Finite Group Databasebeta

← C3×#10590315 #13157330

Subgroups — C3×#10590315 #13157330

Order 1728.

554 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C32×#870C211yesmaximal, characteristic
2C32×#869C211yesmaximal, characteristic
3C3×#63630C211yesmaximal, characteristic
4C3×#63576C212yesmaximal
5C3×#63574C212yesmaximal
6C3×#5081—31nomaximal
7C3×#5064—31nomaximal
8#10590315C311yesmaximal, characteristic, direct product factor
9C2×C32×Dic6—21no—
10C33×C42C2211yescharacteristic
11C2×C3×C3⋊(C3⋊C8)C2212yes—
12C32×C4⋊Dic3—21no—
13C32×C4⋊Dic3C2211yescharacteristic
14C32×C4⋊Dic3C2212yes—
15C32×C4⋊Dic3C2211yescharacteristic
16C32×Dic3⋊C4—21no—
17C32×Dic3⋊C4—21no—
18C32×C4⋊Q8—31no—
19C32×#125—31no—
20C3×#870—21no—
21C3×#870D311yescharacteristic
22C3×#870C2×C311yescharacteristic
23C3×#869—21no—
24C3×#869D311yescharacteristic
25C3×#869C2×C311yescharacteristic
26#63630C2×C311yescharacteristic
27C3×#808—32no—
28C3×#808—32no—
29C3×#807—32no—
30C3×#807—32no—
31C3×#803—31no—
32C3×#803—31no—
33C3×#803—61no—
34#63576C2×C312yes—
35#63574C2×C312yes—
36C2×C33×C4D411yescharacteristic
37C2×C33×C4D411yescharacteristic
38C2×C33×C4C2311yescharacteristic
39C2×C32×Dic3—22no—
40C2×C32×Dic3—21no—
41C2×C32×Dic3—21no—
42C32×Dic6—42no—
43C3×C3⋊(C3⋊C8)Q814yescomplemented
44C3×#501—91no—
45#5081—31no—
46#5064—31no—
47C2×C32×Q8—61no—
48C2×C3×Dic6—22no—
49C2×C3×Dic6—21no—
50C2×C3×Dic6—21no—
51C32×C4⋊C4—31no—
52C32×C4⋊C4—32no—
53C32×C4⋊C4—31no—
54C32×C4⋊C4—61no—
55C32×C4⋊C4—61no—
56C32×C4⋊C4—61no—
57C32×C42—21no—
58C32×C42—21no—
59C32×C42—21no—
60C32×C42—41no—
61C32×C42C2×D311yescharacteristic
62C32×C42C2×D311yescharacteristic
63C32×C42C22×C311yescharacteristic
64C2×C3⋊(C3⋊C8)C22×C312yes—
65C3×C4⋊Dic3—22no—
66C3×C4⋊Dic3—22no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3—21no—
69C3×C4⋊Dic3—21no—
70C3×C4⋊Dic3—21no—
71C3×C4⋊Dic3C2×D311yescharacteristic
72C3×C4⋊Dic3C2×D312yes—
73C3×C4⋊Dic3C2×D311yescharacteristic
74C3×C4⋊Dic3C22×C311yescharacteristic
75C3×C4⋊Dic3C22×C312yes—
76C3×C4⋊Dic3C22×C311yescharacteristic
77C3×Dic3⋊C4—22no—
78C3×Dic3⋊C4—22no—
79C3×Dic3⋊C4—21no—
80C3×Dic3⋊C4—21no—
81C3×Dic3⋊C4—21no—
82C3×Dic3⋊C4—21no—
83C2×C3×C3⋊C8—32no—
84C2×C3×C3⋊C8—32no—
85C2×C3×C3⋊C8—62no—
86C22×C33C2×D411yescharacteristic
87C33×C4—21no—
88C33×C4—21no—
89C33×C4C2×Q812yes—
90C32×Dic3—42no—
91C32×Dic3—41no—
92C32×Dic3—41no—
93C3×C4⋊Q8—31no—
94C3×C4⋊Q8—31no—
95C3×C4⋊Q8—61no—
96C3×#125—31no—
97C3×#125—31no—
98C3×#125—61no—
99#870C3×D311yescharacteristic
100#869C3×D311yescharacteristic
101C3×#107—92no—
102C3×#106—92no—
103C3×#97—91no—
104#808—32no—
105#808—32no—
106#807—32no—
107#807—32no—
108#803—31no—
109#803—31no—
110#803—61no—
111C32×Q8—122no—
112C2×C32×C4—21no—
113C2×C32×C4—21no—
114C2×C32×C4—21no—
115C2×C32×C4—21no—
116C2×C32×C4—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4—21no—
120C2×C32×C4—21no—
121C2×C32×C4—41no—
122C2×C32×C4—41no—
123C2×C32×C4—41no—
124C2×C32×C4—61no—
125C2×C32×C4—61no—
126C2×C32×C4—62no—
127C2×C32×C4#6711yescharacteristic
128C2×C32×C4#6711yescharacteristic
129C2×C32×C4#6711yescharacteristic
130C2×C32×C4#6711yescharacteristic
131C2×C32×C4C3×D411yescharacteristic
132C2×C32×C4C3×D411yescharacteristic
133C2×C32×C4C22×D311yescharacteristic
134C2×C32×C4C22×D311yescharacteristic
135C2×C32×C4C23×C311yescharacteristic
136C2×C3×Dic3—22no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—24no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—22no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—22no—
145C3×Dic6—44no—
146C3×Dic6—42no—
147C3×Dic6—42no—
148C3⋊(C3⋊C8)C3×Q814yescomplemented
149C3×C3⋊C8—34no—
150C3×C3⋊C8—34no—
151C3×C3⋊C8—64no—
152#501—91noSylow
153C2×C33C4⋊Q811yescharacteristic
154C2×C33AGL(1,ℤ/8)11yescharacteristic
155C2×C33#13711yescharacteristic
156C2×C3×Q8—62no—
157C2×C3×Q8—61no—
158C2×C3×Q8—61no—
159C2×Dic6—21no—
160C2×C3×C8—92no—
161C3×C4⋊C4—31no—
162C3×C4⋊C4—32no—
163C3×C4⋊C4—31no—
164C3×C4⋊C4—31no—
165C3×C4⋊C4—32no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—61no—
168C3×C4⋊C4—61no—
169C3×C4⋊C4—62no—
170C3×C4⋊C4—62no—
171C3×C4⋊C4—62no—
172C3×C4⋊C4—62no—
173C3×C4⋊C4—61no—
174C3×C4⋊C4—61no—
175C3×C4⋊C4—61no—
176C3×C4⋊C4—61no—
177C3×C4⋊C4—61no—
178C3×C4⋊C4—61no—
179C3×C42—21no—
180C3×C42—21no—
181C3×C42—21no—
182C3×C42—41no—
183C3×C42D3211yescharacteristic
184C3×C42C2×C3×D311yescharacteristic
185C3×C42C2×C3×D311yescharacteristic
186C4⋊Dic3—21no—
187C4⋊Dic3C2×C3×D311yescharacteristic
188C4⋊Dic3C2×C3×D312yes—
189C4⋊Dic3C2×C3×D311yescharacteristic
190Dic3⋊C4—21no—
191Dic3⋊C4—21no—
192C2×C3⋊C8—32no—
193C2×C3⋊C8—32no—
194C2×C3⋊C8—62no—
195C22×C32—21no—
196C22×C32—21no—
197C22×C32—21no—
198C22×C32—41no—
199C22×C32C2×#6711yescharacteristic
200C22×C32C2×#6711yescharacteristic
201C22×C32C2×C3×D411yescharacteristic
202C32×C4—21no—
203C32×C4—21no—
204C32×C4—22no—
205C32×C4—22no—
206C32×C4—21no—
207C32×C4—21no—
208C32×C4—21no—
209C32×C4—21no—
210C32×C4—22no—
211C32×C4—22no—
212C32×C4—22no—
213C32×C4—42no—
214C32×C4—41no—
215C32×C4—41no—
216C32×C4—41no—
217C32×C4—41no—
218C32×C4—42no—
219C32×C4—42no—
220C32×C4—121no—
221C32×C4—121no—
222C32×C4—122no—
223C32×C4D3×Q812yes—
224C32×C4D3×Q812yes—
225C32×C4C2×C3×Q812yes—
226C3×Dic3—44no—
227C3×Dic3—42no—
228C3×Dic3—42no—
229C3×Dic3—42no—
230C3×Dic3—41no—
231C3×Dic3—41no—
232C3×Dic3—42no—
233C3×Dic3—41no—
234C3×Dic3—41no—
235C4⋊Q8—31no—
236#125—31no—
237#107—92no—
238#106—92no—
239#97—91no—
240C33#50111yescharacteristic, complemented, Sylow
241C3×Q8—124no—
242C3×Q8—122no—
243C3×Q8—122no—
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—21no—
252C2×C3×C4—21no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—62no—
257C2×C3×C4—62no—
258C2×C3×C4—64no—
259C2×C3×C4—61no—
260C2×C3×C4—61no—
261C2×C3×C4—62no—
262C2×C3×C4—61no—
263C2×C3×C4—61no—
264C2×C3×C4—62no—
265C2×C3×C4#62511yescharacteristic
266C2×C3×C4#62511yescharacteristic
267C2×C3×C4C3×#6711yescharacteristic
268C2×C3×C4C3×#6711yescharacteristic
269C2×C3×C4C3×#6711yescharacteristic
270C2×C3×C4C3×#6711yescharacteristic
271C2×C3×C4C2×D3211yescharacteristic
272C2×C3×C4C22×C3×D311yescharacteristic
273C2×C3×C4C22×C3×D311yescharacteristic
274C2×Dic3—21no—
275C2×Dic3—21no—
276C2×Dic3—22no—
277Dic6—42no—
278C3×C8—94no—
279C3⋊C8—34no—
280C3⋊C8—34no—
281C3⋊C8—64no—
282C2×C32—21no—
283C2×C32—21no—
284C2×C32—21no—
285C2×C32—21no—
286C2×C32—21no—
287C2×C32—21no—
288C2×C32—21no—
289C2×C32—21no—
290C2×C32—21no—
291C2×C32—41no—
292C2×C32—41no—
293C2×C32—41no—
294C2×C32#93211yescharacteristic
295C2×C32#94211yescharacteristic
296C2×C32#94411yescharacteristic
297C2×C32#94411yescharacteristic
298C2×C32#94911yescharacteristic
299C2×C32#95111yescharacteristic
300C2×C32C3×C4⋊Q811yescharacteristic
301C2×C32C3×AGL(1,ℤ/8)11yescharacteristic
302C2×C32C3×#13711yescharacteristic
303C2×Q8—61no—
304C2×C8—92no—
305C4⋊C4—31no—
306C4⋊C4—32no—
307C4⋊C4—31no—
308C4⋊C4—61no—
309C4⋊C4—61no—
310C4⋊C4—61no—
311C42C3×D3211yescharacteristic
312C22×C3—21no—
313C22×C3—21no—
314C22×C3—21no—
315C22×C3—41no—
316C22×C3C2×#62511yescharacteristic
317C22×C3C2×C3×#6711yescharacteristic
318C22×C3C2×C3×#6711yescharacteristic
319C3×C4—22no—
320C3×C4—22no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—22no—
328C3×C4—22no—
329C3×C4—22no—
330C3×C4—41no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4—42no—
335C3×C4—42no—
336C3×C4—42no—
337C3×C4—124no—
338C3×C4—122no—
339C3×C4—122no—
340C3×C4—121no—
341C3×C4—121no—
342C3×C4—122no—
343C3×C4—122no—
344C3×C4—121no—
345C3×C4—121no—
346C3×C4#379712yes—
347C3×C4C3×D3×Q812yes—
348C3×C4C3×D3×Q812yes—
349Dic3—42no—
350Dic3—41no—
351Dic3—41no—
352C32—21no—
353C32—21no—
354C32—21no—
355C32—41no—
356C32#506411yescharacteristic, complemented
357C32#508111yescharacteristic, complemented
358C32C3×#50111yescharacteristic, complemented
359Q8—122no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—62no—
363C2×C4C3×#62511yescharacteristic
364C2×C4C3×#62511yescharacteristic
365C2×C4C2×C3×D3211yescharacteristic
366C8—94no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—21no—
373C2×C3—21no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3—41no—
377C2×C3—41no—
378C2×C3—41no—
379C2×C3#6382311yescharacteristic
380C2×C3#6382711yescharacteristic
381C2×C3#6387311yescharacteristic
382C2×C3C3×#93211yescharacteristic
383C2×C3C3×#94211yescharacteristic
384C2×C3C3×#94411yescharacteristic
385C2×C3C3×#94411yescharacteristic
386C2×C3C3×#94911yescharacteristic
387C2×C3C3×#95111yescharacteristic
388C22C2×C3×#62511yescharacteristic
389C4—21no—
390C4—21no—
391C4—122no—
392C4—121no—
393C4—121no—
394C4C3×#379712yes—
395C3—21no—
396C3—21no—
397C3—21no—
398C3—41no—
399C3#1059031511yescharacteristic, direct product factor
400C3C3×#506411yescharacteristic, complemented
401C3C3×#508111yescharacteristic, complemented
402C2C3×#6382311yescharacteristic
403C2C3×#6382711yescharacteristic
404C2C3×#6387311yescharacteristic
subgroupG/H
centerC22×C3C2×#625
commutator, cosocleC2×C32×C4C23×C3
FrattiniC2×C4C2×C3×D32
FittingC33×C42C22
socle, prime radicalC22×C33C2×D4
hypercenterC3×C42D32
hypocenterC32C3×#501
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22×C3 ◁ C3×C42

Complements

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

NH ≅ G/Ntype
8. #10590315C3direct (N × H)
43. C3×C3⋊(C3⋊C8)Q8semidirect
148. C3⋊(C3⋊C8)C3×Q8semidirect
240. C33#501semidirect
356. C32#5064semidirect
357. C32#5081semidirect
358. C32C3×#501semidirect
399. C3#10590315direct (N × H)
400. C3C3×#5064semidirect
401. C3C3×#5081semidirect

Maximal subgroups

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

subgroupindex
C32×#8702
C32×#8692
C3×#636302
C3×#635762
C3×#635742
C3×#50813
C3×#50643
#105903153