Finite Group Databasebeta

← #13166357

Subgroups — #13166357

Order 1728.

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

normal subgroups only

This group has 752 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#11724514C211yesmaximal, characteristic, complemented
3C3×#64286C211yesmaximal, characteristic, complemented
4#11722728C214yesmaximal, complemented
5#10593832—31nomaximal
6C2×C4⋊(C3⋊(C3⋊D3))C411yescharacteristic, complemented
7C2×C3×C4×C3⋊D3C2211yescharacteristic
8C2×C32×Dic6C411yescharacteristic, complemented
9C2×#6292C2212yes—
10C2×#6289C412yescomplemented
11#89018—21no—
12#89018—21no—
13#89018—21no—
14#89018—21no—
15#88994C2212yes—
16C4⋊(C3⋊#157)—21no—
17C4×C3⋊#157—21no—
18C3×C22⋊#157C2212yes—
19C3×C4⋊#157—21no—
20C3×C4×#157—21no—
21C2×#3832—31no—
22C2×#3799—62no—
23#64286—31no—
24#64286D311yescharacteristic, complemented
25#63774—34no—
26C22×C3⋊(C3⋊D3)C2×C412yes—
27C22×C3×C3⋊D3C2311yescharacteristic
28C2×C3⋊#157—22no—
29C2×C3×#157—22no—
30C2×C33×C4C2×C411yescharacteristic
31C4⋊(C3⋊(C3⋊D3))—21no—
32C4⋊(C3⋊(C3⋊D3))—21no—
33C2×C3×C3⋊Dic3C2×C411yescharacteristic
34C3×C4×C3⋊D3D411yescharacteristic
35C3×C4×C3⋊D3D411yescharacteristic
36C3×C4×C3⋊D3D411yescharacteristic
37C3×C4×C3⋊D3D411yescharacteristic
38C2×C32×Dic3C2×C412yes—
39C32×Dic6—21no—
40C32×Dic6—21no—
41#6292—22no—
42#6292—22no—
43#6289—22no—
44#6289—22no—
45#5006—91nomaximal
46C2×C32×Q8—31no—
47#3832—61no—
48#3832—61no—
49#3832—61no—
50#3832—61no—
51C2×C4⋊(C3⋊D3)—32no—
52C2×C4⋊(C3⋊D3)—31no—
53C2×C4⋊(C3⋊D3)—62no—
54C2×C4⋊(C3⋊D3)—122no—
55C2×C4×C3⋊D3—32no—
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—32no—
66C22⋊#157C2×D312yes—
67C4⋊#157—21no—
68C4⋊#157—61no—
69C4×#157—21no—
70C4×#157—61no—
71C22×C33C22×C411yescharacteristic
72C2×C3⋊(C3⋊D3)—22no—
73C2×C3⋊(C3⋊D3)—42no—
74C2×C3×C3⋊D3—41no—
75C2×C3×C3⋊D3C2×D411yescharacteristic
76C2×C3×C3⋊D3C2×D411yescharacteristic
77C3⋊#157—42no—
78C3×#157—42no—
79C33×C4C22⋊C411yescharacteristic
80C33×C4C22⋊C411yescharacteristic
81C3×C3⋊Dic3C22⋊C411yescharacteristic
82C3×C3⋊Dic3C22⋊C411yescharacteristic
83C32×Dic3—22no—
84C2×#239—182no—
85C2×#235—91no—
86C3×#117—91no—
87#926—91no—
88#806—94no—
89C22×C3⋊D3—32no—
90C22×C3⋊D3—64no—
91C22×C3⋊D3—124no—
92C22×C3⋊D3C22×D311yescharacteristic
93C22×C3×D3—62no—
94C2×#157—22no—
95C2×#157—62no—
96C32×Q8—61no—
97C32×Q8—61no—
98C2×C32×C4—22no—
99C2×C32×C4—32no—
100C2×C32×C4—42no—
101C2×C32×C4C4×D311yescharacteristic
102C2×C3⋊Dic3C4×D311yescharacteristic
103C4⋊(C3⋊D3)—62no—
104C4⋊(C3⋊D3)—62no—
105C4⋊(C3⋊D3)—61no—
106C4⋊(C3⋊D3)—61no—
107C4⋊(C3⋊D3)—64no—
108C4⋊(C3⋊D3)—64no—
109C4⋊(C3⋊D3)—124no—
110C4⋊(C3⋊D3)—124no—
111C4×C3⋊D3—62no—
112C4×C3⋊D3—62no—
113C4×C3⋊D3D1211yescharacteristic
114C4×C3⋊D3D1211yescharacteristic
115C4×C3⋊D3#6711yescharacteristic
116C4×C3⋊D3#6711yescharacteristic
117C2×C3×Dic3—24no—
118C2×C3×Dic3—62no—
119C3×C4×D3—62no—
120C3×C4×D3—62no—
121C3×C4×D3—62no—
122C3×C4×D3—62no—
123C3×Dic6—24no—
124C3×Dic6—24no—
125#626—68no—
126#626—68no—
127#624—68no—
128#624—68no—
129#410—271noSylow
130C2×C33—21no—
131C2×C33C2×C22⋊C411yescharacteristic
132C3⋊(C3⋊D3)—42no—
133C3×C3⋊D3—21no—
134C3×C3⋊D3—21no—
135C2×C3×Q8—62no—
136C22×C3×C4—91no—
137C2×#67—92no—
138#239—184no—
139#239—184no—
140#239—184no—
141#239—184no—
142#235—181no—
143#235—181no—
144#235—181no—
145#235—181no—
146C2×D12—91no—
147C2×D12—182no—
148C2×D12—184no—
149C2×D12—362no—
150C2×C4×D3—62no—
151C2×C4×D3—92no—
152C2×C4×D3—184no—
153C2×Dic6#15711yescharacteristic, complemented
154C3×C4⋊C4—181no—
155C3×C22⋊C4—92no—
156C3×C42—181no—
157#217—92no—
158C4⋊Dic3—181no—
159C4×Dic3—181no—
160C22×C32—22no—
161C22×C32—42no—
162C22×C32C2×C4×D311yescharacteristic
163C2×C3⋊D3—41no—
164C2×C3⋊D3—62no—
165C2×C3⋊D3—68no—
166C2×C3⋊D3—128no—
167C2×C3⋊D3—122no—
168C2×C3⋊D3—128no—
169C2×C3⋊D3—248no—
170C2×C3⋊D3C2×D1211yescharacteristic
171C2×C3⋊D3C2×#6711yescharacteristic
172C2×C3×D3—62no—
173C2×C3×D3—62no—
174C2×C3×D3—124no—
175#157—42no—
176#157—122no—
177C32×C4—22no—
178C32×C4—22no—
179C32×C4—42no—
180C32×C4—42no—
181C32×C4—62no—
182C32×C4(C2×D3)⋊C411yescharacteristic
183C32×C4(C2×D3)⋊C411yescharacteristic
184C3⋊Dic3(C2×D3)⋊C411yescharacteristic
185C3⋊Dic3(C2×D3)⋊C411yescharacteristic
186C3×Dic3—28no—
187C3×Dic3—62no—
188C3×Dic3—62no—
189C2×#41—271no—
190#117—91no—
191#117—271no—
192#99—274no—
193C33#41011yescharacteristic, complemented, Sylow
194C23×C3—91no—
195C22×D3—62no—
196C22×D3—92no—
197C22×D3—184no—
198C22×D3—364no—
199C3×Q8—64no—
200C3×Q8—64no—
201C2×C3×C4—22no—
202C2×C3×C4—42no—
203C2×C3×C4—64no—
204C2×C3×C4—91no—
205C2×C3×C4—91no—
206C2×C3×C4—91no—
207C2×C3×C4—91no—
208C2×C3×C4—91no—
209C2×C3×C4—182no—
210C2×C3×C4C2×#15711yescharacteristic
211#67—182no—
212#67—182no—
213C2×Dic3—62no—
214C2×Dic3—182no—
215C2×Dic3C2×#15712yes—
216D12—184no—
217D12—184no—
218D12—188no—
219D12—188no—
220D12—181no—
221D12—181no—
222D12—364no—
223D12—364no—
224C4×D3—62no—
225C4×D3—62no—
226C4×D3—62no—
227C4×D3—62no—
228C4×D3—182no—
229C4×D3—182no—
230C4×D3—188no—
231C4×D3—188no—
232Dic6—21no—
233Dic6—21no—
234C2×C32—21no—
235C2×C32—24no—
236C2×C32—22no—
237C2×C32—44no—
238C2×C32—42no—
239C2×C32C2×(C2×D3)⋊C411yescharacteristic
240C3⋊D3—21no—
241C3⋊D3—21no—
242C3⋊D3—122no—
243C3⋊D3—128no—
244C3⋊D3—248no—
245C3×D3—122no—
246C3×D3—122no—
247#41—541no—
248#41—541no—
249#41—541no—
250#41—541no—
251C2×Q8—31no—
252C2×D4—272no—
253C2×D4—271no—
254C22×C4—91no—
255C22×C4—272no—
256C4⋊C4—181no—
257C4⋊C4—541no—
258C22⋊C4—92no—
259C22⋊C4—272no—
260C42—181no—
261C42—541no—
262C22×C3—22no—
263C22×C3—42no—
264C22×C3—91no—
265C22×C3—91no—
266C22×C3—361no—
267C22×C3C22×#15711yescharacteristic
268C2×D3—62no—
269C2×D3—62no—
270C2×D3—124no—
271C2×D3—188no—
272C2×D3—182no—
273C2×D3—368no—
274C2×D3—368no—
275C2×D3—362no—
276C2×D3—728no—
277C3×C4—22no—
278C3×C4—22no—
279C3×C4—42no—
280C3×C4—42no—
281C3×C4—68no—
282C3×C4—91no—
283C3×C4—91no—
284C3×C4—362no—
285C3×C4C22⋊#15711yescharacteristic
286C3×C4C22⋊#15711yescharacteristic
287Dic3—22no—
288Dic3—62no—
289Dic3—62no—
290Dic3—362no—
291C32—22no—
292C32—42no—
293C32#500611yescharacteristic, complemented
294C23—91no—
295C23—272no—
296Q8—61no—
297Q8—61no—
298D4—542no—
299D4—542no—
300D4—541no—
301D4—541no—
302C2×C4—32no—
303C2×C4—91no—
304C2×C4—91no—
305C2×C4—91no—
306C2×C4—91no—
307C2×C4—91no—
308C2×C4—182no—
309C2×C4—542no—
310C2×C4—542no—
311C2×C4—542no—
312C2×C4D3×#15711yescharacteristic
313C2×C3—24no—
314C2×C3—21no—
315C2×C3—22no—
316C2×C3—44no—
317C2×C3—42no—
318C2×C3—181no—
319C2×C3—181no—
320C2×C3C2×C22⋊#15711yescharacteristic
321D3—122no—
322D3—122no—
323D3—368no—
324D3—362no—
325D3—728no—
326C22—91no—
327C22—91no—
328C22—361no—
329C22—542no—
330C22—1082no—
331C22C2×D3×#15711yescharacteristic
332C4—62no—
333C4—91no—
334C4—91no—
335C4—362no—
336C4—1082no—
337C4(C2×D3)⋊#15711yescharacteristic
338C4(C2×D3)⋊#15711yescharacteristic
339C3—22no—
340C3—42no—
341C3#1059383211yescharacteristic, complemented
342C2—21no—
343C2—181no—
344C2—181no—
345C2—1082no—
346C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC2C2×(C2×D3)⋊#157
commutatorC22×C33C22×C4
FrattiniC22C2×D3×#157
FittingC2×C33×C4C2×C4
socleC2×C33C2×C22⋊C4
prime radicalC2×#89018C2
cosocleC22×C3×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33#410
LCSG ▷ C22×C33 ▷ C2×C33 ▷ C33
UCSC1 ◁ C2 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11724514C2semidirect
3. C3×#64286C2semidirect
4. #11722728C2semidirect
6. C2×C4⋊(C3⋊(C3⋊D3))C4semidirect
8. C2×C32×Dic6C4semidirect
10. C2×#6289C4semidirect
24. #64286D3semidirect
153. C2×Dic6#157semidirect
193. C33#410semidirect
293. C32#5006semidirect
341. C3#10593832semidirect

Maximal subgroups

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

subgroupindex
C2×#890182
#117245142
C3×#642862
#117227282
#105938323
#50069