Finite Group Databasebeta

← C2×#11722709 #13166371

Subgroups — C2×#11722709 #13166371

Order 1728.

712 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 712 conjugacy classes of subgroups — too many to draw. Only its 98 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×C3⋊D3×Dic6C211yesmaximal, characteristic
2C2×C4⋊(C3⋊#157)C211yesmaximal, characteristic
3C2×C3×#3787C211yesmaximal, characteristic
4#11722709C214yesmaximal, direct product factor
5C2×#63784—31nomaximal
6C22×C3⋊#157—21no—
7C2×#6308C411yescharacteristic, complemented
8C2×C3×C4×C3⋊D3C2211yescharacteristic
9C2×C32×Dic6C411yescharacteristic, complemented
10C2×#6293—21no—
11C2×Dic3×C3⋊D3—21no—
12C3⋊D3×Dic6—21no—
13C3⋊D3×Dic6—21no—
14C3⋊D3×Dic6C412yescomplemented
15C3⋊D3×Dic6C2212yes—
16C4⋊(C3⋊#157)—21no—
17C4⋊(C3⋊#157)C2212yes—
18C2×C3×#622—21no—
19C3×#3787—21no—
20C3×#3787C2212yes—
21C2×Q8×C3⋊D3—31no—
22C2×D3×Dic6—62no—
23C2×C4⋊#157—31no—
24C2×#3787D311yescharacteristic
25#63784—34no—
26C22×C3×C3⋊D3D411yescharacteristic
27C2×C3⋊#157—22no—
28C2×C3⋊#157—42no—
29C2×C33×C4C2×C411yescharacteristic
30C2×C3⋊(C3⋊Dic3)—21no—
31#6308—21no—
32#6308C2×C412yescomplemented
33C2×C3×C3⋊Dic3D411yescharacteristic
34C3×C4×C3⋊D3C2×C411yescharacteristic
35C3×C4×C3⋊D3D411yescharacteristic
36C3×C4×C3⋊D3D411yescharacteristic
37C3×C4×C3⋊D3C2311yescharacteristic
38C2×C32×Dic3—21no—
39C32×Dic6—21no—
40C32×Dic6C2×C412yescomplemented
41#6293—22no—
42#6293—22no—
43Dic3×C3⋊D3—22no—
44Dic3×C3⋊D3—22no—
45C3×#622—22no—
46C2×#816—91nomaximal
47C22×#157—61no—
48C2×C32×Q8—31no—
49Q8×C3⋊D3—32no—
50Q8×C3⋊D3—32no—
51Q8×C3⋊D3—61no—
52Q8×C3⋊D3—61no—
53C2×C4×C3⋊D3—61no—
54C2×C4×C3⋊D3C2×D311yescharacteristic
55C2×#634—31no—
56C2×#634—62no—
57C2×#634—61no—
58C2×#634—122no—
59C2×C3×C4×D3—62no—
60C2×C3×Dic6—22no—
61C2×#627—64no—
62C2×D3×Dic3—64no—
63D3×Dic6—64no—
64D3×Dic6—64no—
65D3×Dic6—64no—
66D3×Dic6—64no—
67C2×#622—21no—
68C4⋊#157—32no—
69C4⋊#157—61no—
70#3787—21no—
71#3787C2×D312yes—
72C22×C33C22⋊C411yescharacteristic
73C2×C3×C3⋊D3C22⋊C411yescharacteristic
74C2×C3×C3⋊D3SD1612yes—
75C2×C3×C3⋊D3Q1612yes—
76C2×C3×C3⋊D3C2×D411yescharacteristic
77C3⋊#157—42no—
78C33×C4C22⋊C411yescharacteristic
79C33×C4C22×C411yescharacteristic
80C3⋊(C3⋊Dic3)—22no—
81C3×C3⋊Dic3C22⋊C411yescharacteristic
82C3×C3⋊Dic3C2×D411yescharacteristic
83C32×Dic3—22no—
84C2×D3×Q8—182no—
85C22×Dic6—91no—
86C22×C3×C8—91no—
87C2×C4⋊Dic3—91no—
88#816—94no—
89C22×C3⋊D3D1211yescharacteristic
90C22×C3×D3—62no—
91C2×#157—62no—
92C2×#157—122no—
93C32×Q8—32no—
94C32×Q8—61no—
95C2×C32×C4—22no—
96C2×C32×C4—42no—
97C2×C32×C4—61no—
98C2×C32×C4C4×D311yescharacteristic
99C2×C3⋊Dic3—61no—
100C2×C3⋊Dic3—64no—
101C2×C3⋊Dic3—124no—
102C2×C3⋊Dic3#6711yescharacteristic
103C4×C3⋊D3—62no—
104C4×C3⋊D3—62no—
105C4×C3⋊D3C4×D311yescharacteristic
106C4×C3⋊D3D1211yescharacteristic
107C4×C3⋊D3#6711yescharacteristic
108C4×C3⋊D3C22×D311yescharacteristic
109#634—32no—
110#634—64no—
111#634—61no—
112#634—62no—
113#634—62no—
114#634—64no—
115#634—124no—
116#634—124no—
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#627—68no—
126#627—68no—
127D3×Dic3—68no—
128D3×Dic3—68no—
129#622—22no—
130C2×#103—271noSylow
131C2×C33#10312yescomplemented
132C2×C33C2×C22⋊C411yescharacteristic
133C3×C3⋊D3#10312yescomplemented
134C3×C3⋊D3C2×SD1611yescharacteristic
135C3×C3⋊D3C2×Q1611yescharacteristic
136C2×C3×Q8—62no—
137C22×C3×C4—91no—
138C22×Dic3—181no—
139C22×Dic3—181no—
140D3×Q8—184no—
141D3×Q8—184no—
142D3×Q8—184no—
143D3×Q8—184no—
144C2×C4×D3—62no—
145C2×C4×D3—184no—
146C2×Dic6—91no—
147C2×Dic6—92no—
148C2×Dic6—92no—
149C2×Dic6—182no—
150C2×Dic6—181no—
151C2×Dic6—181no—
152C2×Dic6—181no—
153C2×Dic6—184no—
154C2×Dic6—362no—
155C2×Dic6#15711yescharacteristic, complemented
156C2×C3×C8—92no—
157C2×C3×C8—181no—
158C2×C3×C8—181no—
159C4⋊Dic3—92no—
160C4⋊Dic3—181no—
161C22×C32—22no—
162C22×C32—42no—
163C22×C32(C2×D3)⋊C411yescharacteristic
164C2×C3⋊D3#20412yes—
165C2×C3⋊D3Dic1212yes—
166C2×C3⋊D3(C2×D3)⋊C411yescharacteristic
167C2×C3⋊D3C2×D1211yescharacteristic
168C2×C3×D3—62no—
169C2×C3×D3—64no—
170C2×C3×D3—64no—
171C2×C3×D3—62no—
172#157—122no—
173C32×C4—22no—
174C32×C4—22no—
175C32×C4—42no—
176C32×C4—42no—
177C32×C4—62no—
178C32×C4(C2×D3)⋊C411yescharacteristic
179C32×C4C2×C4×D311yescharacteristic
180C3⋊Dic3—62no—
181C3⋊Dic3—68no—
182C3⋊Dic3—128no—
183C3⋊Dic3(C2×D3)⋊C411yescharacteristic
184C3⋊Dic3C2×#6711yescharacteristic
185C3×Dic3—28no—
186C3×Dic3—62no—
187C3×Dic3—62no—
188C22×Q8—271no—
189C22×C8—91no—
190C2×C4⋊C4—271no—
191#103—274no—
192C33C2×#10311yescharacteristic, complemented, Sylow
193C23×C3—91no—
194C22×D3—62no—
195C3×Q8—64no—
196C3×Q8—64no—
197C2×C3×C4—22no—
198C2×C3×C4—42no—
199C2×C3×C4—64no—
200C2×C3×C4—91no—
201C2×C3×C4—91no—
202C2×C3×C4—91no—
203C2×C3×C4—91no—
204C2×C3×C4—91no—
205C2×C3×C4C2×#15711yescharacteristic
206C2×Dic3—21no—
207C2×Dic3—62no—
208C2×Dic3—182no—
209C2×Dic3—182no—
210C2×Dic3—184no—
211C2×Dic3—181no—
212C2×Dic3—182no—
213C2×Dic3—364no—
214C2×Dic3—362no—
215C4×D3—62no—
216C4×D3—62no—
217C4×D3—62no—
218C4×D3—62no—
219C4×D3—188no—
220C4×D3—188no—
221Dic6—21no—
222Dic6—92no—
223Dic6—184no—
224Dic6—182no—
225Dic6—182no—
226Dic6—188no—
227Dic6—188no—
228Dic6—184no—
229Dic6—181no—
230Dic6—364no—
231Dic6—364no—
232Dic6C2×#15712yescomplemented
233C3×C8—182no—
234C2×C32—22no—
235C2×C32—24no—
236C2×C32—44no—
237C2×C32—42no—
238C2×C32#81612yescomplemented
239C2×C32C2×(C2×D3)⋊C411yescharacteristic
240C3⋊D3#81612yescomplemented
241C3⋊D3C2×#20411yescharacteristic
242C3⋊D3C2×Dic1211yescharacteristic
243C3×D3—64no—
244C3×D3—62no—
245C3×D3—62no—
246C2×Q8—31no—
247C2×Q8—272no—
248C2×Q8—272no—
249C2×Q8—271no—
250C2×Q8—541no—
251C2×Q8—541no—
252C2×Q8—541no—
253C22×C4—91no—
254C22×C4—541no—
255C22×C4—541no—
256C2×C8—92no—
257C2×C8—181no—
258C2×C8—181no—
259C4⋊C4—272no—
260C4⋊C4—541no—
261C22×C3—22no—
262C22×C3—42no—
263C22×C3—91no—
264C22×C3—91no—
265C22×C3—92no—
266C22×C3—92no—
267C22×C3C22⋊#15711yescharacteristic
268C2×D3—62no—
269C2×D3—62no—
270C2×D3—64no—
271C2×D3—64no—
272C3×C4—22no—
273C3×C4—22no—
274C3×C4—42no—
275C3×C4—42no—
276C3×C4—68no—
277C3×C4—91no—
278C3×C4—91no—
279C3×C4C22⋊#15711yescharacteristic
280C3×C4C22×#15711yescharacteristic
281Dic3—22no—
282Dic3—62no—
283Dic3—62no—
284Dic3—188no—
285Dic3—182no—
286Dic3—368no—
287Dic3—362no—
288C32—22no—
289C32—42no—
290C32C2×#81611yescharacteristic, complemented
291C23—91no—
292Q8—32no—
293Q8—61no—
294Q8—272no—
295Q8—542no—
296Q8—542no—
297Q8—541no—
298C2×C4—61no—
299C2×C4—91no—
300C2×C4—91no—
301C2×C4—91no—
302C2×C4—91no—
303C2×C4—91no—
304C2×C4—542no—
305C2×C4—542no—
306C2×C4—541no—
307C2×C4—542no—
308C2×C4—1082no—
309C2×C4D3×#15711yescharacteristic
310C8—182no—
311C2×C3—24no—
312C2×C3—22no—
313C2×C3—44no—
314C2×C3—42no—
315C2×C3—91no—
316C2×C3—91no—
317C2×C3—92no—
318C2×C3#6378412yescomplemented
319C2×C3C2×C22⋊#15711yescharacteristic
320D3—64no—
321D3—62no—
322D3—62no—
323C22—92no—
324C22—92no—
325C22—91no—
326C22—91no—
327C22(C2×D3)⋊#15711yescharacteristic
328C4—62no—
329C4—91no—
330C4—91no—
331C4—542no—
332C4—1082no—
333C4(C2×D3)⋊#15711yescharacteristic
334C4C2×D3×#15711yescharacteristic
335C3—22no—
336C3—42no—
337C3C2×#6378411yescharacteristic, complemented
338C2—91no—
339C2—92no—
340C2—91no—
341C2#1172270912yesdirect product factor
342C2C2×(C2×D3)⋊#15711yescharacteristic
subgroupG/H
centerC22(C2×D3)⋊#157
commutatorC33×C4C22×C4
FrattiniC4C2×D3×#157
FittingC2×C33×C4C2×C4
socleC22×C33C22⋊C4
prime radicalC22×C3×C3⋊D3D4
cosocleC3×C4×C3⋊D3C23
hypercenterC2×C4D3×#157
hypocenterC33C2×#103
LCSG ▷ C33×C4 ▷ C2×C33 ▷ C33
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
4. #11722709C2direct (N × H)
7. C2×#6308C4semidirect
9. C2×C32×Dic6C4semidirect
14. C3⋊D3×Dic6C4semidirect
32. #6308C2×C4semidirect
40. C32×Dic6C2×C4semidirect
131. C2×C33#103semidirect
133. C3×C3⋊D3#103semidirect
155. C2×Dic6#157semidirect
192. C33C2×#103semidirect
232. Dic6C2×#157semidirect
238. C2×C32#816semidirect
240. C3⋊D3#816semidirect
290. C32C2×#816semidirect
318. C2×C3#63784semidirect
337. C3C2×#63784semidirect
341. C2#11722709direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3⋊D3×Dic62
C2×C4⋊(C3⋊#157)2
C2×C3×#37872
#117227092
C2×#637843
C2×#8169