Finite Group Databasebeta

← C3×C4×#3775 #13166794

Subgroups — C3×C4×#3775 #13166794

Order 1728.

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

normal subgroups only

This group has 492 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
1C2×C3×#3775C211yesmaximal, characteristic
2C3×C4×#627C211yesmaximal, characteristic, complemented
3C3×C4×#625C211yesmaximal, characteristic
4C3×C4×#622C211yesmaximal, characteristic, complemented
5C3×#63742C211yesmaximal, characteristic, complemented
6C3×#63740C211yesmaximal, characteristic, complemented
7C3×#63739C211yesmaximal, characteristic
8C4×#3775C311yesmaximal, characteristic, direct product factor
9C2×C32×C4×D3—21no—
10C2×C3×#627C2211yescharacteristic
11C2×C3×#625C2211yescharacteristic
12C2×C3×#622C2211yescharacteristic
13C2×C3×#622C2211yescharacteristic
14C3×#3775C414yesdirect product factor
15C3×C4×C3⋊Dic3C2211yescharacteristic
16C32×C4×Dic3—21no—
17C3×#3724C2211yescharacteristic
18C3×#3724C2211yescharacteristic
19C3×Dic3⋊Dic3—21no—
20C3×(C2×D3)⋊Dic3—21no—
21C2×#3775C2×C311yescharacteristic
22C3×C4×#67—61no—
23C3×C4×Dic6—61no—
24C4×#627C2×C311yescharacteristic, complemented
25C4×#625C2×C311yescharacteristic
26C4×#622C2×C311yescharacteristic, complemented
27#63742C2×C311yescharacteristic, complemented
28#63740C2×C311yescharacteristic, complemented
29#63739C2×C311yescharacteristic
30C22×C32×D3—21no—
31C2×C33×C4D411yescharacteristic
32C2×C3×C3⋊Dic3D411yescharacteristic
33C2×C3×C3⋊Dic3C2311yescharacteristic
34C2×C32×Dic3—21no—
35C2×C32×Dic3—21no—
36C2×C32×Dic3—21no—
37C32×C4×D3—42no—
38C3×#627—21no—
39C3×#627C2×C412yescomplemented
40C3×#625—21no—
41C3×#625C2×C412yes—
42C3×#622—21no—
43C3×#622C2×C412yescomplemented
44C3×C4×SD16—91nomaximal
45C22×C32×C4—61no—
46C2×C3×#67—61no—
47C2×C3×C4×D3—21no—
48C2×C3×C4×D3—21no—
49C2×C3×C4×D3—41no—
50C2×C3×Dic6—61no—
51C2×#627C22×C311yescharacteristic
52C2×#625C22×C311yescharacteristic
53C2×#622C22×C311yescharacteristic
54C2×#622C22×C311yescharacteristic
55#3775C3×C414yesdirect product factor
56C32×C42—61no—
57C4×C3⋊Dic3C22×C311yescharacteristic
58C3×#217—61no—
59C3×(C2×D3)⋊C4—61no—
60C3×C4⋊Dic3—61no—
61C3×Dic3⋊C4—61no—
62C3×Dic3⋊C4—61no—
63C3×Dic3⋊C4—61no—
64C3×C4×Dic3—21no—
65C3×C4×Dic3—21no—
66C3×C4×Dic3—41no—
67C3×C4×Dic3—61no—
68C3×C4×Dic3—61no—
69#3724C22×C311yescharacteristic
70#3724C22×C311yescharacteristic
71Dic3⋊Dic3—21no—
72(C2×D3)⋊Dic3—21no—
73C22×C33C2×D411yescharacteristic
74C2×C32×D3—22no—
75C2×C32×D3—41no—
76C2×C32×D3—41no—
77C33×C4SD1612yescomplemented
78C3×C3⋊Dic3—21no—
79C3×C3⋊Dic3C22×C411yescharacteristic
80C3×C3⋊Dic3#4111yescharacteristic
81C32×Dic3—22no—
82C32×Dic3—41no—
83C32×Dic3—41no—
84C2×C3×SD16—91no—
85C3×C4×Q8—91no—
86C3×C4×D4—91no—
87C4×#67—61no—
88C4×Dic6—61no—
89C3×#106—91no—
90C3×#103—91no—
91C3×#102—91no—
92C3×C4×C8—91no—
93C23×C32—61no—
94C22×C3×D3—21no—
95C22×C3×D3—21no—
96C22×C3×D3—41no—
97C2×C32×C4—21no—
98C2×C32×C4—21no—
99C2×C32×C4—41no—
100C2×C32×C4—41no—
101C2×C32×C4—61no—
102C2×C32×C4—61no—
103C2×C32×C4—61no—
104C2×C32×C4—122no—
105C2×C32×C4C3×D411yescharacteristic
106C2×C3⋊Dic3C3×D411yescharacteristic
107C2×C3⋊Dic3C23×C311yescharacteristic
108C3×#67—62no—
109C3×#67—62no—
110C2×C3×Dic3—21no—
111C2×C3×Dic3—21no—
112C2×C3×Dic3—21no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—41no—
117C2×C3×Dic3—41no—
118C2×C3×Dic3—41no—
119C2×C3×Dic3—61no—
120C2×C3×Dic3—61no—
121C2×C3×Dic3—61no—
122C2×C3×Dic3—61no—
123C3×C4×D3—42no—
124C3×C4×D3—44no—
125C3×C4×D3—42no—
126C3×Dic6—62no—
127C3×Dic6—62no—
128#627—21no—
129#627C2×C3×C412yescomplemented
130#625—21no—
131#625C2×C3×C412yes—
132#622—21no—
133#622C2×C3×C412yescomplemented
134C4×SD16—91noSylow
135C2×C33C4×D411yescharacteristic
136C2×C33C2×SD1611yescharacteristic
137C2×C33#13511yescharacteristic
138C32×D3—42no—
139C2×C3×Q8—91no—
140C2×C3×D4—91no—
141C22×C3×C4—61no—
142C22×C3×C4—61no—
143C22×C3×C4—121no—
144C2×#67—61no—
145C2×C4×D3—21no—
146C2×Dic6—61no—
147C3×SD16—94no—
148C2×C3×C8—91no—
149C2×C3×C8—91no—
150C3×C4⋊C4—91no—
151C3×C4⋊C4—91no—
152C3×C4⋊C4—181no—
153C3×C22⋊C4—181no—
154C3×C42—61no—
155C3×C42—61no—
156C3×C42—91no—
157C3×C42—121no—
158#217—61no—
159(C2×D3)⋊C4—61no—
160C4⋊Dic3—61no—
161Dic3⋊C4—61no—
162Dic3⋊C4—61no—
163Dic3⋊C4—61no—
164C4×Dic3—21no—
165C4×Dic3—61no—
166C4×Dic3—61no—
167C22×C32—21no—
168C22×C32—21no—
169C22×C32—41no—
170C22×C32—41no—
171C22×C32—62no—
172C22×C32—121no—
173C22×C32—121no—
174C22×C32C2×C3×D411yescharacteristic
175C2×C3×D3—22no—
176C2×C3×D3—22no—
177C2×C3×D3—42no—
178C2×C3×D3—41no—
179C2×C3×D3—41no—
180C2×C3×D3—42no—
181C2×C3×D3—42no—
182C2×C3×D3—41no—
183C2×C3×D3—41no—
184C32×C4—22no—
185C32×C4—22no—
186C32×C4—42no—
187C32×C4—42no—
188C32×C4—62no—
189C32×C4—121no—
190C32×C4—121no—
191C32×C4C3×SD1612yescomplemented
192C3⋊Dic3—21no—
193C3⋊Dic3C22×C3×C411yescharacteristic
194C3⋊Dic3C3×#4111yescharacteristic
195C3×Dic3—22no—
196C3×Dic3—22no—
197C3×Dic3—42no—
198C3×Dic3—41no—
199C3×Dic3—41no—
200C3×Dic3—42no—
201C3×Dic3—42no—
202C3×Dic3—41no—
203C3×Dic3—41no—
204C3×Dic3—61no—
205C3×Dic3—61no—
206C3×Dic3—61no—
207C3×Dic3—61no—
208C3×Dic3—121no—
209C3×Dic3—121no—
210C2×SD16—91no—
211C4×Q8—91no—
212C4×D4—91no—
213#106—91no—
214#103—91no—
215#102—91no—
216C4×C8—91no—
217C33C4×SD1611yescharacteristic, complemented, Sylow
218C23×C3—61no—
219C23×C3—61no—
220C23×C3—121no—
221C22×D3—21no—
222C3×Q8—92no—
223C3×Q8—181no—
224C3×D4—92no—
225C3×D4—181no—
226C2×C3×C4—21no—
227C2×C3×C4—21no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—61no—
231C2×C3×C4—61no—
232C2×C3×C4—61no—
233C2×C3×C4—61no—
234C2×C3×C4—61no—
235C2×C3×C4—61no—
236C2×C3×C4—91no—
237C2×C3×C4—91no—
238C2×C3×C4—121no—
239C2×C3×C4—121no—
240C2×C3×C4—121no—
241C2×C3×C4—122no—
242C2×C3×C4—124no—
243C2×C3×C4—122no—
244C2×C3×C4GO+(4,𝔽2)11yescharacteristic
245#67—62no—
246#67—62no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—61no—
251C2×Dic3—61no—
252C2×Dic3—61no—
253C2×Dic3—61no—
254C4×D3—42no—
255Dic6—62no—
256Dic6—62no—
257C3×C8—92no—
258C3×C8—181no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32—21no—
262C2×C32—21no—
263C2×C32—21no—
264C2×C32—21no—
265C2×C32—41no—
266C2×C32—41no—
267C2×C32—41no—
268C2×C32—41no—
269C2×C32—41no—
270C2×C32—41no—
271C2×C32—122no—
272C2×C32C3×C4×D411yescharacteristic
273C2×C32C2×C3×SD1611yescharacteristic
274C2×C32C3×#13511yescharacteristic
275C3×D3—42no—
276C3×D3—44no—
277C3×D3—42no—
278C2×Q8—91no—
279C2×D4—91no—
280C22×C4—61no—
281SD16—94no—
282C2×C8—91no—
283C2×C8—91no—
284C4⋊C4—91no—
285C4⋊C4—91no—
286C4⋊C4—181no—
287C22⋊C4—181no—
288C42—61no—
289C42—91no—
290C22×C3—21no—
291C22×C3—21no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—62no—
295C22×C3—62no—
296C22×C3—122no—
297C22×C3—121no—
298C22×C3—121no—
299C22×C3—122no—
300C22×C3—122no—
301C22×C3—121no—
302C22×C3—121no—
303C22×C3C2×GO+(4,𝔽2)11yescharacteristic
304C2×D3—22no—
305C2×D3—41no—
306C2×D3—41no—
307C3×C4—22no—
308C3×C4—22no—
309C3×C4—42no—
310C3×C4—42no—
311C3×C4—62no—
312C3×C4—62no—
313C3×C4—91no—
314C3×C4—91no—
315C3×C4—122no—
316C3×C4—121no—
317C3×C4—121no—
318C3×C4—122no—
319C3×C4—122no—
320C3×C4—121no—
321C3×C4—121no—
322C3×C4—181no—
323C3×C4#377512yesdirect product factor
324Dic3—22no—
325Dic3—41no—
326Dic3—41no—
327Dic3—61no—
328Dic3—61no—
329Dic3—61no—
330Dic3—61no—
331Dic3—121no—
332Dic3—121no—
333C32—21no—
334C32—21no—
335C32—41no—
336C32—41no—
337C32C3×C4×SD1611yescharacteristic, complemented
338C23—61no—
339Q8—92no—
340Q8—181no—
341D4—92no—
342D4—181no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—91no—
347C2×C4—91no—
348C2×C4—122no—
349C2×C4C3×GO+(4,𝔽2)11yescharacteristic
350C8—92no—
351C8—181no—
352C2×C3—21no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—41no—
359C2×C3—41no—
360C2×C3—41no—
361C2×C3—41no—
362C2×C3—41no—
363C2×C3—41no—
364C2×C3—122no—
365C2×C3—124no—
366C2×C3—122no—
367C2×C3#6422311yescharacteristic
368C2×C3C4×GO+(4,𝔽2)11yescharacteristic
369C2×C3C2×#377511yescharacteristic
370D3—42no—
371C22—62no—
372C22—121no—
373C22—121no—
374C22C2×C3×GO+(4,𝔽2)11yescharacteristic
375C4—62no—
376C4—91no—
377C4—91no—
378C4—121no—
379C4—121no—
380C4—181no—
381C4C3×#377512yesdirect product factor
382C3—21no—
383C3—21no—
384C3—41no—
385C3—41no—
386C3C4×#377511yescharacteristic, direct product factor
387C2—122no—
388C2C3×#6422311yescharacteristic
389C2C3×C4×GO+(4,𝔽2)11yescharacteristic
390C2C2×C3×#377511yescharacteristic
subgroupG/H
center, hypercenterC2×C3×C4GO+(4,𝔽2)
commutatorC3⋊Dic3C22×C3×C4
FrattiniC22C2×C3×GO+(4,𝔽2)
FittingC2×C33×C4D4
socleC22×C33C2×D4
prime radicalC2×C3×#625C22
cosocleC2×C3⋊Dic3C23×C3
hypocenterC32C3×C4×SD16
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×#627C2semidirect
4. C3×C4×#622C2semidirect
5. C3×#63742C2semidirect
6. C3×#63740C2semidirect
8. C4×#3775C3direct (N × H)
14. C3×#3775C4direct (N × H)
24. C4×#627C2×C3semidirect
26. C4×#622C2×C3semidirect
27. #63742C2×C3semidirect
28. #63740C2×C3semidirect
39. C3×#627C2×C4semidirect
43. C3×#622C2×C4semidirect
55. #3775C3×C4direct (N × H)
77. C33×C4SD16semidirect
129. #627C2×C3×C4semidirect
133. #622C2×C3×C4semidirect
191. C32×C4C3×SD16semidirect
217. C33C4×SD16semidirect
323. C3×C4#3775direct (N × H)
337. C32C3×C4×SD16semidirect
381. C4C3×#3775direct (N × H)
386. C3C4×#3775direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#37752
C3×C4×#6272
C3×C4×#6252
C3×C4×#6222
C3×#637422
C3×#637402
C3×#637392
C4×#37753
C3×C4×SD169