Finite Group Databasebeta

← C3×#68750 #11789704

Subgroups — C3×#68750 #11789704

Order 1152.

487 conjugacy classes of subgroups.

The subgroup lattice has 427 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5132C211yesmaximal, characteristic
2C3×#5125C211yesmaximal, characteristic, complemented
3C3×#5005C211yesmaximal, characteristic
4C2×C3×#805C211yesmaximal, characteristic
5C3×C4×#208C211yesmaximal, characteristic, complemented
6C32×#339C211yesmaximal, characteristic, complemented
7C3×#4533C211yesmaximal, characteristic, complemented
8C3×#1932—31nomaximal
9#68750C311yesmaximal, characteristic, direct product factor
10C2×C3×#235C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C32×#117C2211yescharacteristic
13C2×C32×C42C2211yescharacteristic
14C2×C3×#208C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C3×#824C412yescomplemented
18C3×#823C412yescomplemented
19C3×#819—22no—
20C3×#805—21no—
21C3×#805—21no—
22C3×#805—22no—
23C3×#803—21no—
24C3×C4×C3⋊C8—21no—
25C3×#430—31no—
26C3×#421—31no—
27C2×C3×#104—31no—
28C3×#413—31no—
29C3×C4×M16—31no—
30#5132C2×C311yescharacteristic
31#5125C2×C311yescharacteristic, complemented
32#5005C2×C311yescharacteristic
33C2×#805C2×C311yescharacteristic
34C4×#208C2×C311yescharacteristic, complemented
35C3×#344—31no—
36C3×#339—21no—
37C3×#339D311yescharacteristic, complemented
38C3×#339C2×C311yescharacteristic, complemented
39#4533C2×C311yescharacteristic, complemented
40C22×C32×C4—21no—
41C22×C32×C4C2311yescharacteristic
42C2×C3×#67—21no—
43C3×#235—21no—
44C3×#235—21no—
45C3×#235—21no—
46C3×#235—21no—
47C2×C3×D12C2×C411yescharacteristic
48C2×C3×C4×D3—21no—
49C2×C3×Dic6C2×C411yescharacteristic
50C32×M16—21no—
51C32×M16C2×C412yes—
52C2×C32×C8—21no—
53C32×C4⋊C4—21no—
54C32×C22⋊C4—21no—
55C32×C42—21no—
56C32×C42—21no—
57C32×C42D411yescharacteristic
58C32×C42D411yescharacteristic
59C3×#208—21no—
60C3×#208—21no—
61C3×#208—21no—
62C3×#208—21no—
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—31no—
70C2×#235C22×C311yescharacteristic
71C2×C3×M16—21no—
72C2×C3×M16—31no—
73C2×C3×M16—31no—
74C2×C3×M16—31no—
75C2×C3×M16C2×D311yescharacteristic
76C2×C3×M16C22×C311yescharacteristic
77C3×#117—21no—
78C3×#117C2×D311yescharacteristic
79C3×#117C22×C311yescharacteristic
80C2×C3×C42—21no—
81C2×C3×C42C2×D311yescharacteristic
82C2×C3×C42C22×C311yescharacteristic
83C2×#208C22×C311yescharacteristic
84C2×#208C22×C311yescharacteristic
85C2×#208C22×C311yescharacteristic
86C3×#108—62no—
87C3×#104—62no—
88C3×#104—61no—
89C3×#104—61no—
90C3×#101—32no—
91C3×#100—32no—
92C3×#97—61no—
93C3×C4×C8—61no—
94#824C3×C412yescomplemented
95#823C3×C412yescomplemented
96#819—22no—
97#805—21no—
98#805—21no—
99#805—22no—
100#803—21no—
101C4×C3⋊C8—21no—
102C23×C32#4111yescharacteristic
103C22×C3×D3—21no—
104C2×C32×C4—21no—
105C2×C32×C4—21no—
106C2×C32×C4—42no—
107C2×C32×C4—41no—
108C2×C32×C4C22×C411yescharacteristic
109C2×C32×C4C2×D411yescharacteristic
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4#4111yescharacteristic
112C2×C32×C4#4111yescharacteristic
113C2×C32×C4#4111yescharacteristic
114C3×#67—41no—
115C3×#67—41no—
116C2×C3×Dic3—21no—
117C3×D12—21no—
118C3×D12—21no—
119C3×C4×D3—41no—
120C3×C4×D3—41no—
121C3×Dic6—21no—
122C3×Dic6—21no—
123C32×C8—22no—
124C3×C3⋊C8—22no—
125C3×C3⋊C8—22no—
126C3×C3⋊C8—41no—
127#430—31no—
128#421—31no—
129C2×#104—31no—
130#413—31no—
131C4×M16—31no—
132#344—31no—
133#339C3×D311yescharacteristic, complemented
134C3×#41—61no—
135C3×#41—61no—
136C3×#41—61no—
137C3×#41—61no—
138C2×C3×Q8—31no—
139C2×C3×D4—31no—
140C2×C3×D4—61no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—41no—
145C22×C3×C4—61no—
146C22×C3×C4C22×D311yescharacteristic
147C22×C3×C4C23×C311yescharacteristic
148C2×#67—21no—
149#235—21no—
150#235—21no—
151#235—21no—
152#235—21no—
153C2×D12C2×C3×C411yescharacteristic
154C2×C4×D3—21no—
155C2×Dic6C2×C3×C411yescharacteristic
156C3×M16—21no—
157C3×M16—21no—
158C3×M16—22no—
159C3×M16—32no—
160C3×M16—32no—
161C3×M16—41no—
162C3×M16—61no—
163C3×M16—61no—
164C3×M16—61no—
165C3×M16—61no—
166C3×M16C4×D312yes—
167C3×M16C2×C3×C412yes—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8—41no—
171C2×C3×C8—61no—
172C2×C3×C8—61no—
173C2×C3×C8—61no—
174C3×C4⋊C4—22no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C22⋊C4—21no—
178C3×C22⋊C4—21no—
179C3×C22⋊C4—41no—
180C3×C42—21no—
181C3×C42—21no—
182C3×C42—21no—
183C3×C42—21no—
184C3×C42—21no—
185C3×C42—21no—
186C3×C42—41no—
187C3×C42—41no—
188C3×C42D1211yescharacteristic
189C3×C42D1211yescharacteristic
190C3×C42C3×D411yescharacteristic
191C3×C42C3×D411yescharacteristic
192#208—21no—
193#208—21no—
194#208—21no—
195#208—21no—
196#208C2×C3×C412yes—
197#208C3×D412yes—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—21no—
201C22×C32—41no—
202C22×C32#11711yescharacteristic
203C22×C32#12311yescharacteristic
204C22×C32#12411yescharacteristic
205C2×C3×D3—41no—
206C2×C3×D3—81no—
207C32×C4—22no—
208C32×C4—41no—
209C32×C4—81no—
210C32×C4C4×D411yescharacteristic
211C32×C4C4×D411yescharacteristic
212C32×C4#12111yescharacteristic
213C32×C4#12611yescharacteristic
214C3×Dic3—41no—
215C2×#41—31no—
216C2×M16—31no—
217C2×M16—31no—
218C2×M16—31no—
219C2×M16C2×C3×D311yescharacteristic
220#117C2×C3×D311yescharacteristic
221C2×C42C2×C3×D311yescharacteristic
222#108—62no—
223#104—62no—
224#104—61no—
225#104—61no—
226#101—32no—
227#100—32no—
228#97—61no—
229C4×C8—61no—
230C23×C3—21no—
231C23×C3—61no—
232C23×C3#23711yescharacteristic
233C23×C3C3×#4111yescharacteristic
234C22×D3—21no—
235C3×Q8—61no—
236C3×Q8—61no—
237C3×D4—61no—
238C3×D4—61no—
239C3×D4—121no—
240C3×D4—121no—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
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—21no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—42no—
255C2×C3×C4—44no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—61no—
260C2×C3×C4—121no—
261C2×C3×C4—121no—
262C2×C3×C4C2×C4×D311yescharacteristic
263C2×C3×C4C2×D1211yescharacteristic
264C2×C3×C4#23511yescharacteristic
265C2×C3×C4D3×D411yescharacteristic
266C2×C3×C4#23711yescharacteristic
267C2×C3×C4#23911yescharacteristic
268C2×C3×C4C22×C3×C411yescharacteristic
269C2×C3×C4C2×C3×D411yescharacteristic
270C2×C3×C4C2×C3×D411yescharacteristic
271C2×C3×C4C3×#4111yescharacteristic
272C2×C3×C4C3×#4111yescharacteristic
273C2×C3×C4C3×#4111yescharacteristic
274#67—41no—
275#67—41no—
276C2×Dic3—21no—
277D12—21no—
278D12—21no—
279C4×D3—41no—
280C4×D3—41no—
281Dic6—21no—
282Dic6—21no—
283C3×C8—22no—
284C3×C8—22no—
285C3×C8—42no—
286C3×C8—62no—
287C3×C8—62no—
288C3×C8—121no—
289C3⋊C8—22no—
290C3⋊C8—22no—
291C3⋊C8—41no—
292C2×C32—21no—
293C2×C32—21no—
294C2×C32—21no—
295C2×C32#38811yescharacteristic
296C3×D3—81no—
297#41—61no—
298#41—61no—
299#41—61no—
300#41—61no—
301C2×Q8—31no—
302C2×D4—31no—
303C2×D4—61no—
304C22×C4—21no—
305C22×C4—61no—
306C22×C4C22×C3×D311yescharacteristic
307M16—21no—
308M16—32no—
309M16—32no—
310M16—61no—
311M16—61no—
312M16—61no—
313M16—61no—
314M16C3×C4×D312yes—
315C2×C8—21no—
316C2×C8—61no—
317C2×C8—61no—
318C2×C8—61no—
319C4⋊C4—21no—
320C22⋊C4—21no—
321C42—21no—
322C42—21no—
323C42C3×D1211yescharacteristic
324C42C3×D1211yescharacteristic
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—21no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—42no—
331C22×C3—121no—
332C22×C3—241no—
333C22×C3#88511yescharacteristic
334C22×C3#88611yescharacteristic
335C22×C3#89211yescharacteristic
336C22×C3C3×#11711yescharacteristic
337C22×C3C3×#12311yescharacteristic
338C22×C3C3×#12411yescharacteristic
339C2×D3—41no—
340C2×D3—81no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—42no—
348C3×C4—41no—
349C3×C4—41no—
350C3×C4—42no—
351C3×C4—82no—
352C3×C4—81no—
353C3×C4—81no—
354C3×C4—121no—
355C3×C4C4×D1211yescharacteristic
356C3×C4#67⋊C411yescharacteristic
357C3×C4#89511yescharacteristic
358C3×C4#89811yescharacteristic
359C3×C4C3×C4×D411yescharacteristic
360C3×C4C3×C4×D411yescharacteristic
361C3×C4C3×#12111yescharacteristic
362C3×C4C3×#12611yescharacteristic
363Dic3—41no—
364C32#193211yescharacteristic, complemented, Sylow
365C23—61no—
366C23C3×#23711yescharacteristic
367Q8—61no—
368Q8—61no—
369D4—61no—
370D4—61no—
371D4—121no—
372D4—121no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—42no—
376C2×C4—41no—
377C2×C4—61no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4C2×C3×C4×D311yescharacteristic
381C2×C4C2×C3×D1211yescharacteristic
382C2×C4C3×#23511yescharacteristic
383C2×C4C3×D3×D411yescharacteristic
384C2×C4C3×#23711yescharacteristic
385C2×C4C3×#23911yescharacteristic
386C8—22no—
387C8—62no—
388C8—62no—
389C8—121no—
390C2×C3—22no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—41no—
399C2×C3—41no—
400C2×C3—241no—
401C2×C3#467011yescharacteristic
402C2×C3C3×#38811yescharacteristic
403D3—81no—
404C22—41no—
405C22—121no—
406C22—241no—
407C22C3×#88511yescharacteristic
408C22C3×#88611yescharacteristic
409C22C3×#89211yescharacteristic
410C4—22no—
411C4—41no—
412C4—81no—
413C4—121no—
414C4C3×C4×D1211yescharacteristic
415C4C3×#67⋊C411yescharacteristic
416C4C3×#89511yescharacteristic
417C4C3×#89811yescharacteristic
418C3—21no—
419C3#6875011yescharacteristic, direct product factor
420C3C3×#193211yescharacteristic, complemented
421C2—21no—
422C2—21no—
423C2—21no—
424C2—241no—
425C2C3×#467011yescharacteristic
subgroupG/H
centerC3×C4#895
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#339C2
socleC2×C32#388
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC3×#339D3
hypocenterC3C3×#1932
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#339

Complements

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

NH ≅ G/Ntype
2. C3×#5125C2semidirect
5. C3×C4×#208C2semidirect
6. C32×#339C2semidirect
7. C3×#4533C2semidirect
9. #68750C3direct (N × H)
17. C3×#824C4semidirect
18. C3×#823C4semidirect
31. #5125C2×C3semidirect
34. C4×#208C2×C3semidirect
37. C3×#339D3semidirect
38. C3×#339C2×C3semidirect
39. #4533C2×C3semidirect
94. #824C3×C4semidirect
95. #823C3×C4semidirect
133. #339C3×D3semidirect
364. C32#1932semidirect
419. C3#68750direct (N × H)
420. C3C3×#1932semidirect

Maximal subgroups

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

subgroupindex
C3×#51322
C3×#51252
C3×#50052
C2×C3×#8052
C3×C4×#2082
C32×#3392
C3×#45332
C3×#19323
#687503