13146551

Subgroups — #13146551

Order 1728.

631 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 631 conjugacy classes of subgroups — too many to draw. Only its 65 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
1C2×#88655C211yesmaximal, characteristic
2C2×#88626C211yesmaximal, characteristic, complemented
3#11721152C211yesmaximal, characteristic, complemented
4#11720705C214yesmaximal, complemented
5C22×((C3⋊D3)⋊C3)⋊C2—21no—
6C2×C4×#272C2211yescharacteristic
7C2×C4⋊((C3⋊D3)⋊C3)—21no—
8C2×#6199—21no—
9C2×#6198C2212yes—
10#88655—22no—
11#88655C414yescomplemented
12C2×#6188C2212yescomplemented
13#88626—22no—
14C4⋊((C3⋊Dic3)⋊C3)—21no—
15#88449C2212yescomplemented
16C2×D3×D12—61no—
17D3×C4⋊Dic3—61no—
18C22×#272D411yescharacteristic
19C22×(C3⋊D3)⋊C3—22no—
20C2×((C3⋊D3)⋊C3)⋊C2—24no—
21C2×((C3⋊D3)⋊C3)⋊C2—42no—
22C2×((C3⋊D3)⋊C3)⋊C2—42no—
23C2×C4×H(3,𝔽3)D411yescharacteristic
24C2×#1084C2311yescharacteristic
25C4×#272C2×C412yes—
26C4×#272D412yes—
27C2×(C3⋊Dic3)⋊C3—22no—
28C4⋊((C3⋊D3)⋊C3)—24no—
29#6199—42no—
30#6198—22no—
31#6198C2×C414yescomplemented
32#6188—22no—
33C2×#821—91nomaximal
34C22×D32—62no—
35C2×C4⋊(C3⋊D3)—21no—
36C2×C3×D12—61no—
37C2×C3×C4×D3—61no—
38C2×C3×C4×D3—61no—
39C2×#626—62no—
40C2×D3×Dic3—62no—
41D3×D12—64no—
42D3×D12—64no—
43C4⋊(C3⋊Dic3)—21no—
44C3×C4⋊Dic3—61no—
45Dic3⋊Dic3—62no—
46C22×H(3,𝔽3)C2×D411yescharacteristic
47C2×#272C22⋊C412yes—
48C2×#272D812yes—
49C2×#272SD1612yes—
50C2×(C3⋊D3)⋊C3—24no—
51C2×(C3⋊D3)⋊C3—42no—
52C2×(C3⋊D3)⋊C3—42no—
53((C3⋊D3)⋊C3)⋊C2—48no—
54C4×H(3,𝔽3)C22⋊C412yes—
55#1084C22×C411yescharacteristic
56#1084C2×D411yescharacteristic
57(C3⋊Dic3)⋊C3—42no—
58C2×D3×D4—181no—
59C22×D12—91no—
60C22×C3×C8—91no—
61C2×C4⋊Dic3—91no—
62D3×C4⋊C4—181no—
63#821—94no—
64C22×C3⋊D3—22no—
65C22×C3×D3—61no—
66C22×C3×D3—61no—
67C22×C3×D3—62no—
68C2×D32—68no—
69C2×D32—124no—
70C2×D32—124no—
71C2×C32×C4—21no—
72C2×C32×C4—21no—
73C2×C3⋊Dic3—22no—
74C4⋊(C3⋊D3)—24no—
75C2×C3×Dic3—61no—
76C2×C3×Dic3—61no—
77C2×C3×Dic3—62no—
78C3×D12—64no—
79C3×C4×D3—62no—
80C3×C4×D3—62no—
81C3×C4×D3—62no—
82C3×C4×D3—62no—
83#626—64no—
84#626—64no—
85D3×Dic3—124no—
86C2×#102—271noSylow
87C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
88C2×H(3,𝔽3)C2×D811yescharacteristic
89C2×H(3,𝔽3)C2×SD1611yescharacteristic
90#272#10214yescomplemented
91(C3⋊D3)⋊C3—44no—
92C23×D3—181no—
93C23×D3—182no—
94C2×C3×D4—181no—
95C22×C3×C4—91no—
96C2×#67—182no—
97C22×Dic3—181no—
98D3×D4—184no—
99D3×D4—184no—
100C2×D12—61no—
101C2×D12—94no—
102C2×D12—92no—
103C2×D12—181no—
104C2×D12—182no—
105C2×C4×D3—61no—
106C2×C4×D3—61no—
107C2×C4×D3—182no—
108C2×C3×C8—92no—
109C2×C3×C8—182no—
110C3×C4⋊C4—181no—
111C4⋊Dic3—61no—
112C4⋊Dic3—92no—
113C4⋊Dic3—181no—
114Dic3⋊C4—182no—
115C22×C32—21no—
116C22×C32—21no—
117C2×C3⋊D3—24no—
118C2×C3⋊D3—42no—
119C2×C3⋊D3—42no—
120C2×C3×D3—62no—
121C2×C3×D3—62no—
122C2×C3×D3—62no—
123C2×C3×D3—62no—
124C2×C3×D3—62no—
125C2×C3×D3—62no—
126C2×C3×D3—64no—
127C2×C3×D3—122no—
128C2×C3×D3—122no—
129D32—1216no—
130C32×C4—22no—
131C32×C4—22no—
132C3⋊Dic3—42no—
133C3×Dic3—61no—
134C3×Dic3—61no—
135C3×Dic3—61no—
136C3×Dic3—61no—
137C3×Dic3—122no—
138C22×D4—271no—
139C22×C8—91no—
140C2×C4⋊C4—271no—
141#102—274no—
142H(3,𝔽3)C2×#10211yescharacteristic, complemented, Sylow
143C23×C3—91no—
144C23×C3—182no—
145C22×D3—61no—
146C22×D3—61no—
147C22×D3—62no—
148C22×D3—182no—
149C22×D3—184no—
150C22×D3—188no—
151C22×D3—364no—
152C22×D3—364no—
153C22×D3—362no—
154C22×D3—362no—
155C3×D4—184no—
156C2×C3×C4—61no—
157C2×C3×C4—61no—
158C2×C3×C4—91no—
159C2×C3×C4—92no—
160C2×C3×C4—92no—
161C2×C3×C4—182no—
162C2×C3×C4GO+(4,𝔽2)11yescharacteristic
163#67—184no—
164#67—184no—
165C2×Dic3—61no—
166C2×Dic3—61no—
167C2×Dic3—62no—
168C2×Dic3—182no—
169C2×Dic3—362no—
170D12—64no—
171D12—94no—
172D12—184no—
173D12—182no—
174C4×D3—62no—
175C4×D3—62no—
176C4×D3—62no—
177C4×D3—62no—
178C4×D3—364no—
179C3×C8—182no—
180C2×C32—21no—
181C2×C32—21no—
182C2×C32—21no—
183C2×C32—21no—
184C2×C32—21no—
185C2×C32—21no—
186C3⋊D3—44no—
187C3×D3—64no—
188C3×D3—64no—
189C3×D3—124no—
190C24—541no—
191C2×D4—181no—
192C2×D4—274no—
193C2×D4—272no—
194C2×D4—542no—
195C22×C4—91no—
196C22×C4—541no—
197C2×C8—92no—
198C2×C8—182no—
199C4⋊C4—181no—
200C4⋊C4—272no—
201C22×C3—61no—
202C22×C3—61no—
203C22×C3—92no—
204C22×C3—92no—
205C22×C3—92no—
206C22×C3—184no—
207C22×C3—362no—
208C22×C3—362no—
209C22×C3C2×GO+(4,𝔽2)11yescharacteristic
210C2×D3—62no—
211C2×D3—62no—
212C2×D3—62no—
213C2×D3—62no—
214C2×D3—62no—
215C2×D3—62no—
216C2×D3—64no—
217C2×D3—122no—
218C2×D3—122no—
219C2×D3—184no—
220C2×D3—362no—
221C2×D3—362no—
222C2×D3—3616no—
223C2×D3—368no—
224C3×C4—62no—
225C3×C4—62no—
226C3×C4—91no—
227C3×C4—91no—
228C3×C4—362no—
229C3×C4D32⋊C412yes—
230Dic3—61no—
231Dic3—61no—
232Dic3—61no—
233Dic3—61no—
234Dic3—122no—
235Dic3—362no—
236C32—21no—
237C32—21no—
238C23—91no—
239C23—182no—
240C23—544no—
241C23—1082no—
242C23—1082no—
243D4—184no—
244D4—274no—
245D4—542no—
246C2×C4—91no—
247C2×C4—92no—
248C2×C4—92no—
249C2×C4—182no—
250C2×C4—1082no—
251C2×C4#1088⋊C211yescharacteristic
252C8—182no—
253C2×C3—61no—
254C2×C3—61no—
255C2×C3—61no—
256C2×C3—61no—
257C2×C3—61no—
258C2×C3—61no—
259C2×C3—94no—
260C2×C3—364no—
261C2×C3#6422511yescharacteristic
262C2×C3#6422711yescharacteristic
263C2×C3C2×D32⋊C411yescharacteristic
264D3—64no—
265D3—64no—
266D3—124no—
267D3—364no—
268C22—92no—
269C22—92no—
270C22—92no—
271C22—184no—
272C22—362no—
273C22—362no—
274C22—1088no—
275C22C2×#1088⋊C211yescharacteristic
276C4—91no—
277C4—91no—
278C4—362no—
279C4(((C3⋊D3)⋊C3)⋊C2)⋊C412yes—
280C3—61no—
281C3—61no—
282C3#1059336511yescharacteristic
283C2—94no—
284C2—364no—
285C2#1172229511yescharacteristic
286C2#1172229811yescharacteristic
287C2C2×(((C3⋊D3)⋊C3)⋊C2)⋊C411yescharacteristic
subgroupG/H
centerC22C2×#1088⋊C2
commutator#1084C22×C4
Frattini, socleC22×C3C2×GO+(4,𝔽2)
FittingC2×C4×H(3,𝔽3)D4
prime radicalC2×#88655C2
cosocleC2×#1084C23
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)C2×#102
LCSG ▷ #1084 ▷ C2×H(3,𝔽3) ▷ H(3,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. C2×#88626C2semidirect
3. #11721152C2semidirect
4. #11720705C2semidirect
11. #88655C4semidirect
12. C2×#6188C22semidirect
15. #88449C22semidirect
31. #6198C2×C4semidirect
90. #272#102semidirect
142. H(3,𝔽3)C2×#102semidirect

Maximal subgroups

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

subgroupindex
C2×#886552
C2×#886262
#117211522
#117207052
C2×#8219