Finite Group Databasebeta

← C3×#70715 #11794318

Subgroups — C3×#70715 #11794318

Order 1152.

509 conjugacy classes of subgroups.

The subgroup lattice has 432 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C32×#108C211yesmaximal, characteristic, complemented
2C3×#5131C211yesmaximal, characteristic
3C2×C3×#823C211yesmaximal, characteristic
4C3×#5121C211yesmaximal, characteristic, complemented
5C2×C3×#821C211yesmaximal, characteristic
6C3×#5102C211yesmaximal, characteristic, complemented
7C3×#4554C211yesmaximal, characteristic, complemented
8C3×#2063—31nomaximal
9#70715C311yesmaximal, characteristic, direct product factor
10C22×C3×D12C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C22×C32×C8C2211yescharacteristic
14C3×#926C2211yescharacteristic
15C2×C3×C4⋊Dic3C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C32×#108—22no—
18C3×#823—22no—
19C3×#821—22no—
20C3×#821—21no—
21C3×#821—21no—
22C3×#818—21no—
23C3×#817—21no—
24C3×Dic3⋊C8—22no—
25C2×C3×#108—21no—
26C2×C3×#108D311yescharacteristic, complemented
27C2×C3×#108C2×C311yescharacteristic, complemented
28C3×#428—31no—
29C3×#424—31no—
30C3×#418—31no—
31C2×C3×#102—31no—
32C2×C3×#100—31no—
33#5131C2×C311yescharacteristic
34C2×#823C2×C311yescharacteristic
35#5121C2×C311yescharacteristic, complemented
36C2×#821C2×C311yescharacteristic
37#5102C2×C311yescharacteristic, complemented
38C3×#343—31no—
39#4554C2×C311yescharacteristic, complemented
40C23×C3×D3—21no—
41C22×C32×C4C2311yescharacteristic
42C22×C3×Dic3—21no—
43C2×C3×D12—21no—
44C2×C3×D12—21no—
45C2×C3×D12—21no—
46C2×C3×D12—21no—
47C2×C3×D12—21no—
48C2×C3×D12Q812yes—
49C32×M16—21no—
50C32×M16—21no—
51C32×M16—21no—
52C32×M16D412yes—
53C2×C32×C8—21no—
54C2×C32×C8—21no—
55C2×C32×C8—21no—
56C2×C32×C8—21no—
57C2×C32×C8D412yes—
58C3×#217—21no—
59C3×C4⋊Dic3—21no—
60C3×C4⋊Dic3—21no—
61C3×C4⋊Dic3D411yescharacteristic
62C3×C4⋊Dic3D411yescharacteristic
63C3×C4×Dic3—21no—
64C3×#208—21no—
65C3×#208—21no—
66C2×C3×C3⋊C8—21no—
67#2063—31noSylow
68C22×C3×D4—31no—
69C22×D12C22×C311yescharacteristic
70C2×C3×M16—21no—
71C2×C3×M16—21no—
72C2×C3×M16—31no—
73C2×C3×M16C2×D311yescharacteristic
74C2×C3×M16C2×D311yescharacteristic
75C2×C3×M16C22×C311yescharacteristic
76C2×C3×M16C22×C311yescharacteristic
77C22×C3×C8—21no—
78C22×C3×C8C2×D311yescharacteristic
79C22×C3×C8C22×C311yescharacteristic
80C3×#117—31no—
81C2×C3×C4⋊C4—31no—
82#926C22×C311yescharacteristic
83C2×C4⋊Dic3C22×C311yescharacteristic
84C2×#208C22×C311yescharacteristic
85C3×#108—24no—
86C3×#108—22no—
87C3×#108—22no—
88C3×#107—61no—
89C3×#106—61no—
90C3×C4⋊C8—62no—
91C3×#102—61no—
92C3×#102—61no—
93C3×#102—62no—
94C3×#100—62no—
95#823—22no—
96#821—22no—
97#821—21no—
98#821—21no—
99#818—21no—
100#817—21no—
101Dic3⋊C8—22no—
102C23×C32#4111yescharacteristic
103C22×C3×D3—22no—
104C22×C3×D3—41no—
105C22×C3×D3—41no—
106C22×C3×D3—81no—
107C22×C3×D3—81no—
108C2×C32×C4C2×D411yescharacteristic
109C2×C32×C4C2×D411yescharacteristic
110C2×C32×C4C2×D411yescharacteristic
111C2×C32×C4C2×Q811yescharacteristic
112C2×C32×C4#4111yescharacteristic
113C2×C32×C4#4111yescharacteristic
114C2×C3×Dic3—21no—
115C2×C3×Dic3—21no—
116C2×C3×Dic3—41no—
117C2×C3×Dic3—81no—
118C3×D12—22no—
119C3×D12—22no—
120C3×D12—41no—
121C3×D12—41no—
122C32×C8—22no—
123C32×C8—22no—
124C32×C8—41no—
125C3×C3⋊C8—41no—
126C2×#108C3×D311yescharacteristic, complemented
127#428—31no—
128#424—31no—
129#418—31no—
130C2×#102—31no—
131C2×#100—31no—
132#343—31no—
133C24×C3—61no—
134C23×D3—21no—
135C2×C3×D4—32no—
136C2×C3×D4—61no—
137C2×C3×D4—61no—
138C2×C3×D4—61no—
139C2×C3×D4—61no—
140C2×C3×D4—61no—
141C22×C3×C4—21no—
142C22×C3×C4—61no—
143C22×C3×C4C22×D311yescharacteristic
144C22×C3×C4C23×C311yescharacteristic
145C22×Dic3—21no—
146C2×D12—21no—
147C2×D12—21no—
148C2×D12—21no—
149C2×D12—21no—
150C2×D12—21no—
151C2×D12C3×Q812yes—
152C3×M16—21no—
153C3×M16—21no—
154C3×M16—22no—
155C3×M16—21no—
156C3×M16—21no—
157C3×M16—21no—
158C3×M16—21no—
159C3×M16—22no—
160C3×M16—41no—
161C3×M16—41no—
162C3×M16—61no—
163C3×M16—61no—
164C3×M16#6712yes—
165C3×M16C3×D412yes—
166C2×C3×C8—22no—
167C2×C3×C8—22no—
168C2×C3×C8—21no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—21no—
172C2×C3×C8—21no—
173C2×C3×C8—21no—
174C2×C3×C8—21no—
175C2×C3×C8—21no—
176C2×C3×C8—22no—
177C2×C3×C8—41no—
178C2×C3×C8—41no—
179C2×C3×C8—61no—
180C2×C3×C8D1212yes—
181C2×C3×C8C3×D412yes—
182C3×C4⋊C4—31no—
183C3×C4⋊C4—31no—
184C3×C4⋊C4—61no—
185C3×C4⋊C4—61no—
186C3×C22⋊C4—61no—
187C3×C42—61no—
188#217—21no—
189C4⋊Dic3—21no—
190C4⋊Dic3—21no—
191C4⋊Dic3C3×D411yescharacteristic
192C4⋊Dic3C3×D411yescharacteristic
193C4×Dic3—21no—
194#208—21no—
195#208—21no—
196C2×C3⋊C8—21no—
197C22×C32—21no—
198C22×C32—21no—
199C22×C32#12111yescharacteristic
200C22×C32C22⋊Q811yescharacteristic
201C22×C32#12611yescharacteristic
202C2×C3×D3—42no—
203C2×C3×D3—82no—
204C2×C3×D3—81no—
205C2×C3×D3—81no—
206C2×C3×D3—82no—
207C32×C4#12111yescharacteristic
208C32×C4#12111yescharacteristic
209C32×C4C22⋊Q811yescharacteristic
210C32×C4C22⋊Q811yescharacteristic
211C3×Dic3—41no—
212C3×Dic3—41no—
213C3×Dic3—81no—
214C22×D4—31no—
215C2×M16—31no—
216C2×M16C2×C3×D311yescharacteristic
217C2×M16C2×C3×D311yescharacteristic
218C22×C8C2×C3×D311yescharacteristic
219#117—31no—
220C2×C4⋊C4—31no—
221#108—22no—
222#107—61no—
223#106—61no—
224C4⋊C8—62no—
225#102—61no—
226#102—61no—
227#102—62no—
228#100—62no—
229C23×C3—21no—
230C23×C3—62no—
231C23×C3—121no—
232C23×C3—121no—
233C23×C3—241no—
234C23×C3—241no—
235C23×C3#23911yescharacteristic
236C23×C3C3×#4111yescharacteristic
237C22×D3—22no—
238C22×D3—41no—
239C22×D3—41no—
240C22×D3—81no—
241C22×D3—81no—
242C3×D4—62no—
243C3×D4—62no—
244C3×D4—121no—
245C3×D4—121no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—61no—
253C2×C3×C4—61no—
254C2×C3×C4—121no—
255C2×C3×C4—241no—
256C2×C3×C4C2×D1211yescharacteristic
257C2×C3×C4#23511yescharacteristic
258C2×C3×C4D3×D411yescharacteristic
259C2×C3×C4#23711yescharacteristic
260C2×C3×C4D3×Q811yescharacteristic
261C2×C3×C4C2×#6711yescharacteristic
262C2×C3×C4C2×C3×D411yescharacteristic
263C2×C3×C4C2×C3×D411yescharacteristic
264C2×C3×C4C2×C3×D411yescharacteristic
265C2×C3×C4C2×C3×Q811yescharacteristic
266C2×C3×C4C3×#4111yescharacteristic
267C2×C3×C4C3×#4111yescharacteristic
268C2×Dic3—21no—
269C2×Dic3—21no—
270C2×Dic3—41no—
271C2×Dic3—81no—
272D12—22no—
273D12—22no—
274D12—41no—
275D12—41no—
276C3×C8—24no—
277C3×C8—22no—
278C3×C8—22no—
279C3×C8—22no—
280C3×C8—22no—
281C3×C8—42no—
282C3×C8—42no—
283C3×C8—41no—
284C3×C8—41no—
285C3×C8—121no—
286C3⋊C8—41no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32#39611yescharacteristic
290C2×C32#47111yescharacteristic
291C2×C32#47211yescharacteristic
292C3×D3—82no—
293C24—61no—
294C2×D4—32no—
295C2×D4—61no—
296C2×D4—61no—
297C2×D4—61no—
298C2×D4—61no—
299C2×D4—61no—
300C22×C4—61no—
301C22×C4C22×C3×D311yescharacteristic
302M16—21no—
303M16—21no—
304M16—21no—
305M16—61no—
306M16—61no—
307M16C3×#6712yes—
308C2×C8—21no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—61no—
313C2×C8C3×D1212yes—
314C4⋊C4—31no—
315C4⋊C4—31no—
316C4⋊C4—61no—
317C4⋊C4—61no—
318C22⋊C4—61no—
319C42—61no—
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—21no—
326C22×C3—21no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—122no—
330C22×C3—242no—
331C22×C3—242no—
332C22×C3—241no—
333C22×C3—241no—
334C22×C3#89511yescharacteristic
335C22×C3#89811yescharacteristic
336C22×C3#94611yescharacteristic
337C22×C3C3×#12111yescharacteristic
338C22×C3C3×C22⋊Q811yescharacteristic
339C22×C3C3×#12611yescharacteristic
340C2×D3—42no—
341C2×D3—82no—
342C2×D3—81no—
343C2×D3—81no—
344C2×D3—82no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—121no—
350C3×C4—121no—
351C3×C4—241no—
352C3×C4#89611yescharacteristic
353C3×C4#89711yescharacteristic
354C3×C4#93011yescharacteristic
355C3×C4#93911yescharacteristic
356C3×C4C3×#12111yescharacteristic
357C3×C4C3×#12111yescharacteristic
358C3×C4C3×C22⋊Q811yescharacteristic
359C3×C4C3×C22⋊Q811yescharacteristic
360Dic3—41no—
361Dic3—41no—
362Dic3—81no—
363C32#206311yescharacteristic, complemented, Sylow
364C23—62no—
365C23—121no—
366C23—121no—
367C23—241no—
368C23—241no—
369C23C3×#23911yescharacteristic
370D4—62no—
371D4—62no—
372D4—121no—
373D4—121no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—121no—
377C2×C4—241no—
378C2×C4C2×C3×D1211yescharacteristic
379C2×C4C3×#23511yescharacteristic
380C2×C4C3×D3×D411yescharacteristic
381C2×C4C3×#23711yescharacteristic
382C2×C4C3×D3×Q811yescharacteristic
383C2×C4C2×C3×#6711yescharacteristic
384C8—22no—
385C8—22no—
386C8—41no—
387C8—121no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—242no—
398C2×C3#489711yescharacteristic
399C2×C3#489811yescharacteristic
400C2×C3#499411yescharacteristic
401C2×C3C3×#39611yescharacteristic
402C2×C3C3×#47111yescharacteristic
403C2×C3C3×#47211yescharacteristic
404D3—82no—
405C22—21no—
406C22—21no—
407C22—122no—
408C22—242no—
409C22—242no—
410C22—241no—
411C22—241no—
412C22C3×#89511yescharacteristic
413C22C3×#89811yescharacteristic
414C22C3×#94611yescharacteristic
415C4—121no—
416C4—121no—
417C4—241no—
418C4C3×#89611yescharacteristic
419C4C3×#89711yescharacteristic
420C4C3×#93011yescharacteristic
421C4C3×#93911yescharacteristic
422C3—21no—
423C3#7071511yescharacteristic, direct product factor
424C3C3×#206311yescharacteristic, complemented
425C2—21no—
426C2—21no—
427C2—242no—
428C2C3×#489711yescharacteristic
429C2C3×#489811yescharacteristic
430C2C3×#499411yescharacteristic
subgroupG/H
centerC22×C3#895
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#108C2
socleC22×C32#121
prime radicalC22×C3×D12C22
hypercenterC2×C3×#108D3
hypocenterC3C3×#2063
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#108

Complements

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

NH ≅ G/Ntype
1. C2×C32×#108C2semidirect
4. C3×#5121C2semidirect
6. C3×#5102C2semidirect
7. C3×#4554C2semidirect
9. #70715C3direct (N × H)
26. C2×C3×#108D3semidirect
27. C2×C3×#108C2×C3semidirect
35. #5121C2×C3semidirect
37. #5102C2×C3semidirect
39. #4554C2×C3semidirect
126. C2×#108C3×D3semidirect
363. C32#2063semidirect
423. C3#70715direct (N × H)
424. C3C3×#2063semidirect

Maximal subgroups

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

subgroupindex
C2×C32×#1082
C3×#51312
C2×C3×#8232
C3×#51212
C2×C3×#8212
C3×#51022
C3×#45542
C3×#20633
#707153