Finite Group Databasebeta

← C3×#70646 #11785618

Subgroups — C3×#70646 #11785618

Order 1152.

593 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#901C211yesmaximal, characteristic
2C2×C32×#107C211yesmaximal, characteristic, complemented
3C2×C3×#819C211yesmaximal, characteristic, complemented
4C3×#4966C212yesmaximal
5C3×#4530C212yesmaximal, complemented
6C3×#1816—31nomaximal
7#70646C311yesmaximal, characteristic, direct product factor
8C2×C3×#235C2211yescharacteristic
9C22×C32×C8C2211yescharacteristic
10C2×C32×C4⋊C4C2212yes—
11C2×C3×#208C2212yes—
12C2×C3×#208C2211yescharacteristic
13C3×#901—22no—
14C3×#901C414yescomplemented
15C2×C3×ΓL(1,𝔽25)—21no—
16C2×C3×D3×C8—21no—
17C32×#107—21no—
18C32×#107—21no—
19C3×#819—22no—
20C3×#810—22no—
21C3×#809—22no—
22C2×C3×#131—31no—
23C2×#901C2×C311yescharacteristic
24C2×C3×#108—31no—
25C2×C3×#107—21no—
26C2×C3×#107D311yescharacteristic, complemented
27C2×C3×#107C2×C311yescharacteristic, complemented
28C3×#417—32no—
29C2×#819C2×C311yescharacteristic, complemented
30#4966C2×C312yes—
31C3×#343—32no—
32#4530C2×C312yescomplemented
33C22×C32×C4—22no—
34C22×C32×C4C2311yescharacteristic
35C2×C3×#67—21no—
36C3×#235—21no—
37C3×#235—21no—
38C3×#235C2×C412yes—
39C3×#235Q812yes—
40C2×C3×D12D411yescharacteristic
41C2×C3×C4×D3—21no—
42C2×C3×Dic6D411yescharacteristic
43C2×C32×C8C2×C412yes—
44C2×C32×C8D412yes—
45C2×C32×C8D411yescharacteristic
46C2×C32×C8D411yescharacteristic
47C32×C4⋊C4—22no—
48C32×C4⋊C4—22no—
49C3×#208—21no—
50C3×#208—22no—
51C3×#208—22no—
52C3×#208C2×C412yes—
53C2×C3×C3⋊C8—21no—
54C2×C3×C3⋊C8—22no—
55C3×ΓL(1,𝔽25)—24no—
56C3×D3×C8—24no—
57#1816—31noSylow
58C2×C3×#41—31no—
59C2×#235C22×C311yescharacteristic
60C3×#131—34no—
61C3×#131—62no—
62C2×C3×M16—31no—
63C2×C3×M16—32no—
64C2×C3×M16—61no—
65C22×C3×C8—21no—
66C22×C3×C8—61no—
67C22×C3×C8C2×D311yescharacteristic
68C22×C3×C8C22×C311yescharacteristic
69C2×C3×C4⋊C4—22no—
70C2×C3×C4⋊C4C2×D312yes—
71C2×C3×C4⋊C4C22×C312yes—
72C2×#208C22×C312yes—
73C2×#208C22×C311yescharacteristic
74#901—22no—
75#901C3×C414yescomplemented
76C2×ΓL(1,𝔽25)—21no—
77C2×D3×C8—21no—
78C3×#108—62no—
79C3×#107—22no—
80C3×#107—22no—
81C3×#107—21no—
82C3×#107—21no—
83C3×#107—21no—
84C3×#107—21no—
85C3×#103—62no—
86C3×#102—62no—
87#819—22no—
88#810—22no—
89#809—22no—
90C23×C32#4111yescharacteristic
91C22×C3×D3—21no—
92C2×C32×C4—42no—
93C2×C32×C4—82no—
94C2×C32×C4C22×C411yescharacteristic
95C2×C32×C4C2×D411yescharacteristic
96C2×C32×C4C2×D411yescharacteristic
97C2×C32×C4C2×D411yescharacteristic
98C2×C32×C4C2×Q811yescharacteristic
99C2×C32×C4#4111yescharacteristic
100C3×#67—22no—
101C3×#67—22no—
102C2×C3×Dic3—21no—
103C3×D12—21no—
104C3×D12C4⋊C412yes—
105C3×C4×D3—22no—
106C3×C4×D3—22no—
107C3×Dic6—21no—
108C3×Dic6C4⋊C412yes—
109C32×C8C22⋊C412yes—
110C32×C8C22⋊C412yes—
111C3×C3⋊C8—22no—
112C3×C3⋊C8—42no—
113C2×#131—31no—
114C2×#108—31no—
115C2×#107C3×D311yescharacteristic, complemented
116#417—32no—
117#343—32no—
118C3×#41—32no—
119C3×#41—32no—
120C3×#41—61no—
121C3×#41—61no—
122C2×C3×Q8—31no—
123C2×C3×D4—31no—
124C2×C3×D4—61no—
125C22×C3×C4—22no—
126C22×C3×C4—22no—
127C22×C3×C4—21no—
128C22×C3×C4—42no—
129C22×C3×C4—61no—
130C22×C3×C4C22×D311yescharacteristic
131C22×C3×C4C23×C311yescharacteristic
132C2×#67—21no—
133#235—21no—
134#235—21no—
135#235C2×C3×C412yes—
136#235C3×Q812yes—
137C2×D12C3×D411yescharacteristic
138C2×C4×D3—21no—
139C2×Dic6C3×D411yescharacteristic
140C3×M16—32no—
141C3×M16—61no—
142C3×M16—64no—
143C3×M16—62no—
144C3×M16—62no—
145C2×C3×C8—22no—
146C2×C3×C8—22no—
147C2×C3×C8—21no—
148C2×C3×C8—21no—
149C2×C3×C8—61no—
150C2×C3×C8—64no—
151C2×C3×C8—62no—
152C2×C3×C8C4×D312yes—
153C2×C3×C8D1212yes—
154C2×C3×C8#6711yescharacteristic
155C2×C3×C8#6711yescharacteristic
156C2×C3×C8C2×C3×C412yes—
157C2×C3×C8C3×D412yes—
158C2×C3×C8C3×D411yescharacteristic
159C2×C3×C8C3×D411yescharacteristic
160C3×C4⋊C4—24no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—22no—
165C3×C4⋊C4—42no—
166#208—21no—
167#208—22no—
168#208—22no—
169#208C2×C3×C412yes—
170C2×C3⋊C8—21no—
171C2×C3⋊C8—22no—
172ΓL(1,𝔽25)—24no—
173D3×C8—24no—
174C22×C32—21no—
175C22×C32—21no—
176C22×C32#11711yescharacteristic
177C22×C32#12111yescharacteristic
178C22×C32C22⋊Q811yescharacteristic
179C2×C3×D3—22no—
180C2×C3×D3—41no—
181C2×C3×D3—41no—
182C32×C4—82no—
183C32×C4C2×C22⋊C411yescharacteristic
184C32×C4C2×C4⋊C411yescharacteristic
185C32×C4#12111yescharacteristic
186C32×C4C22⋊Q811yescharacteristic
187C3×Dic3—22no—
188C2×#41—31no—
189#131—34no—
190#131—62no—
191C2×M16—31no—
192C2×M16—32no—
193C2×M16—61no—
194C22×C8—61no—
195C22×C8C2×C3×D311yescharacteristic
196C2×C4⋊C4C2×C3×D312yes—
197#108—62no—
198#107—21no—
199#107—21no—
200#103—62no—
201#102—62no—
202C23×C3—21no—
203C23×C3—61no—
204C23×C3#23711yescharacteristic
205C23×C3C3×#4111yescharacteristic
206C22×D3—21no—
207C3×Q8—32no—
208C3×Q8—61no—
209C3×D4—32no—
210C3×D4—62no—
211C3×D4—62no—
212C3×D4—61no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—21no—
218C2×C3×C4—21no—
219C2×C3×C4—44no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—61no—
223C2×C3×C4—62no—
224C2×C3×C4—62no—
225C2×C3×C4—84no—
226C2×C3×C4—82no—
227C2×C3×C4—82no—
228C2×C3×C4C2×C4×D311yescharacteristic
229C2×C3×C4C2×D1211yescharacteristic
230C2×C3×C4D3×D411yescharacteristic
231C2×C3×C4D3×Q811yescharacteristic
232C2×C3×C4#23911yescharacteristic
233C2×C3×C4C2×#6711yescharacteristic
234C2×C3×C4C22×C3×C411yescharacteristic
235C2×C3×C4C2×C3×D411yescharacteristic
236C2×C3×C4C2×C3×D411yescharacteristic
237C2×C3×C4C2×C3×D411yescharacteristic
238C2×C3×C4C2×C3×Q811yescharacteristic
239C2×C3×C4C3×#4111yescharacteristic
240#67—22no—
241#67—22no—
242C2×Dic3—21no—
243D12—21no—
244D12C3×C4⋊C412yes—
245C4×D3—22no—
246C4×D3—22no—
247Dic6—21no—
248Dic6C3×C4⋊C412yes—
249C3×C8—22no—
250C3×C8—22no—
251C3×C8—62no—
252C3×C8—122no—
253C3×C8(C2×D3)⋊C412yes—
254C3×C8(C2×D3)⋊C412yes—
255C3×C8C3×C22⋊C412yes—
256C3×C8C3×C22⋊C412yes—
257C3⋊C8—22no—
258C3⋊C8—42no—
259C2×C32—21no—
260C2×C32—21no—
261C2×C32#38011yescharacteristic
262C2×C32#47211yescharacteristic
263C2×C32#47311yescharacteristic
264C3×D3—42no—
265#41—32no—
266#41—32no—
267#41—61no—
268#41—61no—
269C2×Q8—31no—
270C2×D4—31no—
271C2×D4—61no—
272C22×C4—22no—
273C22×C4—61no—
274C22×C4C22×C3×D311yescharacteristic
275M16—32no—
276M16—61no—
277M16—64no—
278M16—62no—
279M16—62no—
280C2×C8—61no—
281C2×C8—64no—
282C2×C8—62no—
283C2×C8C3×C4×D312yes—
284C2×C8C3×D1212yes—
285C2×C8C3×#6711yescharacteristic
286C2×C8C3×#6711yescharacteristic
287C4⋊C4—22no—
288C4⋊C4—22no—
289C22×C3—22no—
290C22×C3—22no—
291C22×C3—21no—
292C22×C3—21no—
293C22×C3—21no—
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—21no—
297C22×C3—21no—
298C22×C3—62no—
299C22×C3—121no—
300C22×C3—121no—
301C22×C3#89211yescharacteristic
302C22×C3#89711yescharacteristic
303C22×C3#93811yescharacteristic
304C22×C3C3×#11711yescharacteristic
305C22×C3C3×#12111yescharacteristic
306C22×C3C3×C22⋊Q811yescharacteristic
307C2×D3—22no—
308C2×D3—41no—
309C2×D3—41no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—62no—
315C3×C4—84no—
316C3×C4—82no—
317C3×C4—82no—
318C3×C4D3×C4⋊C411yescharacteristic
319C3×C4#89511yescharacteristic
320C3×C4C2×(C2×D3)⋊C411yescharacteristic
321C3×C4#94611yescharacteristic
322C3×C4C2×C3×C22⋊C411yescharacteristic
323C3×C4C2×C3×C4⋊C411yescharacteristic
324C3×C4C3×#12111yescharacteristic
325C3×C4C3×C22⋊Q811yescharacteristic
326Dic3—22no—
327C32#181611yescharacteristic, complemented, Sylow
328C23—61no—
329C23C3×#23711yescharacteristic
330Q8—32no—
331Q8—61no—
332D4—32no—
333D4—62no—
334D4—62no—
335D4—61no—
336C2×C4—42no—
337C2×C4—61no—
338C2×C4—62no—
339C2×C4—62no—
340C2×C4—82no—
341C2×C4C2×C3×C4×D311yescharacteristic
342C2×C4C2×C3×D1211yescharacteristic
343C2×C4C3×D3×D411yescharacteristic
344C2×C4C3×D3×Q811yescharacteristic
345C2×C4C3×#23911yescharacteristic
346C2×C4C2×C3×#6711yescharacteristic
347C8—62no—
348C8—122no—
349C8C3×(C2×D3)⋊C412yes—
350C8C3×(C2×D3)⋊C412yes—
351C2×C3—22no—
352C2×C3—22no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3—21no—
358C2×C3—21no—
359C2×C3—21no—
360C2×C3—122no—
361C2×C3#498711yescharacteristic
362C2×C3#516011yescharacteristic
363C2×C3#519211yescharacteristic
364C2×C3C3×#38011yescharacteristic
365C2×C3C3×#47211yescharacteristic
366C2×C3C3×#47311yescharacteristic
367D3—42no—
368C22—21no—
369C22—21no—
370C22—62no—
371C22—121no—
372C22—121no—
373C22C3×#89211yescharacteristic
374C22C3×#89711yescharacteristic
375C22C3×#93811yescharacteristic
376C4—62no—
377C4—82no—
378C4C3×D3×C4⋊C411yescharacteristic
379C4C3×#89511yescharacteristic
380C4C2×C3×(C2×D3)⋊C411yescharacteristic
381C4C3×#94611yescharacteristic
382C3—21no—
383C3#7064611yescharacteristic, direct product factor
384C3C3×#181611yescharacteristic, complemented
385C2—21no—
386C2—21no—
387C2—122no—
388C2C3×#498711yescharacteristic
389C2C3×#516011yescharacteristic
390C2C3×#519211yescharacteristic
subgroupG/H
centerC22×C3#938
commutatorC2×C3×C4C22×C3×C4
FrattiniC22×C4C22×C3×D3
FittingC2×C32×#107C2
socleC22×C32#121
prime radicalC2×C3×#235C22
cosocleC22×C3×C4C23×C3
hypercenterC2×C3×#107D3
hypocenterC3C3×#1816
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#107

Complements

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

NH ≅ G/Ntype
2. C2×C32×#107C2semidirect
3. C2×C3×#819C2semidirect
5. C3×#4530C2semidirect
7. #70646C3direct (N × H)
14. C3×#901C4semidirect
26. C2×C3×#107D3semidirect
27. C2×C3×#107C2×C3semidirect
29. C2×#819C2×C3semidirect
32. #4530C2×C3semidirect
75. #901C3×C4semidirect
115. C2×#107C3×D3semidirect
327. C32#1816semidirect
383. C3#70646direct (N × H)
384. C3C3×#1816semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9012
C2×C32×#1072
C2×C3×#8192
C3×#49662
C3×#45302
C3×#18163
#706463