Finite Group Databasebeta

← C3×#69719 #11778805

Subgroups — C3×#69719 #11778805

Order 1152.

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

normal subgroups only

This group has 471 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
2C32×#400C211yesmaximal, characteristic
3C3×#4970C212yesmaximal
4C3×#4528C211yesmaximal, characteristic
5C3×#4452C212yesmaximal
6C3×#1617—31nomaximal
7#69719C311yesmaximal, characteristic, direct product factor
8C2×C32×C4⋊C4C2211yescharacteristic
9C2×C32×C4⋊C4C2211yescharacteristic
10C3×#926—21no—
11C3×#924—21no—
12C3×#924C2212yes—
13C3×#890—22no—
14C3×#888—22no—
15C32×#95—21no—
16C32×#95—21no—
17C32×#95C2211yescharacteristic
18C3×#830C2212yes—
19C3×#810—22no—
20C3×#807—22no—
21C3×#544—31no—
22#5500C2×C311yescharacteristic
23C3×#484—32no—
24C3×#400—21no—
25C3×#400D311yescharacteristic
26C3×#400C2×C311yescharacteristic
27#4970C2×C312yes—
28C3×#328—32no—
29C3×#324—31no—
30#4528C2×C311yescharacteristic
31#4452C2×C312yes—
32C22×C32×C4—21no—
33C22×C32×C4—21no—
34C22×C32×C4D411yescharacteristic
35C22×C32×C4D411yescharacteristic
36C22×C32×C4C2311yescharacteristic
37C2×C3×Dic6—21no—
38C2×C3×Dic6D412yes—
39C32×C4⋊C4—22no—
40C32×C4⋊C4—22no—
41C3×#217—22no—
42C3×C4⋊Dic3—21no—
43C3×C4⋊Dic3—21no—
44C3×C4⋊Dic3—21no—
45C3×C4⋊Dic3D412yes—
46C3×Dic3⋊C4—22no—
47C3×Dic3⋊C4—22no—
48C3×C4×Dic3—22no—
49C2×C3×C3⋊C8—22no—
50#1617—31noSylow
51C3×C4⋊Q8—62no—
52C3×#125—62no—
53C3×C22⋊Q8—32no—
54C3×C22⋊Q8—61no—
55C3×#117—61no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4C2×D311yescharacteristic
59C2×C3×C4⋊C4C2×D311yescharacteristic
60C2×C3×C4⋊C4C22×C311yescharacteristic
61C2×C3×C4⋊C4C22×C311yescharacteristic
62#926—21no—
63#924—21no—
64#924C22×C312yes—
65#890—22no—
66#888—22no—
67C3×#107—62no—
68C3×#103—62no—
69C3×C22⋊C8—32no—
70C3×#95—22no—
71C3×#95—22no—
72C3×#95—21no—
73C3×#95—21no—
74C3×#95—21no—
75C3×#95—21no—
76C3×#95—21no—
77C3×#95C2×D311yescharacteristic
78C3×#95C22×C311yescharacteristic
79#830C22×C312yes—
80#810—22no—
81#807—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×C4—41no—
89C2×C32×C4—41no—
90C2×C32×C4—42no—
91C2×C32×C4—41no—
92C2×C32×C4—41no—
93C2×C32×C4Q1612yes—
94C2×C32×C4C2×D412yes—
95C2×C3×Dic3—22no—
96C2×C3×Dic3—22no—
97C3×Dic6—42no—
98C3×Dic6—81no—
99C3×C3⋊C8—42no—
100#544—31no—
101#484—32no—
102#400C3×D311yescharacteristic
103#328—32no—
104#324—31no—
105C2×C3×Q8—32no—
106C2×C3×Q8—61no—
107C22×C3×C4—22no—
108C22×C3×C4—22no—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—21no—
116C22×C3×C4#6711yescharacteristic
117C22×C3×C4#6711yescharacteristic
118C22×C3×C4C3×D411yescharacteristic
119C22×C3×C4C3×D411yescharacteristic
120C22×C3×C4C22×D311yescharacteristic
121C22×C3×C4C23×C311yescharacteristic
122C2×Dic6—21no—
123C2×Dic6C3×D412yes—
124C2×C3×C8—62no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—32no—
130C3×C4⋊C4—42no—
131C3×C4⋊C4—42no—
132C3×C4⋊C4—62no—
133C3×C4⋊C4—61no—
134C3×C4⋊C4—61no—
135C3×C4⋊C4—61no—
136C3×C4⋊C4—62no—
137C3×C22⋊C4—62no—
138C3×C42—62no—
139#217—22no—
140C4⋊Dic3—21no—
141C4⋊Dic3—21no—
142C4⋊Dic3—21no—
143C4⋊Dic3C3×D412yes—
144Dic3⋊C4—22no—
145Dic3⋊C4—22no—
146C4×Dic3—22no—
147C2×C3⋊C8—22no—
148C22×C32—21no—
149C22×C32—21no—
150C22×C32#12011yescharacteristic
151C22×C32C2×Q1611yescharacteristic
152C22×C32#13711yescharacteristic
153C32×C4—22no—
154C32×C4—22no—
155C32×C4—41no—
156C32×C4—81no—
157C32×C4—81no—
158C3×Dic3—42no—
159C3×Dic3—42no—
160C4⋊Q8—62no—
161#125—62no—
162C22⋊Q8—32no—
163C22⋊Q8—61no—
164#117—61no—
165C2×C4⋊C4C2×C3×D311yescharacteristic
166C2×C4⋊C4C2×C3×D311yescharacteristic
167#107—62no—
168#103—62no—
169C22⋊C8—32no—
170#95—21no—
171#95—21no—
172#95C2×C3×D311yescharacteristic
173C23×C3—21no—
174C23×C3C2×#6711yescharacteristic
175C23×C3C2×C3×D411yescharacteristic
176C3×Q8—122no—
177C3×Q8—241no—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—22no—
185C2×C3×C4—22no—
186C2×C3×C4—41no—
187C2×C3×C4—41no—
188C2×C3×C4—41no—
189C2×C3×C4—41no—
190C2×C3×C4—41no—
191C2×C3×C4—42no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—42no—
195C2×C3×C4—42no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—44no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—42no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—41no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—62no—
211C2×C3×C4—62no—
212C2×C3×C4#21612yes—
213C2×C3×C4C3×Q1612yes—
214C2×C3×C4D3×D412yes—
215C2×C3×C4C2×C3×D412yes—
216C2×Dic3—22no—
217C2×Dic3—22no—
218Dic6—42no—
219Dic6—81no—
220C3×C8—122no—
221C3⋊C8—42no—
222C2×C32—21no—
223C2×C32—21no—
224C2×C32#45111yescharacteristic
225C2×C32#45411yescharacteristic
226C2×C32#45811yescharacteristic
227C2×Q8—32no—
228C2×Q8—61no—
229C22×C4—21no—
230C22×C4—21no—
231C22×C4C3×#6711yescharacteristic
232C22×C4C3×#6711yescharacteristic
233C22×C4C22×C3×D311yescharacteristic
234C2×C8—62no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—32no—
238C4⋊C4—61no—
239C4⋊C4—61no—
240C4⋊C4—61no—
241C4⋊C4—62no—
242C4⋊C4—62no—
243C22⋊C4—62no—
244C42—62no—
245C22×C3—22no—
246C22×C3—22no—
247C22×C3—21no—
248C22×C3—21no—
249C22×C3—21no—
250C22×C3—21no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—21no—
254C22×C3#93711yescharacteristic
255C22×C3C2×#21611yescharacteristic
256C22×C3#95111yescharacteristic
257C22×C3C3×#12011yescharacteristic
258C22×C3C2×C3×Q1611yescharacteristic
259C22×C3C3×#13711yescharacteristic
260C3×C4—22no—
261C3×C4—22no—
262C3×C4—22no—
263C3×C4—22no—
264C3×C4—42no—
265C3×C4—42no—
266C3×C4—41no—
267C3×C4—41no—
268C3×C4—81no—
269C3×C4—82no—
270C3×C4—81no—
271C3×C4—81no—
272C3×C4—82no—
273C3×C4—81no—
274C3×C4—81no—
275C3×C4—122no—
276C3×C4—122no—
277Dic3—42no—
278Dic3—42no—
279C32#161711yescharacteristic, complemented, Sylow
280C23C2×C3×#6711yescharacteristic
281Q8—122no—
282Q8—241no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—62no—
294C2×C4—62no—
295C2×C4C3×#21612yes—
296C2×C4C3×D3×D412yes—
297C8—122no—
298C2×C3—22no—
299C2×C3—22no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—21no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—21no—
306C2×C3—21no—
307C2×C3#505011yescharacteristic
308C2×C3#505311yescharacteristic
309C2×C3#506211yescharacteristic
310C2×C3C3×#45111yescharacteristic
311C2×C3C3×#45411yescharacteristic
312C2×C3C3×#45811yescharacteristic
313C22—21no—
314C22—21no—
315C22C3×#93711yescharacteristic
316C22C2×C3×#21611yescharacteristic
317C22C3×#95111yescharacteristic
318C4—22no—
319C4—22no—
320C4—41no—
321C4—81no—
322C4—81no—
323C4—122no—
324C4—122no—
325C3—21no—
326C3#6971911yescharacteristic, direct product factor
327C3C3×#161711yescharacteristic, complemented
328C2—21no—
329C2—21no—
330C2C3×#505011yescharacteristic
331C2C3×#505311yescharacteristic
332C2C3×#506211yescharacteristic
subgroupG/H
centerC22×C3#937
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#400C2
socleC22×C32#120
prime radicalC23×C32C2×D4
hypercenterC3×#400D3
hypocenterC3C3×#1617
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#400

Complements

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

NH ≅ G/Ntype
7. #69719C3direct (N × H)
279. C32#1617semidirect
326. C3#69719direct (N × H)
327. C3C3×#1617semidirect

Maximal subgroups

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

subgroupindex
C3×#55002
C32×#4002
C3×#49702
C3×#45282
C3×#44522
C3×#16173
#697193