Finite Group Databasebeta

← C3×#71246 #11780673

Subgroups — C3×#71246 #11780673

Order 1152.

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

normal subgroups only

This group has 442 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×#5684C211yesmaximal, characteristic
2C32×#489C211yesmaximal, characteristic, complemented
3C3×#5089C211yesmaximal, characteristic, complemented
4C3×#5082C211yesmaximal, characteristic, complemented
5C3×#4540C211yesmaximal, characteristic, complemented
6C3×#4465C211yesmaximal, characteristic, complemented
7C3×#4464C211yesmaximal, characteristic, complemented
8C3×#1662—31nomaximal
9#71246C311yesmaximal, characteristic, direct product factor
10C32×C4⋊Q8C2211yescharacteristic
11C32×#124C2211yescharacteristic
12C3×#946—21no—
13C3×#939—21no—
14C3×#886—21no—
15C3×#884—21no—
16C3×#883—21no—
17C3×#878—21no—
18C3×#875C2211yescharacteristic
19C3×#870C2211yescharacteristic
20C32×#103—21no—
21C32×#102—21no—
22C32×#97C2211yescharacteristic
23C3×#835—21no—
24C3×#835—21no—
25C3×#810—21no—
26C3×#809—21no—
27C3×#803C2211yescharacteristic
28C3×#803C2211yescharacteristic
29C3×#562—31no—
30#5684C2×C311yescharacteristic
31C3×#491—31no—
32C3×#489—21no—
33C3×#489—31no—
34C3×#489D311yescharacteristic, complemented
35C3×#489C2×C311yescharacteristic, complemented
36#5089C2×C311yescharacteristic, complemented
37#5082C2×C311yescharacteristic, complemented
38C3×#330—31no—
39C3×#329—31no—
40C3×#329—31no—
41#4540C2×C311yescharacteristic, complemented
42#4465C2×C311yescharacteristic, complemented
43#4464C2×C311yescharacteristic, complemented
44C2×C32×Q8—21no—
45C2×C32×Q8D411yescharacteristic
46C2×C32×D4D411yescharacteristic
47C2×C3×#67—21no—
48C2×C3×#67—21no—
49C22×C3×Dic3—21no—
50C2×C3×D12D411yescharacteristic
51C2×C3×C4×D3—21no—
52C2×C3×Dic6D411yescharacteristic
53C2×C32×C8—21no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4—21no—
56C32×C22⋊C4—21no—
57C32×C22⋊C4—21no—
58C32×C42C2311yescharacteristic
59C3×#217—21no—
60C3×(C2×D3)⋊C4—21no—
61C3×(C2×D3)⋊C4—21no—
62C3×(C2×D3)⋊C4—21no—
63C3×C4⋊Dic3—21no—
64C3×C4⋊Dic3D411yescharacteristic
65C3×C4⋊Dic3D411yescharacteristic
66C3×Dic3⋊C4—21no—
67C3×Dic3⋊C4—21no—
68C3×Dic3⋊C4—21no—
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8—21no—
71#1662—31noSylow
72C3×C4⋊Q8—21no—
73C3×C4⋊Q8C2×D311yescharacteristic
74C3×C4⋊Q8C22×C311yescharacteristic
75C3×#125—31no—
76C3×#124—21no—
77C3×#124—31no—
78C3×#124C2×D311yescharacteristic
79C3×#124C22×C311yescharacteristic
80C3×#123—61no—
81C3×#123—61no—
82C3×C22⋊Q8—61no—
83C3×C22⋊Q8—61no—
84C3×#121—61no—
85C3×#121—61no—
86#946—21no—
87#939—21no—
88#886—21no—
89#884—21no—
90#883—21no—
91#878—21no—
92#875C22×C311yescharacteristic
93#870C22×C311yescharacteristic
94C3×#103—22no—
95C3×#103—21no—
96C3×#103—21no—
97C3×#103—61no—
98C3×#103—61no—
99C3×#103—61no—
100C3×#102—22no—
101C3×#102—21no—
102C3×#102—21no—
103C3×#102—61no—
104C3×#97—21no—
105C3×#97—31no—
106C3×#97—31no—
107C3×#97C2×D311yescharacteristic
108C3×#97C22×C311yescharacteristic
109#835—21no—
110#835—21no—
111#810—21no—
112#809—21no—
113#803C22×C311yescharacteristic
114#803C22×C311yescharacteristic
115C23×C32—21no—
116C22×C3×D3—21no—
117C32×Q8—41no—
118C32×Q8—42no—
119C32×D4—41no—
120C2×C32×C4—21no—
121C2×C32×C4—41no—
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C3×#67—81no—
126C3×#67—81no—
127C2×C3×Dic3—21no—
128C2×C3×Dic3—21no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—81no—
131C3×D12—41no—
132C3×C4×D3—81no—
133C3×Dic6—41no—
134C32×C8—41no—
135C3×C3⋊C8—41no—
136C3×C3⋊C8—41no—
137#562—31no—
138#491—31no—
139#489—31no—
140#489C3×D311yescharacteristic, complemented
141#330—31no—
142#329—31no—
143#329—31no—
144C2×C3×Q8—21no—
145C2×C3×Q8—21no—
146C2×C3×Q8—21no—
147C2×C3×Q8—31no—
148C2×C3×Q8—41no—
149C2×C3×Q8#6711yescharacteristic
150C2×C3×Q8C3×D411yescharacteristic
151C2×C3×D4—21no—
152C2×C3×D4—31no—
153C2×C3×D4—61no—
154C2×C3×D4—61no—
155C2×C3×D4#6711yescharacteristic
156C2×C3×D4C3×D411yescharacteristic
157C22×C3×C4—61no—
158C22×C3×C4—61no—
159C2×#67—21no—
160C2×#67—21no—
161C22×Dic3—21no—
162C2×D12C3×D411yescharacteristic
163C2×C4×D3—21no—
164C2×Dic6C3×D411yescharacteristic
165C2×C3×C8—22no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—61no—
169C2×C3×C8—61no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—21no—
173C3×C4⋊C4—21no—
174C3×C4⋊C4—21no—
175C3×C4⋊C4—31no—
176C3×C4⋊C4—31no—
177C3×C4⋊C4—41no—
178C3×C4⋊C4—61no—
179C3×C4⋊C4—61no—
180C3×C4⋊C4—61no—
181C3×C4⋊C4—61no—
182C3×C22⋊C4—21no—
183C3×C22⋊C4—21no—
184C3×C22⋊C4—21no—
185C3×C22⋊C4—21no—
186C3×C22⋊C4—41no—
187C3×C22⋊C4—41no—
188C3×C22⋊C4—61no—
189C3×C22⋊C4—61no—
190C3×C22⋊C4—61no—
191C3×C22⋊C4—61no—
192C3×C42—21no—
193C3×C42C22×D311yescharacteristic
194C3×C42C23×C311yescharacteristic
195#217—21no—
196(C2×D3)⋊C4—21no—
197(C2×D3)⋊C4—21no—
198(C2×D3)⋊C4—21no—
199C4⋊Dic3—21no—
200C4⋊Dic3C3×D411yescharacteristic
201C4⋊Dic3C3×D411yescharacteristic
202Dic3⋊C4—21no—
203Dic3⋊C4—21no—
204Dic3⋊C4—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C22×C32—41no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32#12011yescharacteristic
211C2×C3×D3—41no—
212C2×C3×D3—41no—
213C2×C3×D3—41no—
214C32×C4—21no—
215C32×C4—21no—
216C32×C4—21no—
217C32×C4—41no—
218C32×C4—81no—
219C3×Dic3—41no—
220C3×Dic3—41no—
221C3×Dic3—41no—
222C4⋊Q8C2×C3×D311yescharacteristic
223#125—31no—
224#124—31no—
225#124C2×C3×D311yescharacteristic
226#123—61no—
227#123—61no—
228C22⋊Q8—61no—
229C22⋊Q8—61no—
230#121—61no—
231#121—61no—
232#103—21no—
233#103—61no—
234#103—61no—
235#103—61no—
236#102—21no—
237#102—61no—
238#97—31no—
239#97—31no—
240#97C2×C3×D311yescharacteristic
241C23×C3—21no—
242C23×C3—21no—
243C23×C3—41no—
244C23×C3—61no—
245C22×D3—21no—
246C3×Q8—41no—
247C3×Q8—41no—
248C3×Q8—42no—
249C3×Q8—42no—
250C3×Q8—42no—
251C3×Q8—82no—
252C3×Q8—121no—
253C3×D4—41no—
254C3×D4—41no—
255C3×D4—42no—
256C3×D4—121no—
257C3×D4—241no—
258C3×D4—241no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—61no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—241no—
272C2×C3×C4—241no—
273C2×C3×C4D3×D411yescharacteristic
274C2×C3×C4D3×D411yescharacteristic
275C2×C3×C4C2×#6711yescharacteristic
276C2×C3×C4C2×C3×D411yescharacteristic
277C2×C3×C4C2×C3×D411yescharacteristic
278C2×C3×C4C2×C3×D411yescharacteristic
279#67—81no—
280#67—81no—
281C2×Dic3—21no—
282C2×Dic3—21no—
283C2×Dic3—21no—
284C2×Dic3—81no—
285D12—41no—
286C4×D3—81no—
287Dic6—41no—
288C3×C8—42no—
289C3×C8—41no—
290C3×C8—41no—
291C3×C8—121no—
292C3×C8—121no—
293C3⋊C8—41no—
294C3⋊C8—41no—
295C2×C32—81no—
296C2×C32#45411yescharacteristic
297C2×C32#45511yescharacteristic
298C2×C32#45511yescharacteristic
299C3×D3—81no—
300C2×Q8—21no—
301C2×Q8—31no—
302C2×Q8C3×#6711yescharacteristic
303C2×D4—31no—
304C2×D4—61no—
305C2×D4—61no—
306C2×D4C3×#6711yescharacteristic
307C22×C4—61no—
308C22×C4—61no—
309C2×C8—21no—
310C2×C8—61no—
311C2×C8—61no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—31no—
315C4⋊C4—31no—
316C4⋊C4—61no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C22⋊C4—21no—
321C22⋊C4—21no—
322C22⋊C4—61no—
323C22⋊C4—61no—
324C22⋊C4—61no—
325C22⋊C4—61no—
326C42C22×C3×D311yescharacteristic
327C22×C3—21no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3—81no—
337C22×C3—121no—
338C22×C3—121no—
339C22×C3—121no—
340C22×C3#93711yescharacteristic
341C22×C3C3×#12011yescharacteristic
342C2×D3—41no—
343C2×D3—41no—
344C2×D3—41no—
345C3×C4—22no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—41no—
353C3×C4—41no—
354C3×C4—41no—
355C3×C4—41no—
356C3×C4—81no—
357C3×C4—82no—
358C3×C4—81no—
359C3×C4—81no—
360C3×C4—121no—
361C3×C4—121no—
362C3×C4—121no—
363Dic3—41no—
364Dic3—41no—
365Dic3—41no—
366C32#166211yescharacteristic, complemented, Sylow
367C23—21no—
368C23—61no—
369Q8—41no—
370Q8—42no—
371Q8—121no—
372D4—41no—
373D4—121no—
374D4—241no—
375D4—241no—
376C2×C4—21no—
377C2×C4—41no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—241no—
382C2×C4—241no—
383C2×C4C3×D3×D411yescharacteristic
384C2×C4C3×D3×D411yescharacteristic
385C2×C4C2×C3×#6711yescharacteristic
386C8—41no—
387C8—121no—
388C8—121no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—81no—
393C2×C3—81no—
394C2×C3—161no—
395C2×C3—241no—
396C2×C3#506111yescharacteristic
397C2×C3#520111yescharacteristic
398C2×C3#520211yescharacteristic
399C2×C3C3×#45411yescharacteristic
400C2×C3C3×#45511yescharacteristic
401C2×C3C3×#45511yescharacteristic
402D3—81no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—121no—
407C22—121no—
408C22—121no—
409C22C3×#93711yescharacteristic
410C4—21no—
411C4—21no—
412C4—21no—
413C4—41no—
414C4—81no—
415C4—121no—
416C4—121no—
417C4—121no—
418C3—21no—
419C3#7124611yescharacteristic, direct product factor
420C3C3×#166211yescharacteristic, complemented
421C2—81no—
422C2—241no—
423C2C3×#506111yescharacteristic
424C2C3×#520111yescharacteristic
425C2C3×#520211yescharacteristic
subgroupG/H
centerC22×C3#937
commutator, cosocleC3×C42C23×C3
FrattiniC42C22×C3×D3
FittingC32×#489C2
socleC22×C32#120
prime radicalC3×#5684C2
hypercenterC3×#489D3
hypocenterC3C3×#1662
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×#489

Complements

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

NH ≅ G/Ntype
2. C32×#489C2semidirect
3. C3×#5089C2semidirect
4. C3×#5082C2semidirect
5. C3×#4540C2semidirect
6. C3×#4465C2semidirect
7. C3×#4464C2semidirect
9. #71246C3direct (N × H)
34. C3×#489D3semidirect
35. C3×#489C2×C3semidirect
36. #5089C2×C3semidirect
37. #5082C2×C3semidirect
41. #4540C2×C3semidirect
42. #4465C2×C3semidirect
43. #4464C2×C3semidirect
140. #489C3×D3semidirect
366. C32#1662semidirect
419. C3#71246direct (N × H)
420. C3C3×#1662semidirect

Maximal subgroups

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

subgroupindex
C3×#56842
C32×#4892
C3×#50892
C3×#50822
C3×#45402
C3×#44652
C3×#44642
C3×#16623
#712463