Finite Group Databasebeta

← #13166378

Subgroups — #13166378

Order 1728.

744 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 744 conjugacy classes of subgroups — too many to draw. Only its 74 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×#64286C211yesmaximal, characteristic, complemented
4#11722713C214yesmaximal, complemented
5#10593831—31nomaximal
6C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic, complemented
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×Dic6C411yescharacteristic, complemented
9C2×#6292—21no—
10C2×#6289—21no—
11#89018—21no—
12#89018—21no—
13#89018C412yescomplemented
14#89018C2212yes—
15#88993—21no—
16#88989C2212yescomplemented
17#88988—21no—
18C3×C22⋊#157—21no—
19C3×C4⋊#157—21no—
20C3×C4×#157C2212yescomplemented
21C2×#3832—31no—
22C2×#3799—62no—
23#64286D311yescharacteristic, complemented
24#64283—31no—
25#63785—34no—
26C22×C3⋊(C3⋊D3)—21no—
27C22×C3×C3⋊D3D411yescharacteristic
28C2×C3×#157—22no—
29C2×C33×C4C2×C411yescharacteristic
30C4⋊(C3⋊(C3⋊D3))—21no—
31C4⋊(C3⋊(C3⋊D3))C2×C412yes—
32C2×C3×C3⋊Dic3D411yescharacteristic
33C3×C4×C3⋊D3C2×C411yescharacteristic
34C3×C4×C3⋊D3D411yescharacteristic
35C3×C4×C3⋊D3D411yescharacteristic
36C3×C4×C3⋊D3C2311yescharacteristic
37C2×C32×Dic3—21no—
38C32×Dic6—21no—
39C32×Dic6C2×C412yes—
40#6292—22no—
41#6292—22no—
42#6289—22no—
43#6289—22no—
44C3⋊#622—22no—
45#5005—91nomaximal
46C2×C32×Q8—31no—
47#3832—32no—
48#3832—32no—
49#3832—61no—
50#3832—61no—
51C2×C4⋊(C3⋊D3)—31no—
52C2×C4⋊(C3⋊D3)—62no—
53C2×C4⋊(C3⋊D3)—61no—
54C2×C4⋊(C3⋊D3)—122no—
55C2×C4×C3⋊D3—61no—
56C2×C4×C3⋊D3C2×D311yescharacteristic
57C2×C3×C4×D3—62no—
58C2×C3×Dic6—22no—
59C2×#626—64no—
60C2×#624—64no—
61#3799—64no—
62#3799—64no—
63#3799—64no—
64#3799—64no—
65C22⋊#157—21no—
66#3792—61no—
67C4⋊#157—21no—
68C4×#157C2×D312yescomplemented
69#3788—32no—
70#3787—61no—
71C22×C33C22⋊C411yescharacteristic
72C2×C3⋊(C3⋊D3)—22no—
73C2×C3⋊(C3⋊D3)—42no—
74C2×C3×C3⋊D3—41no—
75C2×C3×C3⋊D3C22⋊C411yescharacteristic
76C2×C3×C3⋊D3C2×D411yescharacteristic
77C3×#157—42no—
78C33×C4C22⋊C411yescharacteristic
79C33×C4C22×C411yescharacteristic
80C3×C3⋊Dic3C22⋊C411yescharacteristic
81C3×C3⋊Dic3C2×D411yescharacteristic
82C32×Dic3—22no—
83C2×#239—182no—
84C2×#235—91no—
85C3×#117—91no—
86C2×#208—91no—
87#805—94no—
88C22×C3⋊D3—61no—
89C22×C3⋊D3—64no—
90C22×C3⋊D3—124no—
91C22×C3⋊D3#6711yescharacteristic
92C22×C3×D3—62no—
93C2×#157—22no—
94C32×Q8—32no—
95C32×Q8—61no—
96C2×C32×C4—22no—
97C2×C32×C4—42no—
98C2×C32×C4—61no—
99C2×C32×C4C4×D311yescharacteristic
100C2×C3⋊Dic3D1211yescharacteristic
101C4⋊(C3⋊D3)—32no—
102C4⋊(C3⋊D3)—64no—
103C4⋊(C3⋊D3)—62no—
104C4⋊(C3⋊D3)—62no—
105C4⋊(C3⋊D3)—61no—
106C4⋊(C3⋊D3)—64no—
107C4⋊(C3⋊D3)—124no—
108C4⋊(C3⋊D3)—124no—
109C4×C3⋊D3—62no—
110C4×C3⋊D3—62no—
111C4×C3⋊D3C4×D311yescharacteristic
112C4×C3⋊D3D1211yescharacteristic
113C4×C3⋊D3#6711yescharacteristic
114C4×C3⋊D3C22×D311yescharacteristic
115C2×C3×Dic3—24no—
116C2×C3×Dic3—62no—
117C3×C4×D3—62no—
118C3×C4×D3—62no—
119C3×C4×D3—62no—
120C3×C4×D3—62no—
121C3×Dic6—24no—
122C3×Dic6—24no—
123#626—68no—
124#626—68no—
125#624—68no—
126#624—68no—
127#622—62no—
128#421—271noSylow
129C2×C33—21no—
130C2×C33C2×C22⋊C411yescharacteristic
131C3⋊(C3⋊D3)—42no—
132C3×C3⋊D3—21no—
133C3×C3⋊D3—21no—
134C2×C3×Q8—62no—
135C22×C3×C4—91no—
136C2×#67—181no—
137#239—184no—
138#239—184no—
139#239—184no—
140#239—184no—
141#235—92no—
142#235—92no—
143#235—181no—
144#235—181no—
145C2×D12—91no—
146C2×D12—182no—
147C2×D12—184no—
148C2×D12—362no—
149C2×C4×D3—62no—
150C2×C4×D3—181no—
151C2×C4×D3—184no—
152C2×Dic6#15711yescharacteristic, complemented
153C3×C4⋊C4—181no—
154C3×C22⋊C4—181no—
155C3×C42—92no—
156#208—92no—
157#208—181no—
158C2×C3⋊C8—181no—
159C22×C32—22no—
160C22×C32—42no—
161C22×C32(C2×D3)⋊C411yescharacteristic
162C2×C3⋊D3—41no—
163C2×C3⋊D3—62no—
164C2×C3⋊D3—68no—
165C2×C3⋊D3—128no—
166C2×C3⋊D3—122no—
167C2×C3⋊D3—128no—
168C2×C3⋊D3—248no—
169C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
170C2×C3⋊D3C2×#6711yescharacteristic
171C2×C3×D3—62no—
172C2×C3×D3—62no—
173C2×C3×D3—124no—
174#157—42no—
175C32×C4—22no—
176C32×C4—22no—
177C32×C4—42no—
178C32×C4—42no—
179C32×C4—62no—
180C32×C4(C2×D3)⋊C411yescharacteristic
181C32×C4C2×C4×D311yescharacteristic
182C3⋊Dic3(C2×D3)⋊C411yescharacteristic
183C3⋊Dic3C2×D1211yescharacteristic
184C3×Dic3—28no—
185C3×Dic3—62no—
186C3×Dic3—62no—
187C2×#41—271no—
188C2×M16—271no—
189#117—91no—
190#104—274no—
191C33#42111yescharacteristic, complemented, Sylow
192C23×C3—91no—
193C22×D3—62no—
194C22×D3—184no—
195C22×D3—181no—
196C22×D3—364no—
197C3×Q8—64no—
198C3×Q8—64no—
199C2×C3×C4—22no—
200C2×C3×C4—42no—
201C2×C3×C4—64no—
202C2×C3×C4—91no—
203C2×C3×C4—91no—
204C2×C3×C4—91no—
205C2×C3×C4—91no—
206C2×C3×C4—91no—
207C2×C3×C4—182no—
208C2×C3×C4C2×#15711yescharacteristic
209#67—182no—
210#67—182no—
211C2×Dic3—21no—
212C2×Dic3—62no—
213D12—92no—
214D12—184no—
215D12—184no—
216D12—188no—
217D12—188no—
218D12—181no—
219D12—364no—
220D12—364no—
221C4×D3—62no—
222C4×D3—62no—
223C4×D3—62no—
224C4×D3—62no—
225C4×D3—182no—
226C4×D3—182no—
227C4×D3—188no—
228C4×D3—188no—
229Dic6—21no—
230Dic6C2×#15712yes—
231C3⋊C8—182no—
232C2×C32—21no—
233C2×C32—24no—
234C2×C32—22no—
235C2×C32—44no—
236C2×C32—42no—
237C2×C32C2×(C2×D3)⋊C411yescharacteristic
238C3⋊D3—21no—
239C3⋊D3—21no—
240C3⋊D3—122no—
241C3⋊D3—128no—
242C3⋊D3—248no—
243C3×D3—122no—
244C3×D3—122no—
245#41—272no—
246#41—272no—
247#41—541no—
248#41—541no—
249C2×Q8—31no—
250C2×D4—271no—
251C2×D4—541no—
252C22×C4—91no—
253C22×C4—541no—
254M16—272no—
255M16—541no—
256C2×C8—541no—
257C4⋊C4—181no—
258C22⋊C4—181no—
259C42—92no—
260C22×C3—22no—
261C22×C3—42no—
262C22×C3—91no—
263C22×C3—91no—
264C22×C3—361no—
265C22×C3C22⋊#15711yescharacteristic
266C2×D3—62no—
267C2×D3—62no—
268C2×D3—124no—
269C2×D3—188no—
270C2×D3—182no—
271C2×D3—368no—
272C2×D3—362no—
273C2×D3—368no—
274C2×D3—728no—
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—42no—
278C3×C4—42no—
279C3×C4—68no—
280C3×C4—91no—
281C3×C4—91no—
282C3×C4—362no—
283C3×C4C22⋊#15711yescharacteristic
284C3×C4C22×#15711yescharacteristic
285Dic3—22no—
286Dic3—62no—
287Dic3—62no—
288C32—22no—
289C32—42no—
290C32#500511yescharacteristic, complemented
291C23—91no—
292C23—541no—
293Q8—32no—
294Q8—61no—
295D4—272no—
296D4—542no—
297D4—542no—
298D4—541no—
299C2×C4—61no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—91no—
305C2×C4—182no—
306C2×C4—542no—
307C2×C4—542no—
308C2×C4D3×#15711yescharacteristic
309C8—542no—
310C2×C3—24no—
311C2×C3—21no—
312C2×C3—22no—
313C2×C3—44no—
314C2×C3—42no—
315C2×C3—181no—
316C2×C3—181no—
317C2×C3C2×C22⋊#15711yescharacteristic
318D3—122no—
319D3—122no—
320D3—368no—
321D3—362no—
322D3—728no—
323C22—91no—
324C22—91no—
325C22—361no—
326C22—542no—
327C22—1082no—
328C22(C2×D3)⋊#15711yescharacteristic
329C4—62no—
330C4—91no—
331C4—91no—
332C4—362no—
333C4(C2×D3)⋊#15711yescharacteristic
334C4C2×D3×#15711yescharacteristic
335C3—22no—
336C3—42no—
337C3#1059383111yescharacteristic, complemented
338C2—21no—
339C2—181no—
340C2—181no—
341C2—1082no—
342C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC2C2×(C2×D3)⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC2×#89018C2
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#421
LCSG ▷ C33×C4 ▷ 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×#64286C2semidirect
4. #11722713C2semidirect
6. C2×C4⋊(C3⋊(C3⋊D3))C4semidirect
8. C2×C32×Dic6C4semidirect
13. #89018C4semidirect
16. #88989C22semidirect
20. C3×C4×#157C22semidirect
23. #64286D3semidirect
68. C4×#157C2×D3semidirect
152. C2×Dic6#157semidirect
191. C33#421semidirect
290. C32#5005semidirect
337. C3#10593831semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
#117245112
C3×#642862
#117227132
#105938313
#50059