Finite Group Databasebeta

← #13180966

Subgroups — #13180966

Order 1728.

952 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 952 conjugacy classes of subgroups — too many to draw. Only its 170 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×#89018C211yesmaximal, characteristic
2#11724511C211yesmaximal, characteristic, complemented
3C3×#64283C211yesmaximal, characteristic, complemented
4#11724269C214yesmaximal, complemented
5#11724253C214yesmaximal, complemented
6#11724252C214yesmaximal, complemented
7#10596644—31nomaximal
8C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic
9C2×C3×C4×C3⋊D3C2211yescharacteristic
10C2×C32×Dic6C411yescharacteristic
11C2×#6292C412yes—
12C2×#6289C2212yes—
13#89018C412yes—
14#89018C412yes—
15#89018C2212yes—
16#89018C2212yes—
17#88993C2212yes—
18#88989C2212yescomplemented
19#88988C2212yescomplemented
20C3×#3792C2212yes—
21C3×#3788C2212yescomplemented
22C3×#3787C2212yescomplemented
23#88926C2218yescomplemented
24#88924C2218yescomplemented
25C2×#3832—31no—
26C2×#3799—62no—
27#64289—34no—
28#64289—34no—
29#64283—31no—
30#64283—34no—
31#64283D311yescharacteristic, complemented
32C22×C3⋊(C3⋊D3)C2×C412yes—
33C22×C3×C3⋊D3C2×C411yescharacteristic
34C2×C33×C4C2×C411yescharacteristic
35C4⋊(C3⋊(C3⋊D3))C2×C412yes—
36C4⋊(C3⋊(C3⋊D3))C2×C412yes—
37C2×C3×C3⋊Dic3C2311yescharacteristic
38C3×C4×C3⋊D3C2×C411yescharacteristic
39C3×C4×C3⋊D3C2×C411yescharacteristic
40C3×C4×C3⋊D3C2311yescharacteristic
41C3×C4×C3⋊D3C2311yescharacteristic
42C2×C32×Dic3C2×C412yes—
43C32×Dic6C2×C412yes—
44C32×Dic6C2×C412yes—
45#6292C2×C414yes—
46#6292C2×C414yes—
47#6289C2×C414yes—
48#6289C2314yes—
49C3⋊#622C2314yes—
50C3×#622C2314yes—
51#5745—91nomaximal
52C2×C32×Q8—31no—
53#3832—32no—
54#3832—32no—
55#3832—32no—
56#3832—32no—
57C2×C4⋊(C3⋊D3)—32no—
58C2×C4⋊(C3⋊D3)—31no—
59C2×C4⋊(C3⋊D3)—62no—
60C2×C4⋊(C3⋊D3)—122no—
61C2×C4×C3⋊D3—32no—
62C2×C4×C3⋊D3C2×D311yescharacteristic
63C2×C3×C4×D3—62no—
64C2×C3×Dic6—22no—
65C2×#626—64no—
66C2×#624—64no—
67#3799—64no—
68#3799—64no—
69#3799—64no—
70#3799—64no—
71#3792—32no—
72#3792C2×D312yes—
73#3788—32no—
74#3788—38no—
75#3788C2×D312yescomplemented
76#3787—32no—
77#3787—38no—
78#3787C2×D312yescomplemented
79C22×C33C22×C411yescharacteristic
80C2×C3⋊(C3⋊D3)—42no—
81C2×C3⋊(C3⋊D3)C22×C414yes—
82C2×C3×C3⋊D3—41no—
83C2×C3×C3⋊D3C22×C411yescharacteristic
84C2×C3×C3⋊D3C22×C411yescharacteristic
85C33×C4C22×C411yescharacteristic
86C33×C4C22×C411yescharacteristic
87C3×C3⋊Dic3C22×C411yescharacteristic
88C3×C3⋊Dic3C2411yescharacteristic
89C32×Dic3C22×C414yes—
90C2×#239—182no—
91C2×#235—91no—
92C2×C3×M16—91no—
93C2×#208—91no—
94#907—94no—
95D3×M16—94no—
96#901—94no—
97C22×C3⋊D3—32no—
98C22×C3⋊D3—64no—
99C22×C3⋊D3—124no—
100C22×C3⋊D3C4×D311yescharacteristic
101C22×C3×D3—62no—
102C32×Q8—32no—
103C32×Q8—32no—
104C2×C32×C4—22no—
105C2×C32×C4—32no—
106C2×C32×C4—42no—
107C2×C32×C4C4×D311yescharacteristic
108C2×C3⋊Dic3C22×D311yescharacteristic
109C4⋊(C3⋊D3)—34no—
110C4⋊(C3⋊D3)—34no—
111C4⋊(C3⋊D3)—32no—
112C4⋊(C3⋊D3)—32no—
113C4⋊(C3⋊D3)—64no—
114C4⋊(C3⋊D3)—64no—
115C4⋊(C3⋊D3)—124no—
116C4⋊(C3⋊D3)—124no—
117C4×C3⋊D3—34no—
118C4×C3⋊D3—34no—
119C4×C3⋊D3C4×D311yescharacteristic
120C4×C3⋊D3C4×D311yescharacteristic
121C4×C3⋊D3C22×D311yescharacteristic
122C4×C3⋊D3C22×D311yescharacteristic
123C2×C3×Dic3—24no—
124C2×C3×Dic3—62no—
125C3×C4×D3—62no—
126C3×C4×D3—62no—
127C3×C4×D3—62no—
128C3×C4×D3—62no—
129C3×Dic6—24no—
130C3×Dic6—24no—
131#626—68no—
132#626—68no—
133#624—68no—
134#624—68no—
135#622—34no—
136#622C22×D314yes—
137#568—271noSylow
138C2×C33—21no—
139C2×C33C23×C411yescharacteristic
140C3⋊(C3⋊D3)—24no—
141C3×C3⋊D3—21no—
142C3×C3⋊D3—21no—
143C2×C3×Q8—62no—
144C22×C3×C4—91no—
145C2×#67—92no—
146#239—184no—
147#239—184no—
148#239—184no—
149#239—184no—
150#235—92no—
151#235—92no—
152#235—92no—
153#235—92no—
154C2×D12—91no—
155C2×D12—182no—
156C2×D12—184no—
157C2×D12—362no—
158C2×C4×D3—62no—
159C2×C4×D3—92no—
160C2×C4×D3—184no—
161C2×Dic6#15711yescharacteristic
162C3×M16—92no—
163C3×M16—92no—
164C2×C3×C8—92no—
165#208—92no—
166#208—92no—
167C2×C3⋊C8—92no—
168ΓL(1,𝔽25)—98no—
169D3×C8—98no—
170C22×C32—22no—
171C22×C32—42no—
172C22×C32C2×C4×D311yescharacteristic
173C2×C3⋊D3—34no—
174C2×C3⋊D3—41no—
175C2×C3⋊D3—68no—
176C2×C3⋊D3—128no—
177C2×C3⋊D3—122no—
178C2×C3⋊D3—128no—
179C2×C3⋊D3—248no—
180C2×C3⋊D3C2×C4×D311yescharacteristic
181C2×C3⋊D3C2×C4×D311yescharacteristic
182C2×C3×D3—62no—
183C2×C3×D3—62no—
184C2×C3×D3—124no—
185C32×C4—22no—
186C32×C4—22no—
187C32×C4—34no—
188C32×C4—42no—
189C32×C4—42no—
190C32×C4C2×C4×D311yescharacteristic
191C32×C4C2×C4×D311yescharacteristic
192C3⋊Dic3C2×C4×D311yescharacteristic
193C3⋊Dic3C23×D311yescharacteristic
194C3×Dic3—28no—
195C3×Dic3—62no—
196C3×Dic3—62no—
197C2×#41—271no—
198#131—274no—
199#131—274no—
200C2×M16—91no—
201C2×M16—274no—
202C2×M16—271no—
203C33#56811yescharacteristic, complemented, Sylow
204C23×C3—91no—
205C22×D3—62no—
206C22×D3—92no—
207C22×D3—184no—
208C22×D3—364no—
209C3×Q8—64no—
210C3×Q8—64no—
211C2×C3×C4—22no—
212C2×C3×C4—42no—
213C2×C3×C4—64no—
214C2×C3×C4—91no—
215C2×C3×C4—91no—
216C2×C3×C4—91no—
217C2×C3×C4—91no—
218C2×C3×C4—91no—
219C2×C3×C4C2×#15711yescharacteristic
220#67—94no—
221#67—94no—
222C2×Dic3—62no—
223C2×Dic3C2×#15712yes—
224D12—92no—
225D12—92no—
226D12—184no—
227D12—184no—
228D12—188no—
229D12—188no—
230D12—364no—
231D12—364no—
232C4×D3—62no—
233C4×D3—62no—
234C4×D3—62no—
235C4×D3—62no—
236C4×D3—94no—
237C4×D3—94no—
238C4×D3—188no—
239C4×D3—188no—
240Dic6C2×#15712yes—
241Dic6C2×#15712yes—
242C3×C8—94no—
243C3⋊C8—94no—
244C2×C32—21no—
245C2×C32—24no—
246C2×C32—22no—
247C2×C32—44no—
248C2×C32—42no—
249C2×C32C22×C4×D311yescharacteristic
250C3⋊D3—21no—
251C3⋊D3—21no—
252C3⋊D3—64no—
253C3⋊D3—128no—
254C3⋊D3—248no—
255C3×D3—122no—
256C3×D3—122no—
257#41—272no—
258#41—272no—
259#41—272no—
260#41—272no—
261C2×Q8—31no—
262C2×D4—272no—
263C2×D4—271no—
264C22×C4—91no—
265C22×C4—272no—
266M16—92no—
267M16—92no—
268M16—278no—
269M16—272no—
270M16—272no—
271C2×C8—92no—
272C2×C8—278no—
273C2×C8—272no—
274C22×C3—22no—
275C22×C3—42no—
276C22×C3—91no—
277C22×C3—91no—
278C22×C3—361no—
279C22×C3C22×#15711yescharacteristic
280C2×D3—62no—
281C2×D3—62no—
282C2×D3—94no—
283C2×D3—124no—
284C2×D3—188no—
285C2×D3—368no—
286C2×D3—362no—
287C2×D3—368no—
288C2×D3—728no—
289C3×C4—22no—
290C3×C4—22no—
291C3×C4—42no—
292C3×C4—42no—
293C3×C4—68no—
294C3×C4—91no—
295C3×C4—91no—
296C3×C4C22×#15711yescharacteristic
297C3×C4C22×#15711yescharacteristic
298Dic3—62no—
299Dic3—62no—
300Dic3C22×#15714yes—
301C32—22no—
302C32—42no—
303C32#574511yescharacteristic, complemented
304C23—91no—
305C23—272no—
306Q8—32no—
307Q8—32no—
308D4—274no—
309D4—274no—
310D4—272no—
311D4—272no—
312C2×C4—32no—
313C2×C4—91no—
314C2×C4—91no—
315C2×C4—91no—
316C2×C4—91no—
317C2×C4—91no—
318C2×C4—274no—
319C2×C4—274no—
320C2×C4D3×#15711yescharacteristic
321C8—94no—
322C8—274no—
323C2×C3—24no—
324C2×C3—21no—
325C2×C3—22no—
326C2×C3—44no—
327C2×C3—42no—
328C2×C3—181no—
329C2×C3—181no—
330C2×C3C23×#15711yescharacteristic
331D3—122no—
332D3—122no—
333D3—184no—
334D3—368no—
335D3—728no—
336C22—91no—
337C22—91no—
338C22—274no—
339C22—361no—
340C22—1082no—
341C22C2×D3×#15711yescharacteristic
342C4—34no—
343C4—91no—
344C4—91no—
345C4C2×D3×#15711yescharacteristic
346C4C2×D3×#15711yescharacteristic
347C3—22no—
348C3—42no—
349C3#1059664411yescharacteristic, complemented
350C2—21no—
351C2—181no—
352C2—181no—
353C2—544no—
354C2C22×D3×#15711yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×#157
commutator, socleC2×C33C23×C4
FittingC2×C33×C4C2×C4
prime radicalC2×#89018C2
cosocleC3×C3⋊Dic3C24
hypercenterC2×C4D3×#157
hypocenterC33#568
LCSG ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724511C2semidirect
3. C3×#64283C2semidirect
4. #11724269C2semidirect
5. #11724253C2semidirect
6. #11724252C2semidirect
18. #88989C22semidirect
19. #88988C22semidirect
21. C3×#3788C22semidirect
22. C3×#3787C22semidirect
23. #88926C22semidirect
24. #88924C22semidirect
31. #64283D3semidirect
75. #3788C2×D3semidirect
78. #3787C2×D3semidirect
203. C33#568semidirect
303. C32#5745semidirect
349. C3#10596644semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
#117245112
C3×#642832
#117242692
#117242532
#117242522
#105966443
#57459