Finite Group Databasebeta

← C3×#68139 #11778362

Subgroups — C3×#68139 #11778362

Order 1152.

515 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 106 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×#5500C211yesmaximal, characteristic
2C3×#4979C211yesmaximal, characteristic
3C3×#4739C212yesmaximal
4C32×#324C211yesmaximal, characteristic
5C3×#4467C212yesmaximal
6C3×#1605—31nomaximal
7#68139C311yesmaximal, characteristic, direct product factor
8C22×C3×Dic6C2211yescharacteristic
9C2×C32×C4⋊C4C2211yescharacteristic
10C3×#926—21no—
11C3×#924—21no—
12C3×#924C2212yes—
13C2×C3×Dic3⋊C4—21no—
14C2×C3×Dic12—22no—
15C3×#890—22no—
16C3×#888—22no—
17C32×C22⋊C8C2212yes—
18C3×C3⋊#95—21no—
19C3×C3⋊#95C2211yescharacteristic
20C3×#816—22no—
21C3×#544—31no—
22#5500C2×C311yescharacteristic
23C3×#451—32no—
24C3×#395—31no—
25#4979C2×C311yescharacteristic
26#4739C2×C312yes—
27C3×#328—32no—
28C3×#324—21no—
29C3×#324D311yescharacteristic
30C3×#324C2×C311yescharacteristic
31#4467C2×C312yes—
32C22×C32×C4D411yescharacteristic
33C22×C32×C4D411yescharacteristic
34C22×C32×C4C2311yescharacteristic
35C22×C3×Dic3—21no—
36C22×C3×Dic3—21no—
37C2×C3×Dic6—22no—
38C2×C3×Dic6—21no—
39C2×C3×Dic6—42no—
40C2×C3×Dic6D412yes—
41C2×C32×C8—22no—
42C32×C4⋊C4—22no—
43C3×#217—22no—
44C3×C4⋊Dic3—21no—
45C3×C4⋊Dic3—21no—
46C3×C4⋊Dic3—21no—
47C3×C4⋊Dic3D412yes—
48C3×Dic3⋊C4—22no—
49C3×Dic3⋊C4—22no—
50C3×Dic3⋊C4—42no—
51C3×C4×Dic3—22no—
52C3×Dic12—44no—
53#1605—31noSylow
54C22×C3×Q8—31no—
55C22×Dic6C22×C311yescharacteristic
56C2×C3×Q16—62no—
57C3×C4⋊Q8—62no—
58C3×#125—62no—
59C3×C22⋊Q8—32no—
60C3×C22⋊Q8—61no—
61C3×#117—61no—
62C2×C3×C4⋊C4—21no—
63C2×C3×C4⋊C4—61no—
64C2×C3×C4⋊C4C2×D311yescharacteristic
65C2×C3×C4⋊C4C22×C311yescharacteristic
66#926—21no—
67#924—21no—
68#924C22×C312yes—
69C2×Dic3⋊C4—21no—
70C2×Dic12—22no—
71#890—22no—
72#888—22no—
73C3×#103—62no—
74C3×C22⋊C8—22no—
75C3×C22⋊C8C2×D312yes—
76C3×C22⋊C8C22×C312yes—
77C3×#95—31no—
78C3×#95—61no—
79C3⋊#95—21no—
80C3⋊#95C22×C311yescharacteristic
81#816—22no—
82C23×C32C2×D411yescharacteristic
83C2×C32×C4—21no—
84C2×C32×C4—21no—
85C2×C32×C4—21no—
86C2×C32×C4—41no—
87C2×C32×C4—41no—
88C2×C32×C4Q1612yescomplemented
89C2×C32×C4C2×D412yes—
90C2×C3×Dic3—22no—
91C2×C3×Dic3—22no—
92C2×C3×Dic3—22no—
93C2×C3×Dic3—22no—
94C2×C3×Dic3—41no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3—41no—
97C2×C3×Dic3—41no—
98C3×Dic6—24no—
99C3×Dic6—42no—
100C3×Dic6—42no—
101C3×Dic6—81no—
102C32×C8—42no—
103#544—31no—
104#451—32no—
105#395—31no—
106#328—32no—
107#324C3×D311yescharacteristic
108C2×C3×Q8—32no—
109C2×C3×Q8—61no—
110C2×C3×Q8—62no—
111C2×C3×Q8—122no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—61no—
116C22×C3×C4—61no—
117C22×C3×C4D1211yescharacteristic
118C22×C3×C4D1211yescharacteristic
119C22×C3×C4C3×D411yescharacteristic
120C22×C3×C4C3×D411yescharacteristic
121C22×C3×C4C22×D311yescharacteristic
122C22×C3×C4C23×C311yescharacteristic
123C22×Dic3—21no—
124C22×Dic3—21no—
125C2×Dic6—22no—
126C2×Dic6—21no—
127C2×Dic6—42no—
128C2×Dic6C3×D412yes—
129C3×Q16—124no—
130C2×C3×C8—22no—
131C2×C3×C8—22no—
132C2×C3×C8—42no—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—32no—
136C3×C4⋊C4—42no—
137C3×C4⋊C4—62no—
138C3×C4⋊C4—61no—
139C3×C4⋊C4—61no—
140C3×C4⋊C4—61no—
141C3×C4⋊C4—62no—
142C3×C4⋊C4—122no—
143C3×C22⋊C4—62no—
144C3×C42—62no—
145#217—22no—
146C4⋊Dic3—21no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3C3×D412yes—
150Dic3⋊C4—22no—
151Dic3⋊C4—22no—
152Dic3⋊C4—42no—
153C4×Dic3—22no—
154Dic12—44no—
155C22×C32—21no—
156C22×C32—21no—
157C22×C32#12011yescharacteristic
158C22×C32C2×Q1611yescharacteristic
159C22×C32#13711yescharacteristic
160C32×C4—22no—
161C32×C4—22no—
162C32×C4—41no—
163C3×Dic3—42no—
164C3×Dic3—42no—
165C3×Dic3—42no—
166C3×Dic3—42no—
167C22×Q8—31no—
168C2×Q16—62no—
169C4⋊Q8—62no—
170#125—62no—
171C22⋊Q8—32no—
172C22⋊Q8—61no—
173#117—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4C2×C3×D311yescharacteristic
176#103—62no—
177C22⋊C8C2×C3×D312yes—
178#95—31no—
179#95—61no—
180C23×C3—21no—
181C23×C3C2×D1211yescharacteristic
182C23×C3C2×C3×D411yescharacteristic
183C3×Q8—64no—
184C3×Q8—122no—
185C3×Q8—122no—
186C3×Q8—241no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—22no—
194C2×C3×C4—22no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—62no—
203C2×C3×C4—62no—
204C2×C3×C4—62no—
205C2×C3×C4—62no—
206C2×C3×C4—81no—
207C2×C3×C4—81no—
208C2×C3×C4—121no—
209C2×C3×C4—121no—
210C2×C3×C4—121no—
211C2×C3×C4—121no—
212C2×C3×C4Dic1212yescomplemented
213C2×C3×C4C3×Q1612yescomplemented
214C2×C3×C4D3×D412yes—
215C2×C3×C4C2×C3×D412yes—
216C2×Dic3—22no—
217C2×Dic3—22no—
218C2×Dic3—22no—
219C2×Dic3—22no—
220C2×Dic3—41no—
221C2×Dic3—41no—
222C2×Dic3—41no—
223C2×Dic3—41no—
224Dic6—24no—
225Dic6—42no—
226Dic6—42no—
227Dic6—81no—
228C3×C8—42no—
229C3×C8—42no—
230C3×C8—82no—
231C2×C32—21no—
232C2×C32—21no—
233C2×C32#45111yescharacteristic
234C2×C32#45611yescharacteristic
235C2×C32B(4,𝔽2)11yescharacteristic
236C2×Q8—32no—
237C2×Q8—61no—
238C2×Q8—62no—
239C2×Q8—122no—
240C22×C4—61no—
241C22×C4—61no—
242C22×C4C3×D1211yescharacteristic
243C22×C4C3×D1211yescharacteristic
244C22×C4C22×C3×D311yescharacteristic
245Q16—124no—
246C2×C8—22no—
247C4⋊C4—22no—
248C4⋊C4—32no—
249C4⋊C4—61no—
250C4⋊C4—61no—
251C4⋊C4—61no—
252C4⋊C4—62no—
253C4⋊C4—62no—
254C4⋊C4—122no—
255C22⋊C4—62no—
256C42—62no—
257C22×C3—21no—
258C22×C3—21no—
259C22×C3—21no—
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—21no—
263C22×C3—21no—
264C22×C3—41no—
265C22×C3—41no—
266C22×C3#88211yescharacteristic
267C22×C3C2×Dic1211yescharacteristic
268C22×C3#90911yescharacteristic
269C22×C3C3×#12011yescharacteristic
270C22×C3C2×C3×Q1611yescharacteristic
271C22×C3C3×#13711yescharacteristic
272C3×C4—22no—
273C3×C4—22no—
274C3×C4—22no—
275C3×C4—22no—
276C3×C4—42no—
277C3×C4—42no—
278C3×C4—41no—
279C3×C4—41no—
280C3×C4—81no—
281C3×C4—122no—
282C3×C4—122no—
283C3×C4—122no—
284C3×C4—122no—
285Dic3—42no—
286Dic3—42no—
287Dic3—42no—
288Dic3—42no—
289C32#160511yescharacteristic, complemented, Sylow
290C23C2×C3×D1211yescharacteristic
291Q8—64no—
292Q8—122no—
293Q8—122no—
294Q8—241no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—62no—
301C2×C4—62no—
302C2×C4—62no—
303C2×C4—62no—
304C2×C4—121no—
305C2×C4—121no—
306C2×C4—121no—
307C2×C4—121no—
308C2×C4C3×Dic1212yescomplemented
309C2×C4C3×D3×D412yes—
310C8—42no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—21no—
314C2×C3—21no—
315C2×C3—21no—
316C2×C3—21no—
317C2×C3—21no—
318C2×C3—41no—
319C2×C3—41no—
320C2×C3#473911yescharacteristic
321C2×C3#474111yescharacteristic
322C2×C3#475211yescharacteristic
323C2×C3C3×#45111yescharacteristic
324C2×C3C3×#45611yescharacteristic
325C2×C3C3×B(4,𝔽2)11yescharacteristic
326C22—21no—
327C22—21no—
328C22C3×#88211yescharacteristic
329C22C2×C3×Dic1211yescharacteristic
330C22C3×#90911yescharacteristic
331C4—22no—
332C4—22no—
333C4—41no—
334C4—122no—
335C4—122no—
336C4—122no—
337C4—122no—
338C3—21no—
339C3#6813911yescharacteristic, direct product factor
340C3C3×#160511yescharacteristic, complemented
341C2—21no—
342C2—21no—
343C2C3×#473911yescharacteristic
344C2C3×#474111yescharacteristic
345C2C3×#475211yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#324C2
socleC22×C32#120
prime radicalC23×C32C2×D4
hypercenterC3×#324D3
hypocenterC3C3×#1605
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#324

Complements

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

NH ≅ G/Ntype
7. #68139C3direct (N × H)
88. C2×C32×C4Q16semidirect
212. C2×C3×C4Dic12semidirect
213. C2×C3×C4C3×Q16semidirect
289. C32#1605semidirect
308. C2×C4C3×Dic12semidirect
339. C3#68139direct (N × H)
340. C3C3×#1605semidirect

Maximal subgroups

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

subgroupindex
C3×#55002
C3×#49792
C3×#47392
C32×#3242
C3×#44672
C3×#16053
#681393