Finite Group Databasebeta

← #13146545

Subgroups — #13146545

Order 1728.

499 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 57 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×#88650C211yesmaximal, characteristic
2C2×#88627C211yesmaximal, characteristic, complemented
3#11721152C211yesmaximal, characteristic, complemented
4#11720705C212yesmaximal, complemented
5#11720702C212yesmaximal, complemented
6C2×C4×#272C2211yescharacteristic
7C2×C4×(C3⋊D3)⋊C3—21no—
8C2×#6200—21no—
9C2×#6199—21no—
10C2×#6198C2211yescharacteristic
11C2×#6196C2211yescharacteristic
12#88650—22no—
13#88650C414yescomplemented
14C2×#6188C2212yes—
15#88627—22no—
16C4⋊((C3⋊Dic3)⋊C3)—21no—
17#88449C2212yes—
18C2×#3798—61no—
19D3×C4⋊Dic3—61no—
20C22×#272D411yescharacteristic
21C22×(C3⋊D3)⋊C3—21no—
22C2×C4×H(3,𝔽3)D411yescharacteristic
23C2×#1084C2311yescharacteristic
24C4×#272C2×C412yes—
25C4×#272D412yes—
26C2×(C3⋊Dic3)⋊C3—21no—
27C2×(C3⋊Dic3)⋊C3—22no—
28C4×(C3⋊D3)⋊C3—24no—
29#6200—24no—
30#6199—42no—
31#6198—21no—
32#6198C2×C412yescomplemented
33#6196—21no—
34#6196C2×C412yescomplemented
35#6188—22no—
36#5133—91nomaximal
37C2×C4×C3⋊D3—21no—
38C2×C3×C4×D3—61no—
39C2×C3×C4×D3—61no—
40C2×C3×C4×D3—61no—
41C2×#627—61no—
42C2×#626—61no—
43C2×#626—61no—
44C2×#625—61no—
45C2×D3×Dic3—62no—
46#3798—64no—
47#3798—64no—
48C4⋊(C3⋊Dic3)—21no—
49C3×C4⋊Dic3—61no—
50Dic3⋊Dic3—62no—
51C22×H(3,𝔽3)C2×D411yescharacteristic
52C2×#272—21no—
53C2×#272—21no—
54C2×#272C22⋊C412yes—
55C2×(C3⋊D3)⋊C3—22no—
56C2×(C3⋊D3)⋊C3—41no—
57C2×(C3⋊D3)⋊C3—41no—
58C4×H(3,𝔽3)C22⋊C412yes—
59#1084C22×C411yescharacteristic
60#1084C2×D411yescharacteristic
61(C3⋊Dic3)⋊C3—22no—
62(C3⋊Dic3)⋊C3—42no—
63C2×#235—91no—
64C2×#235—181no—
65C2×C3×M16—91no—
66C2×C4⋊Dic3—91no—
67D3×C4⋊C4—181no—
68#821—92no—
69#816—92no—
70C22×C3⋊D3—21no—
71C22×C3×D3—61no—
72C22×C3×D3—61no—
73C22×C3×D3—61no—
74C2×C32×C4—21no—
75C2×C32×C4—21no—
76C2×C3⋊Dic3—21no—
77C2×C3⋊Dic3—22no—
78C4×C3⋊D3—24no—
79C2×C3×Dic3—61no—
80C2×C3×Dic3—61no—
81C2×C3×Dic3—61no—
82C2×C3×Dic3—62no—
83C3×C4×D3—62no—
84C3×C4×D3—62no—
85C3×C4×D3—62no—
86C3×C4×D3—62no—
87C3×C4×D3—64no—
88#627—62no—
89#627—62no—
90#626—62no—
91#626—62no—
92#626—64no—
93#625—64no—
94D3×Dic3—124no—
95#417—271noSylow
96C2×H(3,𝔽3)C2×C22⋊C411yescharacteristic
97C2×H(3,𝔽3)AGL(1,ℤ/8)11yescharacteristic
98C2×H(3,𝔽3)#13711yescharacteristic
99#272—22no—
100(C3⋊D3)⋊C3—42no—
101C22×C3×C4—91no—
102C22×C3×C4—181no—
103C2×#67—181no—
104C2×#67—181no—
105C2×#67—181no—
106C22×Dic3—181no—
107#235—94no—
108#235—184no—
109#235—182no—
110#235—184no—
111C2×D12—91no—
112C2×D12—181no—
113C2×C4×D3—61no—
114C2×C4×D3—61no—
115C2×C4×D3—61no—
116C2×C4×D3—181no—
117C2×C4×D3—182no—
118C2×Dic6—91no—
119C2×Dic6—181no—
120C3×M16—182no—
121C2×C3×C8—92no—
122C3×C4⋊C4—181no—
123C4⋊Dic3—61no—
124C4⋊Dic3—92no—
125C4⋊Dic3—181no—
126Dic3⋊C4—182no—
127C22×C32—21no—
128C22×C32—21no—
129C2×C3⋊D3—22no—
130C2×C3⋊D3—41no—
131C2×C3⋊D3—41no—
132C2×C3×D3—62no—
133C2×C3×D3—62no—
134C2×C3×D3—62no—
135C2×C3×D3—62no—
136C2×C3×D3—62no—
137C2×C3×D3—121no—
138C2×C3×D3—121no—
139C2×C3×D3—121no—
140C2×C3×D3—121no—
141C32×C4—22no—
142C32×C4—22no—
143C3⋊Dic3—22no—
144C3⋊Dic3—42no—
145C3×Dic3—61no—
146C3×Dic3—61no—
147C3×Dic3—61no—
148C3×Dic3—61no—
149C3×Dic3—62no—
150C3×Dic3—122no—
151C2×#41—271no—
152C2×M16—91no—
153C2×C4⋊C4—271no—
154#103—272no—
155#102—272no—
156H(3,𝔽3)#41711yescharacteristic, complemented, Sylow
157C23×C3—91no—
158C23×C3—181no—
159C22×D3—61no—
160C22×D3—61no—
161C22×D3—61no—
162C22×D3—181no—
163C2×C3×C4—61no—
164C2×C3×C4—61no—
165C2×C3×C4—91no—
166C2×C3×C4—92no—
167C2×C3×C4—92no—
168C2×C3×C4—184no—
169C2×C3×C4—181no—
170C2×C3×C4—182no—
171C2×C3×C4GO+(4,𝔽2)11yescharacteristic
172#67—182no—
173#67—184no—
174#67—182no—
175#67—184no—
176C2×Dic3—61no—
177C2×Dic3—61no—
178C2×Dic3—61no—
179C2×Dic3—62no—
180C2×Dic3—181no—
181C2×Dic3—182no—
182C2×Dic3—362no—
183D12—92no—
184D12—181no—
185D12—184no—
186C4×D3—62no—
187C4×D3—62no—
188C4×D3—62no—
189C4×D3—62no—
190C4×D3—64no—
191C4×D3—184no—
192C4×D3—364no—
193Dic6—92no—
194Dic6—182no—
195Dic6—181no—
196Dic6—182no—
197C3×C8—182no—
198C2×C32—21no—
199C2×C32—21no—
200C2×C32—21no—
201C2×C32—21no—
202C2×C32—21no—
203C2×C32—21no—
204C3⋊D3—42no—
205C3×D3—64no—
206C3×D3—122no—
207C3×D3—122no—
208#41—274no—
209#41—542no—
210C2×Q8—271no—
211C2×D4—271no—
212C2×D4—541no—
213C22×C4—91no—
214C22×C4—181no—
215C22×C4—541no—
216M16—182no—
217C2×C8—92no—
218C4⋊C4—181no—
219C4⋊C4—272no—
220C22×C3—61no—
221C22×C3—61no—
222C22×C3—92no—
223C22×C3—182no—
224C22×C3—181no—
225C22×C3—181no—
226C22×C3—361no—
227C22×C3—361no—
228C22×C3C2×GO+(4,𝔽2)11yescharacteristic
229C2×D3—62no—
230C2×D3—62no—
231C2×D3—62no—
232C2×D3—62no—
233C2×D3—62no—
234C2×D3—121no—
235C2×D3—121no—
236C2×D3—121no—
237C2×D3—121no—
238C2×D3—182no—
239C2×D3—361no—
240C2×D3—361no—
241C3×C4—62no—
242C3×C4—62no—
243C3×C4—91no—
244C3×C4—91no—
245C3×C4—182no—
246C3×C4—362no—
247C3×C4D32⋊C412yes—
248Dic3—61no—
249Dic3—61no—
250Dic3—61no—
251Dic3—61no—
252Dic3—62no—
253Dic3—122no—
254Dic3—182no—
255Dic3—362no—
256C32—21no—
257C32—21no—
258C23—91no—
259C23—181no—
260Q8—272no—
261Q8—541no—
262D4—272no—
263D4—541no—
264D4—544no—
265C2×C4—91no—
266C2×C4—92no—
267C2×C4—92no—
268C2×C4—181no—
269C2×C4—184no—
270C2×C4—182no—
271C2×C4—1082no—
272C2×C4#1088⋊C211yescharacteristic
273C8—182no—
274C2×C3—61no—
275C2×C3—61no—
276C2×C3—61no—
277C2×C3—61no—
278C2×C3—61no—
279C2×C3—61no—
280C2×C3—182no—
281C2×C3—362no—
282C2×C3#6422411yescharacteristic
283C2×C3#6422611yescharacteristic
284C2×C3C2×D32⋊C411yescharacteristic
285D3—64no—
286D3—122no—
287D3—122no—
288D3—362no—
289C22—92no—
290C22—181no—
291C22—181no—
292C22—182no—
293C22—361no—
294C22—361no—
295C22C2×#1088⋊C211yescharacteristic
296C4—91no—
297C4—91no—
298C4—182no—
299C4—362no—
300C4(((C3⋊D3)⋊C3)⋊C2)⋊C412yes—
301C3—61no—
302C3—61no—
303C3#1059336011yescharacteristic
304C2—182no—
305C2—362no—
306C2#1172229411yescharacteristic
307C2#1172229711yescharacteristic
308C2C2×(((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×#88650C2
cosocleC2×#1084C23
hypercenterC2×C4#1088⋊C2
hypocenterH(3,𝔽3)#417
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×#88627C2semidirect
3. #11721152C2semidirect
4. #11720705C2semidirect
5. #11720702C2semidirect
13. #88650C4semidirect
32. #6198C2×C4semidirect
34. #6196C2×C4semidirect
156. H(3,𝔽3)#417semidirect

Maximal subgroups

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

subgroupindex
C2×#886502
C2×#886272
#117211522
#117207052
#117207022
#51339