Finite Group Databasebeta

← C3×#70045 #11787762

Subgroups — C3×#70045 #11787762

Order 1152.

743 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5832C211yesmaximal, characteristic
2C3×#5819C211yesmaximal, characteristic, complemented
3C32×#410C211yesmaximal, characteristic, complemented
4C3×#5005C212yesmaximal, complemented
5C3×#4532C212yesmaximal, complemented
6C3×#1884—31nomaximal
7#70045C311yesmaximal, characteristic, direct product factor
8C2×C32×#41C2211yescharacteristic
9C2×C3×#235C2211yescharacteristic
10C32×#117C2212yes—
11C3×#948—24no—
12C3×#947C411yescharacteristic, complemented
13C3×#944C411yescharacteristic, complemented
14C3×#940C411yescharacteristic, complemented
15C3×#939—22no—
16C3×#936C411yescharacteristic, complemented
17C3×C4×#67—22no—
18C2×C3×C4×Dic3C2211yescharacteristic
19C2×C3×#208—22no—
20C2×C3×#208C2212yes—
21C32×#99—22no—
22C3×#805—24no—
23C3×#568—31no—
24C3×#532—31no—
25#5832C2×C311yescharacteristic
26#5819C2×C311yescharacteristic, complemented
27C3×#421—32no—
28C3×#410—21no—
29C3×#410D311yescharacteristic, complemented
30C3×#410C2×C311yescharacteristic, complemented
31#5005C2×C312yescomplemented
32C3×#339—32no—
33#4532C2×C312yescomplemented
34C32×#41—22no—
35C32×#41—22no—
36C2×C32×Q8C2×C411yescharacteristic
37C2×C32×D4C2×C411yescharacteristic
38C2×C32×D4D412yes—
39C22×C32×C4D412yes—
40C22×C32×C4C2311yescharacteristic
41C2×C3×#67—22no—
42C2×C3×#67—21no—
43C22×C3×Dic3—21no—
44C3×#235—22no—
45C3×#235—22no—
46C2×C3×D12C2×C411yescharacteristic
47C2×C3×C4×D3—21no—
48C2×C3×Dic6C2×C411yescharacteristic
49C32×C4⋊C4—22no—
50C32×C22⋊C4—22no—
51C32×C42—22no—
52C3×#217—22no—
53C3×(C2×D3)⋊C4—22no—
54C3×Dic3⋊C4—22no—
55C3×C4×Dic3—21no—
56C3×C4×Dic3C2×C411yescharacteristic
57C3×C4×Dic3C2×C411yescharacteristic
58C3×#208—24no—
59C3×#208—22no—
60C3×#208D414yescomplemented
61C2×C3×C3⋊C8—24no—
62C2×C3×C3⋊C8—22no—
63#1884—31noSylow
64C2×C3×#41—21no—
65C2×C3×#41—31no—
66C2×C3×#41C2×D311yescharacteristic
67C2×C3×#41C22×C311yescharacteristic
68C2×#235C22×C311yescharacteristic
69C3×#131—64no—
70C2×C3×M16—32no—
71C2×C3×M16—62no—
72C3×C4⋊Q8—31no—
73C3×#127—31no—
74C3×#124—31no—
75C3×#124—31no—
76C3×#121—62no—
77C3×C4×D4—62no—
78C3×#117—22no—
79C3×#117C2×D312yes—
80C3×#117C22×C312yes—
81C2×C3×C42—31no—
82#948—24no—
83#947C3×C411yescharacteristic, complemented
84#944C3×C411yescharacteristic, complemented
85#940C3×C411yescharacteristic, complemented
86#939—22no—
87#936C3×C411yescharacteristic, complemented
88C4×#67—22no—
89C2×C4×Dic3C22×C311yescharacteristic
90C2×#208—22no—
91C2×#208C22×C312yes—
92C3×#104—64no—
93C3×#99—24no—
94C3×#99—22no—
95C3×#99—22no—
96#805—24no—
97C23×C32C22⋊C412yes—
98C23×C32C2×D411yescharacteristic
99C22×C3×D3—21no—
100C32×Q8—22no—
101C32×D4—24no—
102C32×D4—22no—
103C2×C32×C4—22no—
104C2×C32×C4—22no—
105C2×C32×C4—42no—
106C2×C32×C4C22⋊C412yes—
107C2×C32×C4C22×C411yescharacteristic
108C2×C32×C4C2×D411yescharacteristic
109C2×C32×C4C2×D412yes—
110C2×C32×C4#4112yes—
111C3×#67—44no—
112C3×#67—42no—
113C2×C3×Dic3—21no—
114C2×C3×Dic3—21no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—42no—
117C3×D12—22no—
118C3×C4×D3—41no—
119C3×C4×D3—41no—
120C3×Dic6—22no—
121C3×C3⋊C8—24no—
122#568—31no—
123#532—31no—
124#421—32no—
125#410C3×D311yescharacteristic, complemented
126#339—32no—
127C3×#41—22no—
128C3×#41—22no—
129C3×#41—22no—
130C3×#41—22no—
131C3×#41—42no—
132C3×#41—42no—
133C3×#41—62no—
134C3×#41—62no—
135C2×C3×Q8—21no—
136C2×C3×Q8—31no—
137C2×C3×Q8C4×D311yescharacteristic
138C2×C3×Q8C2×C3×C411yescharacteristic
139C2×C3×D4—21no—
140C2×C3×D4—22no—
141C2×C3×D4—31no—
142C2×C3×D4—62no—
143C2×C3×D4—61no—
144C2×C3×D4C4×D311yescharacteristic
145C2×C3×D4#6712yes—
146C2×C3×D4C2×C3×C411yescharacteristic
147C2×C3×D4C3×D412yes—
148C22×C3×C4—22no—
149C22×C3×C4—21no—
150C22×C3×C4—61no—
151C22×C3×C4—61no—
152C22×C3×C4D1212yes—
153C22×C3×C4C3×D412yes—
154C22×C3×C4C22×D311yescharacteristic
155C22×C3×C4C23×C311yescharacteristic
156C2×#67—22no—
157C2×#67—21no—
158C22×Dic3—21no—
159#235—22no—
160#235—22no—
161C2×D12C2×C3×C411yescharacteristic
162C2×C4×D3—21no—
163C2×Dic6C2×C3×C411yescharacteristic
164C3×M16—34no—
165C3×M16—64no—
166C3×M16—62no—
167C2×C3×C8—64no—
168C2×C3×C8—62no—
169C3×C4⋊C4—22no—
170C3×C4⋊C4—22no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—62no—
173C3×C22⋊C4—24no—
174C3×C22⋊C4—22no—
175C3×C22⋊C4—22no—
176C3×C22⋊C4—62no—
177C3×C22⋊C4—62no—
178C3×C42—22no—
179C3×C42—22no—
180C3×C42—31no—
181C3×C42—31no—
182C3×C42—42no—
183C3×C42—61no—
184#217—22no—
185(C2×D3)⋊C4—22no—
186Dic3⋊C4—22no—
187C4×Dic3—21no—
188C4×Dic3C2×C3×C411yescharacteristic
189C4×Dic3C2×C3×C411yescharacteristic
190#208—24no—
191#208—22no—
192#208C3×D414yescomplemented
193C2×C3⋊C8—24no—
194C2×C3⋊C8—22no—
195C22×C32—22no—
196C22×C32—41no—
197C22×C32—42no—
198C22×C32C2×C22⋊C411yescharacteristic
199C22×C32#12112yes—
200C2×C3×D3—42no—
201C2×C3×D3—41no—
202C32×C4—22no—
203C32×C4—82no—
204C32×C4C4×D412yes—
205C32×C4#12011yescharacteristic
206C32×C4#12311yescharacteristic
207C3×Dic3—22no—
208C3×Dic3—41no—
209C3×Dic3—41no—
210C2×#41—31no—
211C2×#41C2×C3×D311yescharacteristic
212#131—64no—
213C2×M16—32no—
214C2×M16—62no—
215C4⋊Q8—31no—
216#127—31no—
217#124—31no—
218#124—31no—
219#121—62no—
220C4×D4—62no—
221#117C2×C3×D312yes—
222C2×C42—31no—
223#104—64no—
224#99—22no—
225C23×C3—22no—
226C23×C3—21no—
227C23×C3—61no—
228C23×C3(C2×D3)⋊C412yes—
229C23×C3C3×C22⋊C412yes—
230C23×C3C2×#6711yescharacteristic
231C23×C3C2×C3×D411yescharacteristic
232C22×D3—21no—
233C3×Q8—22no—
234C3×Q8—22no—
235C3×Q8—42no—
236C3×Q8—62no—
237C3×D4—24no—
238C3×D4—24no—
239C3×D4—22no—
240C3×D4—22no—
241C3×D4—42no—
242C3×D4—44no—
243C3×D4—62no—
244C3×D4—124no—
245C3×D4—122no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—22no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—44no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—61no—
263C2×C3×C4—122no—
264C2×C3×C4—121no—
265C2×C3×C4—121no—
266C2×C3×C4(C2×D3)⋊C412yes—
267C2×C3×C4C3×C22⋊C412yes—
268C2×C3×C4C2×C4×D311yescharacteristic
269C2×C3×C4C2×D1211yescharacteristic
270C2×C3×C4D3×D412yes—
271C2×C3×C4#23712yes—
272C2×C3×C4C22×C3×C411yescharacteristic
273C2×C3×C4C2×C3×D411yescharacteristic
274C2×C3×C4C2×C3×D412yes—
275C2×C3×C4C3×#4112yes—
276#67—44no—
277#67—42no—
278C2×Dic3—21no—
279C2×Dic3—21no—
280C2×Dic3—21no—
281C2×Dic3—42no—
282D12—22no—
283C4×D3—41no—
284C4×D3—41no—
285Dic6—22no—
286C3×C8—64no—
287C3⋊C8—24no—
288C2×C32—21no—
289C2×C32—22no—
290C2×C32—42no—
291C2×C32#38611yescharacteristic
292C3×D3—81no—
293#41—22no—
294#41—22no—
295#41—62no—
296#41—62no—
297C2×Q8—31no—
298C2×Q8C3×C4×D311yescharacteristic
299C2×D4—31no—
300C2×D4—62no—
301C2×D4—61no—
302C2×D4C3×C4×D311yescharacteristic
303C2×D4C3×#6712yes—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4C3×D1212yes—
307C22×C4C22×C3×D311yescharacteristic
308M16—34no—
309M16—64no—
310M16—62no—
311C2×C8—64no—
312C2×C8—62no—
313C4⋊C4—22no—
314C4⋊C4—62no—
315C22⋊C4—22no—
316C22⋊C4—62no—
317C22⋊C4—62no—
318C42—22no—
319C42—31no—
320C42—31no—
321C42—61no—
322C22×C3—22no—
323C22×C3—22no—
324C22×C3—21no—
325C22×C3—22no—
326C22×C3—42no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—42no—
330C22×C3—42no—
331C22×C3—81no—
332C22×C3—82no—
333C22×C3—122no—
334C22×C3—121no—
335C22×C3C2×(C2×D3)⋊C411yescharacteristic
336C22×C3#93912yes—
337C22×C3C2×C3×C22⋊C411yescharacteristic
338C22×C3C3×#12112yes—
339C2×D3—42no—
340C2×D3—41no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—22no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—42no—
347C3×C4—62no—
348C3×C4—82no—
349C3×C4—82no—
350C3×C4—84no—
351C3×C4—121no—
352C3×C4—121no—
353C3×C4#67⋊C412yes—
354C3×C4#88211yescharacteristic
355C3×C4#88611yescharacteristic
356C3×C4C3×C4×D412yes—
357C3×C4C3×#12011yescharacteristic
358C3×C4C3×#12311yescharacteristic
359Dic3—22no—
360Dic3—41no—
361Dic3—41no—
362C32#188411yescharacteristic, complemented, Sylow
363C23—61no—
364C23C3×(C2×D3)⋊C412yes—
365C23C2×C3×#6711yescharacteristic
366Q8—22no—
367Q8—62no—
368D4—24no—
369D4—22no—
370D4—62no—
371D4—124no—
372D4—122no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—42no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—122no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4C3×(C2×D3)⋊C412yes—
383C2×C4C2×C3×C4×D311yescharacteristic
384C2×C4C2×C3×D1211yescharacteristic
385C2×C4C3×D3×D412yes—
386C2×C4C3×#23712yes—
387C8—64no—
388C2×C3—21no—
389C2×C3—22no—
390C2×C3—21no—
391C2×C3—22no—
392C2×C3—21no—
393C2×C3—42no—
394C2×C3—41no—
395C2×C3—42no—
396C2×C3—42no—
397C2×C3—82no—
398C2×C3—241no—
399C2×C3#495811yescharacteristic
400C2×C3C3×#38611yescharacteristic
401D3—81no—
402C22—22no—
403C22—41no—
404C22—42no—
405C22—122no—
406C22—121no—
407C22C2×C3×(C2×D3)⋊C411yescharacteristic
408C22C3×#93912yes—
409C4—22no—
410C4—62no—
411C4—82no—
412C4—121no—
413C4—121no—
414C4C3×#67⋊C412yes—
415C4C3×#88211yescharacteristic
416C4C3×#88611yescharacteristic
417C3—21no—
418C3#7004511yescharacteristic, direct product factor
419C3C3×#188411yescharacteristic, complemented
420C2—21no—
421C2—22no—
422C2—42no—
423C2—241no—
424C2C3×#495811yescharacteristic
subgroupG/H
centerC3×C4#882
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC32×#410C2
socleC2×C32#386
prime radicalC3×#5832C2
cosocleC22×C3×C4C23×C3
hypercenterC3×#410D3
hypocenterC3C3×#1884
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C3×#410

Complements

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

NH ≅ G/Ntype
2. C3×#5819C2semidirect
3. C32×#410C2semidirect
4. C3×#5005C2semidirect
5. C3×#4532C2semidirect
7. #70045C3direct (N × H)
12. C3×#947C4semidirect
13. C3×#944C4semidirect
14. C3×#940C4semidirect
16. C3×#936C4semidirect
26. #5819C2×C3semidirect
29. C3×#410D3semidirect
30. C3×#410C2×C3semidirect
31. #5005C2×C3semidirect
33. #4532C2×C3semidirect
60. C3×#208D4semidirect
83. #947C3×C4semidirect
84. #944C3×C4semidirect
85. #940C3×C4semidirect
87. #936C3×C4semidirect
125. #410C3×D3semidirect
192. #208C3×D4semidirect
362. C32#1884semidirect
418. C3#70045direct (N × H)
419. C3C3×#1884semidirect

Maximal subgroups

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

subgroupindex
C3×#58322
C3×#58192
C32×#4102
C3×#50052
C3×#45322
C3×#18843
#700453