Finite Group Databasebeta

← C2×#11722779 #13167302

Subgroups — C2×#11722779 #13167302

Order 1728.

552 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 111 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
1C22×C3⋊#622C211yesmaximal, characteristic
2C2×#88815C211yesmaximal, characteristic
3C2×C3×#3724C211yesmaximal, characteristic
4#11722779C214yesmaximal, direct product factor
5C2×#63743—31nomaximal
6C22×C3×C3⋊Dic3—21no—
7C22×C3×C3⋊Dic3C2211yescharacteristic
8C22×C32×Dic3—21no—
9C2×C3⋊#622C414yescomplemented
10C2×C3⋊#622C2212yes—
11#88815—21no—
12#88815C2212yes—
13C3×#3724—21no—
14C3×#3724C2212yes—
15C22×#622—31no—
16C2×C3×Dic3⋊C4—61no—
17C2×#3724—31no—
18C2×#3724D311yescharacteristic
19C2×Dic3⋊Dic3—61no—
20#63743—34no—
21C23×C33D411yescharacteristic
22C2×C3×C3⋊Dic3—22no—
23C2×C3×C3⋊Dic3—42no—
24C2×C3×C3⋊Dic3C2×C412yes—
25C2×C3×C3⋊Dic3D411yescharacteristic
26C2×C3×C3⋊Dic3Q812yes—
27C2×C3×C3⋊Dic3C2311yescharacteristic
28C2×C32×Dic3—22no—
29C2×C32×Dic3—42no—
30C3⋊#622C2×C414yescomplemented
31C2×#807—91nomaximal
32C22×C32×C4—61no—
33C22×C3⋊Dic3—21no—
34C22×C3⋊Dic3C2×D311yescharacteristic
35C22×C3×Dic3—21no—
36C22×C3×Dic3—21no—
37C22×C3×Dic3—41no—
38C22×C3×Dic3—61no—
39C22×C3×Dic3—61no—
40C22×C3×Dic3—61no—
41C22×C3×Dic3—61no—
42C22×C3×Dic3—121no—
43C2×#622—34no—
44C2×#622—32no—
45C3×Dic3⋊C4—62no—
46C3×Dic3⋊C4—62no—
47#3724—21no—
48#3724—32no—
49#3724—61no—
50#3724C2×D312yes—
51Dic3⋊Dic3—62no—
52Dic3⋊Dic3—62no—
53C22×C33C4⋊C412yes—
54C22×C33D812yes—
55C22×C33Q1612yes—
56C22×C33C2×D411yescharacteristic
57C3×C3⋊Dic3—42no—
58C3×C3⋊Dic3C4⋊C412yes—
59C3×C3⋊Dic3C22×C411yescharacteristic
60C3×C3⋊Dic3C2×Q811yescharacteristic
61C32×Dic3—42no—
62C2×C3×C4⋊C4—91no—
63C2×C4⋊Dic3—91no—
64C2×Dic3⋊C4—61no—
65C2×Dic3⋊C4—181no—
66C22×C3⋊C8—91no—
67#807—94no—
68C23×C32—21no—
69C23×C32—21no—
70C23×C32—41no—
71C23×C32—41no—
72C23×C32#6711yescharacteristic
73C2×C32×C4—62no—
74C2×C32×C4—122no—
75C2×C3⋊Dic3—22no—
76C2×C3⋊Dic3—42no—
77C2×C3⋊Dic3Dic612yes—
78C2×C3⋊Dic3C4×D312yes—
79C2×C3⋊Dic3#6711yescharacteristic
80C2×C3⋊Dic3C22×D311yescharacteristic
81C2×C3×Dic3—22no—
82C2×C3×Dic3—22no—
83C2×C3×Dic3—42no—
84C2×C3×Dic3—42no—
85C2×C3×Dic3—44no—
86C2×C3×Dic3—42no—
87C2×C3×Dic3—61no—
88C2×C3×Dic3—61no—
89C2×C3×Dic3—61no—
90C2×C3×Dic3—61no—
91C2×C3×Dic3—62no—
92C2×C3×Dic3—62no—
93C2×C3×Dic3—62no—
94C2×C3×Dic3—62no—
95C2×C3×Dic3—62no—
96C2×C3×Dic3—62no—
97C2×C3×Dic3—122no—
98C2×C3×Dic3—122no—
99C2×C3×Dic3—124no—
100C2×C3×Dic3—122no—
101#622—34no—
102C2×#107—271noSylow
103C2×C33#10714yescomplemented
104C2×C33C2×C4⋊C411yescharacteristic
105C2×C33C2×D811yescharacteristic
106C2×C33C2×Q1611yescharacteristic
107C22×C3×C4—61no—
108C22×C3×C4—61no—
109C22×C3×C4—91no—
110C22×C3×C4—121no—
111C22×C3×C4—181no—
112C22×Dic3—21no—
113C22×Dic3—61no—
114C22×Dic3—61no—
115C22×Dic3—61no—
116C22×Dic3—61no—
117C22×Dic3—121no—
118C3×C4⋊C4—92no—
119C3×C4⋊C4—181no—
120C4⋊Dic3—92no—
121C4⋊Dic3—181no—
122Dic3⋊C4—62no—
123Dic3⋊C4—62no—
124Dic3⋊C4—182no—
125Dic3⋊C4—182no—
126C2×C3⋊C8—92no—
127C2×C3⋊C8—94no—
128C22×C32—21no—
129C22×C32—21no—
130C22×C32—22no—
131C22×C32—22no—
132C22×C32—22no—
133C22×C32—22no—
134C22×C32—22no—
135C22×C32—22no—
136C22×C32—42no—
137C22×C32—42no—
138C22×C32—42no—
139C22×C32—42no—
140C22×C32—42no—
141C22×C32—42no—
142C22×C32—41no—
143C22×C32—41no—
144C22×C32Dic3⋊C412yes—
145C22×C32#21312yes—
146C22×C32#21612yes—
147C22×C32C2×#6711yescharacteristic
148C32×C4—122no—
149C3⋊Dic3—42no—
150C3⋊Dic3Dic3⋊C412yes—
151C3⋊Dic3C2×Dic611yescharacteristic
152C3⋊Dic3C2×C4×D311yescharacteristic
153C3×Dic3—42no—
154C3×Dic3—44no—
155C3×Dic3—42no—
156C3×Dic3—62no—
157C3×Dic3—62no—
158C3×Dic3—61no—
159C3×Dic3—61no—
160C3×Dic3—61no—
161C3×Dic3—61no—
162C3×Dic3—122no—
163C3×Dic3—124no—
164C3×Dic3—122no—
165C22×C8—271no—
166C2×C4⋊C4—91no—
167C2×C4⋊C4—271no—
168#107—274no—
169C33C2×#10711yescharacteristic, complemented, Sylow
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—41no—
173C23×C3—41no—
174C23×C3GO+(4,𝔽2)11yescharacteristic
175C2×C3×C4—62no—
176C2×C3×C4—62no—
177C2×C3×C4—91no—
178C2×C3×C4—91no—
179C2×C3×C4—92no—
180C2×C3×C4—92no—
181C2×C3×C4—122no—
182C2×C3×C4—122no—
183C2×C3×C4—124no—
184C2×C3×C4—122no—
185C2×C3×C4—182no—
186C2×C3×C4—362no—
187C2×Dic3—22no—
188C2×Dic3—42no—
189C2×Dic3—62no—
190C2×Dic3—62no—
191C2×Dic3—62no—
192C2×Dic3—62no—
193C2×Dic3—61no—
194C2×Dic3—61no—
195C2×Dic3—61no—
196C2×Dic3—61no—
197C2×Dic3—62no—
198C2×Dic3—62no—
199C2×Dic3—122no—
200C2×Dic3—122no—
201C2×Dic3—124no—
202C2×Dic3—122no—
203C3⋊C8—94no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—21no—
210C2×C32—24no—
211C2×C32—24no—
212C2×C32—44no—
213C2×C32—44no—
214C2×C32—41no—
215C2×C32—41no—
216C2×C32—41no—
217C2×C32—41no—
218C2×C32—41no—
219C2×C32—41no—
220C2×C32#80714yescomplemented
221C2×C32C2×Dic3⋊C411yescharacteristic
222C2×C32C2×#21311yescharacteristic
223C2×C32C2×#21611yescharacteristic
224C22×C4—61no—
225C22×C4—91no—
226C22×C4—181no—
227C2×C8—272no—
228C2×C8—274no—
229C4⋊C4—92no—
230C4⋊C4—181no—
231C4⋊C4—272no—
232C4⋊C4—541no—
233C22×C3—21no—
234C22×C3—21no—
235C22×C3—22no—
236C22×C3—22no—
237C22×C3—22no—
238C22×C3—22no—
239C22×C3—22no—
240C22×C3—22no—
241C22×C3—41no—
242C22×C3—41no—
243C22×C3—42no—
244C22×C3—42no—
245C22×C3—42no—
246C22×C3—42no—
247C22×C3—42no—
248C22×C3—42no—
249C22×C3#377312yes—
250C22×C3#377412yes—
251C22×C3#377612yes—
252C22×C3C2×GO+(4,𝔽2)11yescharacteristic
253C3×C4—91no—
254C3×C4—91no—
255C3×C4—92no—
256C3×C4—122no—
257C3×C4—124no—
258C3×C4—122no—
259C3×C4—362no—
260Dic3—42no—
261Dic3—62no—
262Dic3—62no—
263Dic3—61no—
264Dic3—61no—
265Dic3—61no—
266Dic3—61no—
267Dic3—122no—
268Dic3—124no—
269Dic3—122no—
270C32—21no—
271C32—21no—
272C32—41no—
273C32—41no—
274C32C2×#80711yescharacteristic, complemented
275C23D32⋊D311yescharacteristic
276C2×C4—62no—
277C2×C4—91no—
278C2×C4—91no—
279C2×C4—92no—
280C2×C4—92no—
281C2×C4—122no—
282C2×C4—182no—
283C2×C4—362no—
284C8—274no—
285C2×C3—24no—
286C2×C3—24no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—21no—
292C2×C3—21no—
293C2×C3—44no—
294C2×C3—44no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—41no—
298C2×C3—41no—
299C2×C3—41no—
300C2×C3—41no—
301C2×C3#6374314yescomplemented
302C2×C3C2×#377311yescharacteristic
303C2×C3C2×#377411yescharacteristic
304C2×C3C2×#377611yescharacteristic
305C22#8893412yes—
306C22#8893512yes—
307C22#8893812yes—
308C22C2×D32⋊D311yescharacteristic
309C4—91no—
310C4—91no—
311C4—92no—
312C4—122no—
313C4—362no—
314C3—21no—
315C3—21no—
316C3—41no—
317C3—41no—
318C3C2×#6374311yescharacteristic, complemented
319C2#1172277914yesdirect product factor
320C2C2×#8893411yescharacteristic
321C2C2×#8893511yescharacteristic
322C2C2×#8893811yescharacteristic
subgroupG/H
center, hypercenterC23D32⋊D3
commutatorC3×C3⋊Dic3C22×C4
FrattiniC22C2×D32⋊D3
Fitting, socle, prime radicalC23×C33D4
cosocleC2×C3×C3⋊Dic3C23
hypocenterC33C2×#107
LCSG ▷ C3×C3⋊Dic3 ▷ C2×C33 ▷ C33
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
4. #11722779C2direct (N × H)
9. C2×C3⋊#622C4semidirect
30. C3⋊#622C2×C4semidirect
103. C2×C33#107semidirect
169. C33C2×#107semidirect
220. C2×C32#807semidirect
274. C32C2×#807semidirect
301. C2×C3#63743semidirect
318. C3C2×#63743semidirect
319. C2#11722779direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3⋊#6222
C2×#888152
C2×C3×#37242
#117227792
C2×#637433
C2×#8079