Finite Group Databasebeta

← C3×#70378 #11789692

Subgroups — C3×#70378 #11789692

Order 1152.

517 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#104C211yesmaximal, characteristic
2C3×#5237C211yesmaximal, characteristic
3C3×#5236C211yesmaximal, characteristic
4C3×#5124C211yesmaximal, characteristic, complemented
5C3×C4×#208C211yesmaximal, characteristic, complemented
6C3×#4556C211yesmaximal, characteristic, complemented
7C3×#4523C211yesmaximal, characteristic, complemented
8C3×#1932—31nomaximal
9#70378C311yesmaximal, characteristic, direct product factor
10C2×C32×#41C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×C32×C42C2211yescharacteristic
13C3×#926C2211yescharacteristic
14C2×C3×#208C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C32×#104—21no—
18C32×#104—21no—
19C3×#837—22no—
20C3×#836C412yescomplemented
21C3×#833C412yescomplemented
22C3×#822—22no—
23C3×#803—21no—
24C3×C4×C3⋊C8—21no—
25C3×#430—31no—
26C3×#421—31no—
27C2×C3×#104—21no—
28C2×C3×#104D311yescharacteristic
29C2×C3×#104C2×C311yescharacteristic
30C3×#413—31no—
31C3×C4×M16—31no—
32#5237C2×C311yescharacteristic
33#5236C2×C311yescharacteristic
34#5124C2×C311yescharacteristic, complemented
35C4×#208C2×C311yescharacteristic, complemented
36C3×#344—31no—
37C3×#339—31no—
38#4556C2×C311yescharacteristic, complemented
39#4523C2×C311yescharacteristic, complemented
40C32×#41—21no—
41C32×#41—21no—
42C32×#41—21no—
43C32×#41—21no—
44C2×C32×Q8C2×C411yescharacteristic
45C2×C32×D4—21no—
46C2×C32×D4C2×C411yescharacteristic
47C22×C32×C4—21no—
48C22×C32×C4—21no—
49C22×C32×C4C2311yescharacteristic
50C32×M16—21no—
51C32×M16—21no—
52C2×C32×C8—21no—
53C32×C42—21no—
54C32×C42D411yescharacteristic
55C32×C42D411yescharacteristic
56C3×#217—21no—
57C3×C4⋊Dic3—21no—
58C3×C4×Dic3—21no—
59C3×#208—21no—
60C3×#208—21no—
61C3×#208—21no—
62C3×#208C2×C412yes—
63C3×#208C2×C412yes—
64C3×#208D412yes—
65C2×C3×C3⋊C8—21no—
66C2×C3×C3⋊C8—21no—
67C2×C3×C3⋊C8—21no—
68#1932—31noSylow
69C2×C3×#41—21no—
70C2×C3×#41C2×D311yescharacteristic
71C2×C3×#41C22×C311yescharacteristic
72C2×C3×M16—21no—
73C2×C3×M16—31no—
74C2×C3×M16—31no—
75C2×C3×M16—31no—
76C2×C3×M16C2×D311yescharacteristic
77C2×C3×M16C22×C311yescharacteristic
78C3×#117—31no—
79C2×C3×C42—21no—
80C2×C3×C42C2×D311yescharacteristic
81C2×C3×C42C22×C311yescharacteristic
82#926C22×C311yescharacteristic
83C2×#208C22×C311yescharacteristic
84C2×#208C22×C311yescharacteristic
85C2×#208C22×C311yescharacteristic
86C3×#108—62no—
87C3×#104—22no—
88C3×#104—22no—
89C3×#104—21no—
90C3×#104—21no—
91C3×#104—21no—
92C3×#104—21no—
93C3×#104—62no—
94C3×#101—32no—
95C3×#100—32no—
96C3×#97—61no—
97C3×C4×C8—61no—
98#837—22no—
99#836C3×C412yescomplemented
100#833C3×C412yescomplemented
101#822—22no—
102#803—21no—
103C4×C3⋊C8—21no—
104C23×C32—21no—
105C23×C32#4111yescharacteristic
106C32×Q8—21no—
107C32×Q8—21no—
108C32×D4—21no—
109C32×D4—21no—
110C32×D4—41no—
111C32×D4—41no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—42no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4C22×C411yescharacteristic
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4#4111yescharacteristic
122C2×C32×C4#4111yescharacteristic
123C2×C32×C4#4111yescharacteristic
124C2×C3×Dic3—41no—
125C32×C8—41no—
126C3×C3⋊C8—22no—
127C3×C3⋊C8—22no—
128C3×C3⋊C8—22no—
129#430—31no—
130#421—31no—
131C2×#104C3×D311yescharacteristic
132#413—31no—
133C4×M16—31no—
134#344—31no—
135#339—31no—
136C3×#41—22no—
137C3×#41—22no—
138C3×#41—21no—
139C3×#41—21no—
140C3×#41—21no—
141C3×#41—21no—
142C3×#41—21no—
143C3×#41—21no—
144C3×#41—21no—
145C3×#41—21no—
146C3×#41—41no—
147C3×#41—41no—
148C2×C3×Q8—21no—
149C2×C3×Q8C4×D311yescharacteristic
150C2×C3×Q8C2×C3×C411yescharacteristic
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—21no—
154C2×C3×D4—41no—
155C2×C3×D4C4×D311yescharacteristic
156C2×C3×D4C2×C3×C411yescharacteristic
157C22×C3×C4—21no—
158C22×C3×C4—21no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—21no—
162C22×C3×C4—41no—
163C22×C3×C4—41no—
164C22×C3×C4C22×D311yescharacteristic
165C22×C3×C4C23×C311yescharacteristic
166C3×M16—22no—
167C3×M16—21no—
168C3×M16—21no—
169C3×M16—21no—
170C3×M16—21no—
171C3×M16—32no—
172C3×M16—32no—
173C3×M16—32no—
174C3×M16—41no—
175C3×M16—61no—
176C3×M16—61no—
177C3×M16—61no—
178C2×C3×C8—21no—
179C2×C3×C8—21no—
180C2×C3×C8—41no—
181C2×C3×C8—61no—
182C2×C3×C8—61no—
183C2×C3×C8—61no—
184C3×C4⋊C4—61no—
185C3×C22⋊C4—61no—
186C3×C42—21no—
187C3×C42—21no—
188C3×C42—21no—
189C3×C42—21no—
190C3×C42—41no—
191C3×C42—61no—
192C3×C42#6711yescharacteristic
193C3×C42#6711yescharacteristic
194C3×C42C3×D411yescharacteristic
195C3×C42C3×D411yescharacteristic
196#217—21no—
197C4⋊Dic3—21no—
198C4×Dic3—21no—
199#208—21no—
200#208—21no—
201#208—21no—
202#208C2×C3×C412yes—
203#208C2×C3×C412yes—
204#208C3×D412yes—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—21no—
207C2×C3⋊C8—21no—
208C22×C32—41no—
209C22×C32—41no—
210C22×C32—81no—
211C22×C32#11711yescharacteristic
212C22×C32#12311yescharacteristic
213C22×C32#12411yescharacteristic
214C32×C4—22no—
215C32×C4—41no—
216C32×C4—41no—
217C32×C4C4×D411yescharacteristic
218C32×C4C4×D411yescharacteristic
219C32×C4#12111yescharacteristic
220C32×C4#12611yescharacteristic
221C3×Dic3—81no—
222C2×#41C2×C3×D311yescharacteristic
223C2×M16—31no—
224C2×M16—31no—
225C2×M16—31no—
226C2×M16C2×C3×D311yescharacteristic
227#117—31no—
228C2×C42C2×C3×D311yescharacteristic
229#108—62no—
230#104—21no—
231#104—21no—
232#104—62no—
233#101—32no—
234#100—32no—
235#97—61no—
236C4×C8—61no—
237C23×C3—21no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—41no—
241C23×C3#23711yescharacteristic
242C23×C3C3×#4111yescharacteristic
243C3×Q8—22no—
244C3×Q8—21no—
245C3×Q8—21no—
246C3×Q8—21no—
247C3×Q8—21no—
248C3×Q8—41no—
249C3×D4—22no—
250C3×D4—21no—
251C3×D4—21no—
252C3×D4—21no—
253C3×D4—21no—
254C3×D4—41no—
255C3×D4—42no—
256C3×D4—42no—
257C3×D4—41no—
258C3×D4—41no—
259C3×D4—41no—
260C3×D4—41no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—21no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—42no—
278C2×C3×C4—42no—
279C2×C3×C4—42no—
280C2×C3×C4—44no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—121no—
286C2×C3×C4C2×C4×D311yescharacteristic
287C2×C3×C4#23511yescharacteristic
288C2×C3×C4D3×D411yescharacteristic
289C2×C3×C4#23711yescharacteristic
290C2×C3×C4#23711yescharacteristic
291C2×C3×C4C2×#6711yescharacteristic
292C2×C3×C4C22×C3×C411yescharacteristic
293C2×C3×C4C2×C3×D411yescharacteristic
294C2×C3×C4C2×C3×D411yescharacteristic
295C2×C3×C4C3×#4111yescharacteristic
296C2×C3×C4C3×#4111yescharacteristic
297C2×C3×C4C3×#4111yescharacteristic
298C2×Dic3—41no—
299C3×C8—42no—
300C3×C8—41no—
301C3×C8—41no—
302C3×C8—62no—
303C3×C8—62no—
304C3×C8—62no—
305C3⋊C8—22no—
306C3⋊C8—22no—
307C3⋊C8—22no—
308C2×C32—21no—
309C2×C32—21no—
310C2×C32—21no—
311C2×C32—81no—
312C2×C32#38811yescharacteristic
313#41—21no—
314#41—21no—
315#41—21no—
316#41—21no—
317C2×Q8C3×C4×D311yescharacteristic
318C2×D4—21no—
319C2×D4C3×C4×D311yescharacteristic
320C22×C4—21no—
321C22×C4—21no—
322C22×C4C22×C3×D311yescharacteristic
323M16—21no—
324M16—21no—
325M16—32no—
326M16—32no—
327M16—32no—
328M16—61no—
329M16—61no—
330M16—61no—
331C2×C8—21no—
332C2×C8—61no—
333C2×C8—61no—
334C2×C8—61no—
335C4⋊C4—61no—
336C22⋊C4—61no—
337C42—21no—
338C42—61no—
339C42C3×#6711yescharacteristic
340C42C3×#6711yescharacteristic
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—42no—
347C22×C3—42no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—81no—
351C22×C3—81no—
352C22×C3—82no—
353C22×C3#87711yescharacteristic
354C22×C3#88511yescharacteristic
355C22×C3#93511yescharacteristic
356C22×C3C3×#11711yescharacteristic
357C22×C3C3×#12311yescharacteristic
358C22×C3C3×#12411yescharacteristic
359C3×C4—22no—
360C3×C4—22no—
361C3×C4—21no—
362C3×C4—21no—
363C3×C4—21no—
364C3×C4—21no—
365C3×C4—42no—
366C3×C4—41no—
367C3×C4—41no—
368C3×C4—42no—
369C3×C4—42no—
370C3×C4—41no—
371C3×C4—41no—
372C3×C4—241no—
373C3×C4#87911yescharacteristic
374C3×C4#67⋊C411yescharacteristic
375C3×C4C4×#6711yescharacteristic
376C3×C4#93811yescharacteristic
377C3×C4C3×C4×D411yescharacteristic
378C3×C4C3×C4×D411yescharacteristic
379C3×C4C3×#12111yescharacteristic
380C3×C4C3×#12611yescharacteristic
381Dic3—81no—
382C32#193211yescharacteristic, complemented, Sylow
383C23—21no—
384C23C3×#23711yescharacteristic
385Q8—21no—
386Q8—21no—
387D4—21no—
388D4—21no—
389D4—41no—
390D4—41no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—121no—
398C2×C4C2×C3×C4×D311yescharacteristic
399C2×C4C3×#23511yescharacteristic
400C2×C4C3×D3×D411yescharacteristic
401C2×C4C3×#23711yescharacteristic
402C2×C4C3×#23711yescharacteristic
403C2×C4C2×C3×#6711yescharacteristic
404C8—41no—
405C8—62no—
406C8—62no—
407C8—62no—
408C2×C3—22no—
409C2×C3—21no—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—21no—
413C2×C3—21no—
414C2×C3—21no—
415C2×C3—21no—
416C2×C3—41no—
417C2×C3—41no—
418C2×C3—82no—
419C2×C3—81no—
420C2×C3—81no—
421C2×C3#494411yescharacteristic
422C2×C3C3×#38811yescharacteristic
423C22—41no—
424C22—41no—
425C22—81no—
426C22C3×#87711yescharacteristic
427C22C3×#88511yescharacteristic
428C22C3×#93511yescharacteristic
429C4—22no—
430C4—41no—
431C4—41no—
432C4—241no—
433C4C3×#87911yescharacteristic
434C4C3×#67⋊C411yescharacteristic
435C4C3×C4×#6711yescharacteristic
436C4C3×#93811yescharacteristic
437C3—21no—
438C3#7037811yescharacteristic, direct product factor
439C3C3×#193211yescharacteristic, complemented
440C2—21no—
441C2—21no—
442C2—21no—
443C2—81no—
444C2C3×#494411yescharacteristic
subgroupG/H
centerC3×C4#938
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#104C2
socleC2×C32#388
prime radicalC2×C32×#41C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#104D3
hypocenterC3C3×#1932
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C2×C3×#104

Complements

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

NH ≅ G/Ntype
4. C3×#5124C2semidirect
5. C3×C4×#208C2semidirect
6. C3×#4556C2semidirect
7. C3×#4523C2semidirect
9. #70378C3direct (N × H)
20. C3×#836C4semidirect
21. C3×#833C4semidirect
34. #5124C2×C3semidirect
35. C4×#208C2×C3semidirect
38. #4556C2×C3semidirect
39. #4523C2×C3semidirect
99. #836C3×C4semidirect
100. #833C3×C4semidirect
382. C32#1932semidirect
438. C3#70378direct (N × H)
439. C3C3×#1932semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1042
C3×#52372
C3×#52362
C3×#51242
C3×C4×#2082
C3×#45562
C3×#45232
C3×#19323
#703783