Finite Group Databasebeta

← C3×#10590029 #13154879

Subgroups — C3×#10590029 #13154879

Order 1728.

372 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 372 conjugacy classes of subgroups — too many to draw. Only its 58 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
1C3×C4×#625C211yesmaximal, characteristic
2C3×C4×#622C211yesmaximal, characteristic, complemented
3C3×#63572C211yesmaximal, characteristic, complemented
4#10590029C311yesmaximal, characteristic, direct product factor
5C2×C32×C4×D3—21no—
6C2×C3×#625C411yescharacteristic
7C2×C3×#622—21no—
8C3×C4×C3⋊Dic3C2211yescharacteristic
9C2×C32×C3⋊C8—21no—
10C3×#3724C411yescharacteristic
11C3×(C2×D3)⋊Dic3—21no—
12C3×C4×#67—61no—
13C4×#625C2×C311yescharacteristic
14C4×#622C2×C311yescharacteristic, complemented
15C3×Dic3⋊C8—61no—
16#63572C2×C311yescharacteristic, complemented
17C22×C32×D3—21no—
18C2×C33×C4D411yescharacteristic
19C2×C3×C3⋊Dic3C2×C411yescharacteristic
20C2×C3×C3⋊Dic3D411yescharacteristic
21C2×C32×Dic3—21no—
22C32×C4×D3—41no—
23C32×C4×D3—41no—
24C3×#625—21no—
25C3×#625C812yescomplemented
26C3×#622—22no—
27C32×C3⋊C8—41no—
28C3×D4⋊C8—91nomaximal
29C22×C32×C4—61no—
30C2×C3×#67—61no—
31C2×C3×C4×D3—21no—
32C2×C3×C4×D3—21no—
33C2×C3×C4×D3—41no—
34C2×#625C3×C411yescharacteristic
35C2×#622—21no—
36C2×C32×C8—61no—
37C4×C3⋊Dic3C22×C311yescharacteristic
38C3×#217—61no—
39C3×(C2×D3)⋊C4—61no—
40C3×Dic3⋊C4—61no—
41C3×C4×Dic3—61no—
42C3×C4×Dic3—61no—
43C2×C3×C3⋊C8—21no—
44C2×C3×C3⋊C8—21no—
45C2×C3×C3⋊C8—41no—
46#3724C3×C411yescharacteristic
47(C2×D3)⋊Dic3—21no—
48C22×C33C22⋊C411yescharacteristic
49C2×C32×D3—22no—
50C2×C32×D3—41no—
51C2×C32×D3—41no—
52C33×C4D811yescharacteristic
53C33×C4SD1611yescharacteristic
54C3×C3⋊Dic3—21no—
55C3×C3⋊Dic3C2×C811yescharacteristic
56C3×C3⋊Dic3M1611yescharacteristic
57C32×Dic3—41no—
58C3×C4×D4—91no—
59C4×#67—61no—
60C3×C4⋊C8—91no—
61C3×C4×C8—91no—
62Dic3⋊C8—61no—
63C23×C32—61no—
64C22×C3×D3—21no—
65C22×C3×D3—21no—
66C22×C3×D3—41no—
67C2×C32×C4—21no—
68C2×C32×C4—21no—
69C2×C32×C4—41no—
70C2×C32×C4—41no—
71C2×C32×C4—61no—
72C2×C32×C4—121no—
73C2×C32×C4—121no—
74C2×C32×C4C3×D411yescharacteristic
75C2×C3⋊Dic3C2×C3×C411yescharacteristic
76C2×C3⋊Dic3C3×D411yescharacteristic
77C3×#67—62no—
78C3×#67—62no—
79C2×C3×Dic3—21no—
80C2×C3×Dic3—21no—
81C2×C3×Dic3—41no—
82C2×C3×Dic3—61no—
83C2×C3×Dic3—61no—
84C2×C3×Dic3—61no—
85C2×C3×Dic3—61no—
86C3×C4×D3—41no—
87C3×C4×D3—41no—
88C3×C4×D3—42no—
89C3×C4×D3—42no—
90C3×C4×D3—41no—
91C3×C4×D3—41no—
92#625—21no—
93#625C3×C812yescomplemented
94#622—22no—
95C32×C8—121no—
96C3×C3⋊C8—41no—
97C3×C3⋊C8—42no—
98C3×C3⋊C8—41no—
99D4⋊C8—91noSylow
100C2×C33C22⋊C811yescharacteristic
101C2×C33#10211yescharacteristic
102C2×C33#10411yescharacteristic
103C32×D3—42no—
104C2×C3×D4—91no—
105C22×C3×C4—61no—
106C22×C3×C4—61no—
107C22×C3×C4—121no—
108C2×#67—61no—
109C2×C4×D3—21no—
110C2×C3×C8—61no—
111C2×C3×C8—61no—
112C2×C3×C8—121no—
113C2×C3×C8—181no—
114C3×C4⋊C4—91no—
115C3×C22⋊C4—181no—
116C3×C42—91no—
117#217—61no—
118(C2×D3)⋊C4—61no—
119Dic3⋊C4—61no—
120C4×Dic3—61no—
121C4×Dic3—61no—
122C2×C3⋊C8—21no—
123C22×C32—21no—
124C22×C32—21no—
125C22×C32—41no—
126C22×C32—41no—
127C22×C32—62no—
128C22×C32—121no—
129C22×C32—121no—
130C22×C32C3×C22⋊C411yescharacteristic
131C2×C3×D3—22no—
132C2×C3×D3—22no—
133C2×C3×D3—42no—
134C2×C3×D3—41no—
135C2×C3×D3—41no—
136C2×C3×D3—42no—
137C2×C3×D3—42no—
138C2×C3×D3—41no—
139C2×C3×D3—41no—
140C32×C4—21no—
141C32×C4—21no—
142C32×C4—21no—
143C32×C4—21no—
144C32×C4—41no—
145C32×C4—41no—
146C32×C4—41no—
147C32×C4—41no—
148C32×C4—121no—
149C32×C4C3×D811yescharacteristic
150C32×C4C3×SD1611yescharacteristic
151C3⋊Dic3—21no—
152C3⋊Dic3C2×C3×C811yescharacteristic
153C3⋊Dic3C3×M1611yescharacteristic
154C3×Dic3—41no—
155C3×Dic3—42no—
156C3×Dic3—41no—
157C3×Dic3—61no—
158C3×Dic3—61no—
159C3×Dic3—61no—
160C3×Dic3—61no—
161C3×Dic3—121no—
162C3×Dic3—121no—
163C4×D4—91no—
164C4⋊C8—91no—
165C4×C8—91no—
166C33D4⋊C811yescharacteristic, complemented, Sylow
167C23×C3—61no—
168C23×C3—61no—
169C23×C3—121no—
170C22×D3—21no—
171C3×D4—92no—
172C3×D4—181no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—41no—
176C2×C3×C4—41no—
177C2×C3×C4—61no—
178C2×C3×C4—61no—
179C2×C3×C4—91no—
180C2×C3×C4—91no—
181C2×C3×C4—121no—
182C2×C3×C4—121no—
183C2×C3×C4—121no—
184C2×C3×C4—122no—
185C2×C3×C4—122no—
186C2×C3×C4—121no—
187C2×C3×C4—121no—
188C2×C3×C4GO+(4,𝔽2)11yescharacteristic
189#67—62no—
190#67—62no—
191C2×Dic3—21no—
192C2×Dic3—61no—
193C2×Dic3—61no—
194C2×Dic3—61no—
195C2×Dic3—61no—
196C4×D3—41no—
197C4×D3—41no—
198C3×C8—121no—
199C3×C8—122no—
200C3×C8—121no—
201C3×C8—182no—
202C3⋊C8—41no—
203C2×C32—21no—
204C2×C32—21no—
205C2×C32—21no—
206C2×C32—21no—
207C2×C32—21no—
208C2×C32—21no—
209C2×C32—41no—
210C2×C32—41no—
211C2×C32—41no—
212C2×C32—41no—
213C2×C32—41no—
214C2×C32—41no—
215C2×C32—122no—
216C2×C32C3×C22⋊C811yescharacteristic
217C2×C32C3×#10211yescharacteristic
218C2×C32C3×#10411yescharacteristic
219C3×D3—42no—
220C3×D3—44no—
221C3×D3—42no—
222C2×D4—91no—
223C22×C4—61no—
224C2×C8—61no—
225C2×C8—181no—
226C4⋊C4—91no—
227C22⋊C4—181no—
228C42—91no—
229C22×C3—21no—
230C22×C3—21no—
231C22×C3—41no—
232C22×C3—41no—
233C22×C3—62no—
234C22×C3—62no—
235C22×C3—122no—
236C22×C3—121no—
237C22×C3—121no—
238C22×C3—122no—
239C22×C3—122no—
240C22×C3—121no—
241C22×C3—121no—
242C22×C3D32⋊C411yescharacteristic
243C2×D3—22no—
244C2×D3—41no—
245C2×D3—41no—
246C3×C4—21no—
247C3×C4—21no—
248C3×C4—21no—
249C3×C4—21no—
250C3×C4—41no—
251C3×C4—41no—
252C3×C4—41no—
253C3×C4—41no—
254C3×C4—91no—
255C3×C4—91no—
256C3×C4—121no—
257C3×C4—122no—
258C3×C4—121no—
259C3×C4—181no—
260C3×C4#377411yescharacteristic
261C3×C4#377511yescharacteristic
262Dic3—41no—
263Dic3—61no—
264Dic3—61no—
265Dic3—61no—
266Dic3—61no—
267Dic3—121no—
268Dic3—121no—
269C32—21no—
270C32—21no—
271C32—41no—
272C32—41no—
273C32C3×D4⋊C811yescharacteristic, complemented
274C23—61no—
275D4—92no—
276D4—181no—
277C2×C4—61no—
278C2×C4—91no—
279C2×C4—91no—
280C2×C4—121no—
281C2×C4—121no—
282C2×C4C3×GO+(4,𝔽2)11yescharacteristic
283C8—121no—
284C8—182no—
285C2×C3—21no—
286C2×C3—21no—
287C2×C3—21no—
288C2×C3—21no—
289C2×C3—21no—
290C2×C3—21no—
291C2×C3—41no—
292C2×C3—41no—
293C2×C3—41no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—122no—
298C2×C3—124no—
299C2×C3—122no—
300C2×C3D32⋊C811yescharacteristic
301C2×C3#6373111yescharacteristic
302C2×C3#6373911yescharacteristic
303D3—42no—
304C22—62no—
305C22—121no—
306C22—121no—
307C22C3×D32⋊C411yescharacteristic
308C4—91no—
309C4—91no—
310C4—121no—
311C4—181no—
312C4C3×#377411yescharacteristic
313C4C3×#377511yescharacteristic
314C3—21no—
315C3—21no—
316C3—41no—
317C3—41no—
318C3#1059002911yescharacteristic, direct product factor
319C2—122no—
320C2C3×D32⋊C811yescharacteristic
321C2C3×#6373111yescharacteristic
322C2C3×#6373911yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4GO+(4,𝔽2)
commutatorC3⋊Dic3C2×C3×C8
FrattiniC2×C4C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C22⋊C4
prime radicalC2×C3×#625C4
cosocleC4×C3⋊Dic3C22×C3
hypocenterC32C3×D4⋊C8
LCSG ▷ C3⋊Dic3 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
2. C3×C4×#622C2semidirect
3. C3×#63572C2semidirect
4. #10590029C3direct (N × H)
14. C4×#622C2×C3semidirect
16. #63572C2×C3semidirect
25. C3×#625C8semidirect
93. #625C3×C8semidirect
166. C33D4⋊C8semidirect
273. C32C3×D4⋊C8semidirect
318. C3#10590029direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×#6252
C3×C4×#6222
C3×#635722
#105900293
C3×D4⋊C89