Finite Group Databasebeta

← C3×#71253 #11780732

Subgroups — C3×#71253 #11780732

Order 1152.

413 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 90 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×#5699C211yesmaximal, characteristic
2C32×#491C211yesmaximal, characteristic, complemented
3C3×#5089C211yesmaximal, characteristic
4C3×#5081C211yesmaximal, characteristic, complemented
5C3×#4543C211yesmaximal, characteristic, complemented
6C3×#4465C211yesmaximal, characteristic, complemented
7C3×#4464C211yesmaximal, characteristic
8C3×#1664—31nomaximal
9#71253C311yesmaximal, characteristic, direct product factor
10C32×C4⋊Q8C2211yescharacteristic
11C32×#125C2211yescharacteristic
12C3×#898—21no—
13C3×#898—21no—
14C3×#897—21no—
15C3×#896—21no—
16C3×#894—21no—
17C3×#888—21no—
18C3×#875C2211yescharacteristic
19C3×#870C2211yescharacteristic
20C32×#103—21no—
21C32×#103—21no—
22C32×#97C2211yescharacteristic
23C3×#810—21no—
24C3×#809—21no—
25C3×#808—21no—
26C3×#807—21no—
27C3×#803C2211yescharacteristic
28C3×#803C2211yescharacteristic
29C3×#563—31no—
30#5699C2×C311yescharacteristic
31C3×#501—31no—
32C3×#491—21no—
33C3×#491D311yescharacteristic, complemented
34C3×#491C2×C311yescharacteristic, complemented
35C3×#489—31no—
36#5089C2×C311yescharacteristic
37#5081C2×C311yescharacteristic, complemented
38C3×#330—31no—
39C3×#330—31no—
40C3×#329—31no—
41#4543C2×C311yescharacteristic, complemented
42#4465C2×C311yescharacteristic, complemented
43#4464C2×C311yescharacteristic
44C2×C32×Q8—21no—
45C2×C3×D12D411yescharacteristic
46C2×C3×C4×D3—21no—
47C2×C3×Dic6—21no—
48C2×C3×Dic6D411yescharacteristic
49C2×C32×C8—21no—
50C32×C4⋊C4—21no—
51C32×C4⋊C4—21no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4—21no—
54C32×C4⋊C4D411yescharacteristic
55C32×C4⋊C4D411yescharacteristic
56C32×C42C2311yescharacteristic
57C3×(C2×D3)⋊C4—21no—
58C3×(C2×D3)⋊C4—21no—
59C3×(C2×D3)⋊C4—21no—
60C3×(C2×D3)⋊C4—21no—
61C3×(C2×D3)⋊C4—21no—
62C3×C4⋊Dic3—21no—
63C3×C4⋊Dic3D411yescharacteristic
64C3×C4⋊Dic3D411yescharacteristic
65C3×Dic3⋊C4—21no—
66C3×Dic3⋊C4—21no—
67C3×Dic3⋊C4—21no—
68C3×C4×Dic3—21no—
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8—21no—
71#1664—31noSylow
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8—61no—
74C3×C4⋊Q8C2×D311yescharacteristic
75C3×C4⋊Q8C22×C311yescharacteristic
76C3×#126—61no—
77C3×#126—61no—
78C3×#125—21no—
79C3×#125—31no—
80C3×#125C2×D311yescharacteristic
81C3×#125C22×C311yescharacteristic
82C3×#124—31no—
83C3×#123—61no—
84C3×C22⋊Q8—61no—
85C3×C22⋊Q8—61no—
86#898—21no—
87#898—21no—
88#897—21no—
89#896—21no—
90#894—21no—
91#888—21no—
92#875C22×C311yescharacteristic
93#870C22×C311yescharacteristic
94C3×#107—61no—
95C3×#106—61no—
96C3×#103—22no—
97C3×#103—22no—
98C3×#103—21no—
99C3×#103—21no—
100C3×#103—21no—
101C3×#103—21no—
102C3×#103—61no—
103C3×#102—61no—
104C3×#97—21no—
105C3×#97—31no—
106C3×#97—31no—
107C3×#97C2×D311yescharacteristic
108C3×#97C22×C311yescharacteristic
109#810—21no—
110#809—21no—
111#808—21no—
112#807—21no—
113#803C22×C311yescharacteristic
114#803C22×C311yescharacteristic
115C22×C3×D3—21no—
116C32×Q8—42no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—41no—
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×D411yescharacteristic
123C2×C3×Dic3—21no—
124C2×C3×Dic3—21no—
125C2×C3×Dic3—21no—
126C3×D12—41no—
127C3×C4×D3—81no—
128C3×Dic6—41no—
129C3×Dic6—81no—
130C32×C8—41no—
131C3×C3⋊C8—22no—
132C3×C3⋊C8—41no—
133#563—31no—
134#501—31no—
135#491C3×D311yescharacteristic, complemented
136#489—31no—
137#330—31no—
138#330—31no—
139#329—31no—
140C2×C3×Q8—22no—
141C2×C3×Q8—21no—
142C2×C3×Q8—21no—
143C2×C3×Q8—31no—
144C2×C3×Q8—61no—
145C2×C3×D4—31no—
146C22×C3×C4—61no—
147C2×D12C3×D411yescharacteristic
148C2×C4×D3—21no—
149C2×Dic6—21no—
150C2×Dic6C3×D411yescharacteristic
151C2×C3×C8—22no—
152C2×C3×C8—21no—
153C2×C3×C8—21no—
154C2×C3×C8—61no—
155C2×C3×C8—61no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4—21no—
161C3×C4⋊C4—21no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—31no—
168C3×C4⋊C4—41no—
169C3×C4⋊C4—41no—
170C3×C4⋊C4—41no—
171C3×C4⋊C4—41no—
172C3×C4⋊C4—61no—
173C3×C4⋊C4—61no—
174C3×C4⋊C4—61no—
175C3×C4⋊C4—61no—
176C3×C4⋊C4#6711yescharacteristic
177C3×C4⋊C4#6711yescharacteristic
178C3×C4⋊C4C3×D411yescharacteristic
179C3×C4⋊C4C3×D411yescharacteristic
180C3×C22⋊C4—61no—
181C3×C22⋊C4—61no—
182C3×C22⋊C4—61no—
183C3×C22⋊C4—61no—
184C3×C22⋊C4—61no—
185C3×C42—21no—
186C3×C42—61no—
187C3×C42C22×D311yescharacteristic
188C3×C42C23×C311yescharacteristic
189(C2×D3)⋊C4—21no—
190(C2×D3)⋊C4—21no—
191(C2×D3)⋊C4—21no—
192(C2×D3)⋊C4—21no—
193(C2×D3)⋊C4—21no—
194C4⋊Dic3—21no—
195C4⋊Dic3C3×D411yescharacteristic
196C4⋊Dic3C3×D411yescharacteristic
197Dic3⋊C4—21no—
198Dic3⋊C4—21no—
199Dic3⋊C4—21no—
200C4×Dic3—21no—
201C2×C3⋊C8—21no—
202C2×C3⋊C8—21no—
203C22×C32#12011yescharacteristic
204C2×C3×D3—41no—
205C2×C3×D3—41no—
206C2×C3×D3—41no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—41no—
211C32×C4—41no—
212C32×C4—81no—
213C3×Dic3—41no—
214C3×Dic3—41no—
215C3×Dic3—41no—
216C4⋊Q8—61no—
217C4⋊Q8C2×C3×D311yescharacteristic
218#126—61no—
219#126—61no—
220#125—31no—
221#125C2×C3×D311yescharacteristic
222#124—31no—
223#123—61no—
224C22⋊Q8—61no—
225C22⋊Q8—61no—
226#107—61no—
227#106—61no—
228#103—21no—
229#103—21no—
230#103—61no—
231#102—61no—
232#97—31no—
233#97—31no—
234#97C2×C3×D311yescharacteristic
235C23×C3—61no—
236C22×D3—21no—
237C3×Q8—44no—
238C3×Q8—42no—
239C3×Q8—42no—
240C3×Q8—121no—
241C3×Q8—241no—
242C3×D4—121no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—61no—
256C2×C3×C4—61no—
257C2×C3×C4—61no—
258C2×C3×C4—241no—
259C2×C3×C4D3×D411yescharacteristic
260C2×C3×C4D3×D411yescharacteristic
261C2×C3×C4C2×#6711yescharacteristic
262C2×C3×C4C2×C3×D411yescharacteristic
263C2×C3×C4C2×C3×D411yescharacteristic
264C2×C3×C4C2×C3×D411yescharacteristic
265C2×Dic3—21no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268D12—41no—
269C4×D3—81no—
270Dic6—41no—
271Dic6—81no—
272C3×C8—42no—
273C3×C8—41no—
274C3×C8—41no—
275C3×C8—62no—
276C3×C8—121no—
277C3⋊C8—22no—
278C3⋊C8—41no—
279C2×C32#45411yescharacteristic
280C2×C32#45511yescharacteristic
281C2×C32#45611yescharacteristic
282C3×D3—81no—
283C2×Q8—21no—
284C2×Q8—31no—
285C2×Q8—61no—
286C2×D4—31no—
287C22×C4—61no—
288C2×C8—21no—
289C2×C8—61no—
290C2×C8—61no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—21no—
294C4⋊C4—21no—
295C4⋊C4—31no—
296C4⋊C4—31no—
297C4⋊C4—61no—
298C4⋊C4—61no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C4⋊C4C3×#6711yescharacteristic
302C4⋊C4C3×#6711yescharacteristic
303C22⋊C4—61no—
304C22⋊C4—61no—
305C22⋊C4—61no—
306C22⋊C4—61no—
307C22⋊C4—61no—
308C42—61no—
309C42C22×C3×D311yescharacteristic
310C22×C3—21no—
311C22×C3—121no—
312C22×C3—121no—
313C22×C3—121no—
314C22×C3#93711yescharacteristic
315C22×C3C3×#12011yescharacteristic
316C2×D3—41no—
317C2×D3—41no—
318C2×D3—41no—
319C3×C4—22no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—41no—
327C3×C4—41no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—41no—
331C3×C4—41no—
332C3×C4—81no—
333C3×C4—81no—
334C3×C4—82no—
335C3×C4—81no—
336C3×C4—81no—
337C3×C4—121no—
338C3×C4—121no—
339C3×C4—121no—
340Dic3—41no—
341Dic3—41no—
342Dic3—41no—
343C32#166411yescharacteristic, complemented, Sylow
344C23—61no—
345Q8—42no—
346Q8—121no—
347Q8—241no—
348D4—121no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—41no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—241no—
356C2×C4C3×D3×D411yescharacteristic
357C2×C4C3×D3×D411yescharacteristic
358C2×C4C2×C3×#6711yescharacteristic
359C8—41no—
360C8—62no—
361C8—121no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3—21no—
365C2×C3—241no—
366C2×C3#506211yescharacteristic
367C2×C3#520211yescharacteristic
368C2×C3#520511yescharacteristic
369C2×C3C3×#45411yescharacteristic
370C2×C3C3×#45511yescharacteristic
371C2×C3C3×#45611yescharacteristic
372D3—81no—
373C22—121no—
374C22—121no—
375C22—121no—
376C22C3×#93711yescharacteristic
377C4—21no—
378C4—21no—
379C4—21no—
380C4—41no—
381C4—41no—
382C4—81no—
383C4—121no—
384C4—121no—
385C4—121no—
386C3—21no—
387C3#7125311yescharacteristic, direct product factor
388C3C3×#166411yescharacteristic, complemented
389C2—241no—
390C2C3×#506211yescharacteristic
391C2C3×#520211yescharacteristic
392C2C3×#520511yescharacteristic
subgroupG/H
centerC22×C3#937
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#491C2
socleC22×C32#120
prime radicalC2×C3×D12D4
hypercenterC3×#491D3
hypocenterC3C3×#1664
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#491

Complements

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

NH ≅ G/Ntype
2. C32×#491C2semidirect
4. C3×#5081C2semidirect
5. C3×#4543C2semidirect
6. C3×#4465C2semidirect
9. #71253C3direct (N × H)
33. C3×#491D3semidirect
34. C3×#491C2×C3semidirect
37. #5081C2×C3semidirect
41. #4543C2×C3semidirect
42. #4465C2×C3semidirect
135. #491C3×D3semidirect
343. C32#1664semidirect
387. C3#71253direct (N × H)
388. C3C3×#1664semidirect

Maximal subgroups

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

subgroupindex
C3×#56992
C32×#4912
C3×#50892
C3×#50812
C3×#45432
C3×#44652
C3×#44642
C3×#16643
#712533