Finite Group Databasebeta

← C3×#9265 #10627872

Subgroups — C3×#9265 #10627872

Order 768.

438 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×#3211C211yesmaximal, characteristic
2C3×#2968C211yesmaximal, characteristic, complemented
3C3×#1532C211yesmaximal, characteristic, complemented
4C2×C3×#329C211yesmaximal, characteristic, complemented
5C3×#1469C211yesmaximal, characteristic, complemented
6C3×#1302C211yesmaximal, characteristic, complemented
7C3×#1281C211yesmaximal, characteristic, complemented
8#9265C311yesmaximal, characteristic, direct product factor, Sylow
9C3×#529C2211yescharacteristic
10C2×C3×#124C2211yescharacteristic
11C3×#495—21no—
12C3×#488—21no—
13C3×#464—21no—
14C3×#463—21no—
15C3×#451C411yescharacteristic
16C3×#450C411yescharacteristic
17C3×#448C411yescharacteristic
18C3×#447C411yescharacteristic
19C3×#446—21no—
20C3×#435—21no—
21C3×#435—21no—
22C3×C8×D4—21no—
23C3×#433—21no—
24C3×#432C2211yescharacteristic
25C3×#431C2211yescharacteristic
26C3×#431C2211yescharacteristic
27C3×#405—21no—
28C2×C3×#97C2211yescharacteristic
29C2×C3×#97C2211yescharacteristic
30C3×#330—21no—
31C3×#329—21no—
32C3×#329—21no—
33C3×#329—21no—
34C3×#323—21no—
35C3×#323—21no—
36#3211C2×C311yescharacteristic
37#2968C2×C311yescharacteristic, complemented
38#1532C2×C311yescharacteristic, complemented
39C2×#329C2×C311yescharacteristic, complemented
40#1469C2×C311yescharacteristic, complemented
41#1302C2×C311yescharacteristic, complemented
42#1281C2×C311yescharacteristic, complemented
43C22×C3×Q8C2×C411yescharacteristic
44C22×C3×D4C2×C411yescharacteristic
45C2×C3×Q16—21no—
46C2×C3×SD16—21no—
47C2×C3×SD16—21no—
48C2×C3×D8—21no—
49C2×C3×M16—21no—
50C22×C3×C8—21no—
51C22×C3×C8—21no—
52C22×C3×C8—21no—
53C3×#126—21no—
54C3×#125D411yescharacteristic
55C3×#124—21no—
56C3×#124—21no—
57C3×#124—21no—
58C3×#124—21no—
59C3×#124D411yescharacteristic
60C3×#123—21no—
61C3×C22⋊Q8C2×C411yescharacteristic
62C3×#121C2×C411yescharacteristic
63C3×C4×Q8D411yescharacteristic
64C3×C4×D4—21no—
65C3×C4×D4D411yescharacteristic
66C3×#117—21no—
67C2×C3×C22⋊C4—21no—
68C2×C3×C22⋊C4—21no—
69C2×C3×C42C2311yescharacteristic
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×#103—21no—
76C3×#103—21no—
77C3×#102—21no—
78C3×#102—21no—
79C3×C22⋊C8—21no—
80C3×C22⋊C8—21no—
81C3×C22⋊C8—21no—
82C3×C22⋊C8C2×C411yescharacteristic
83C3×C22⋊C8C2×C411yescharacteristic
84C3×C22⋊C8D411yescharacteristic
85C3×C22⋊C8D411yescharacteristic
86C3×#97—21no—
87C3×#97—21no—
88C3×#97—21no—
89C3×#97—21no—
90C3×#97—21no—
91C3×#97D411yescharacteristic
92C3×#97D411yescharacteristic
93C3×C4×C8—21no—
94C3×C4×C8—21no—
95C3×C4×C8—21no—
96#529C22×C311yescharacteristic
97C2×#124C22×C311yescharacteristic
98#495—21no—
99#488—21no—
100#464—21no—
101#463—21no—
102#451C3×C411yescharacteristic
103#450C3×C411yescharacteristic
104#448C3×C411yescharacteristic
105#447C3×C411yescharacteristic
106#446—21no—
107#435—21no—
108#435—21no—
109C8×D4—21no—
110#433—21no—
111#432C22×C311yescharacteristic
112#431C22×C311yescharacteristic
113#431C22×C311yescharacteristic
114#405—21no—
115C2×#97C22×C311yescharacteristic
116C2×#97C22×C311yescharacteristic
117#330—21no—
118#329—21no—
119#329—21no—
120#329—21no—
121#323—21no—
122#323—21no—
123C24×C3—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—81no—
127C2×C3×Q8C22⋊C411yescharacteristic
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—81no—
132C2×C3×D4C22⋊C411yescharacteristic
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4C22×C411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C3×Q16—81no—
139C3×SD16—81no—
140C3×SD16—81no—
141C3×D8—81no—
142C3×M16—42no—
143C2×C3×C8—21no—
144C2×C3×C8—22no—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—21no—
148C2×C3×C8—21no—
149C2×C3×C8—21no—
150C2×C3×C8—42no—
151C2×C3×C8—42no—
152C2×C3×C8—41no—
153C2×C3×C8—41no—
154C2×C3×C8—41no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—21no—
159C3×C4⋊C4C22⋊C411yescharacteristic
160C3×C4⋊C4C22⋊C411yescharacteristic
161C3×C22⋊C4—21no—
162C3×C22⋊C4—21no—
163C3×C22⋊C4—21no—
164C3×C22⋊C4—21no—
165C3×C22⋊C4—21no—
166C3×C22⋊C4—41no—
167C3×C22⋊C4—41no—
168C3×C22⋊C4—41no—
169C3×C22⋊C4—41no—
170C3×C42—21no—
171C3×C42C2×D411yescharacteristic
172C3×C42C2×D411yescharacteristic
173C3×C42C2×D411yescharacteristic
174C3×C42#4111yescharacteristic
175C22×Q8C2×C3×C411yescharacteristic
176C22×D4C2×C3×C411yescharacteristic
177C2×Q16—21no—
178C2×SD16—21no—
179C2×SD16—21no—
180C2×D8—21no—
181C2×M16—21no—
182C22×C8—21no—
183C22×C8—21no—
184C22×C8—21no—
185#126—21no—
186#125C3×D411yescharacteristic
187#124—21no—
188#124—21no—
189#124—21no—
190#124—21no—
191#124C3×D411yescharacteristic
192#123—21no—
193C22⋊Q8C2×C3×C411yescharacteristic
194#121C2×C3×C411yescharacteristic
195C4×Q8C3×D411yescharacteristic
196C4×D4—21no—
197C4×D4C3×D411yescharacteristic
198#117—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—21no—
201C2×C42C23×C311yescharacteristic
202C4⋊C8—21no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207#103—21no—
208#103—21no—
209#102—21no—
210#102—21no—
211C22⋊C8—21no—
212C22⋊C8—21no—
213C22⋊C8—21no—
214C22⋊C8C2×C3×C411yescharacteristic
215C22⋊C8C2×C3×C411yescharacteristic
216C22⋊C8C3×D411yescharacteristic
217C22⋊C8C3×D411yescharacteristic
218#97—21no—
219#97—21no—
220#97—21no—
221#97—21no—
222#97—21no—
223#97C3×D411yescharacteristic
224#97C3×D411yescharacteristic
225C4×C8—21no—
226C4×C8—21no—
227C4×C8—21no—
228C23×C3—21no—
229C23×C3—41no—
230C23×C3—41no—
231C23×C3—41no—
232C23×C3—81no—
233C23×C3—81no—
234C23×C3C4×D411yescharacteristic
235C3×Q8—41no—
236C3×Q8—41no—
237C3×Q8—41no—
238C3×Q8—81no—
239C3×D4—41no—
240C3×D4—41no—
241C3×D4—41no—
242C3×D4—41no—
243C3×D4—41no—
244C3×D4—81no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—81no—
256C2×C3×C4C2×C22⋊C411yescharacteristic
257C2×C3×C4C4×D411yescharacteristic
258C2×C3×C4#12011yescharacteristic
259C2×C3×C4#12111yescharacteristic
260C2×C3×C4#12111yescharacteristic
261C2×C3×C4#12311yescharacteristic
262C3×C8—22no—
263C3×C8—42no—
264C3×C8—41no—
265C3×C8—41no—
266C3×C8—41no—
267C3×C8—41no—
268C3×C8—41no—
269C3×C8—81no—
270C24—21no—
271C2×Q8—21no—
272C2×Q8—21no—
273C2×Q8—81no—
274C2×Q8C3×C22⋊C411yescharacteristic
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—81no—
279C2×D4C3×C22⋊C411yescharacteristic
280C22×C4—21no—
281C22×C4—21no—
282C22×C4C22×C3×C411yescharacteristic
283C22×C4C2×C3×D411yescharacteristic
284C22×C4C3×#4111yescharacteristic
285Q16—81no—
286SD16—81no—
287SD16—81no—
288D8—81no—
289M16—42no—
290C2×C8—21no—
291C2×C8—22no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—42no—
298C2×C8—42no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4C3×C22⋊C411yescharacteristic
307C4⋊C4C3×C22⋊C411yescharacteristic
308C22⋊C4—21no—
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C42—21no—
318C42C2×C3×D411yescharacteristic
319C42C2×C3×D411yescharacteristic
320C42C2×C3×D411yescharacteristic
321C42C3×#4111yescharacteristic
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—161no—
331C22×C3—161no—
332C22×C3#38611yescharacteristic
333C22×C3#44311yescharacteristic
334C22×C3#44411yescharacteristic
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—41no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4—81no—
345C23—21no—
346C23—41no—
347C23—41no—
348C23—41no—
349C23—81no—
350C23—81no—
351C23C3×C4×D411yescharacteristic
352Q8—41no—
353Q8—41no—
354Q8—41no—
355Q8—81no—
356D4—41no—
357D4—41no—
358D4—41no—
359D4—41no—
360D4—41no—
361D4—81no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4C2×C3×C22⋊C411yescharacteristic
374C2×C4C3×C4×D411yescharacteristic
375C2×C4C3×#12011yescharacteristic
376C2×C4C3×#12111yescharacteristic
377C2×C4C3×#12111yescharacteristic
378C2×C4C3×#12311yescharacteristic
379C8—22no—
380C8—42no—
381C8—41no—
382C8—41no—
383C8—41no—
384C8—41no—
385C8—41no—
386C8—81no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—81no—
390C2×C3—161no—
391C2×C3#187611yescharacteristic
392C2×C3#190611yescharacteristic
393C2×C3#191411yescharacteristic
394C22—21no—
395C22—21no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—81no—
400C22—81no—
401C22—81no—
402C22—161no—
403C22—161no—
404C22C3×#38611yescharacteristic
405C22C3×#44311yescharacteristic
406C22C3×#44411yescharacteristic
407C4—21no—
408C4—21no—
409C4—21no—
410C4—21no—
411C4—21no—
412C4—21no—
413C4—41no—
414C4—41no—
415C4—41no—
416C4—81no—
417C3#926511yescharacteristic, direct product factor, Sylow
418C2—21no—
419C2—21no—
420C2—81no—
421C2—161no—
422C2C3×#187611yescharacteristic
423C2C3×#190611yescharacteristic
424C2C3×#191411yescharacteristic
subgroupG/H
center, socleC22×C3#386
commutatorC22×C4C22×C3×C4
Frattini, cosocleC2×C42C23×C3
prime radicalC3×#447C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2968C2semidirect
3. C3×#1532C2semidirect
4. C2×C3×#329C2semidirect
5. C3×#1469C2semidirect
6. C3×#1302C2semidirect
7. C3×#1281C2semidirect
8. #9265C3direct (N × H)
37. #2968C2×C3semidirect
38. #1532C2×C3semidirect
39. C2×#329C2×C3semidirect
40. #1469C2×C3semidirect
41. #1302C2×C3semidirect
42. #1281C2×C3semidirect
417. C3#9265direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#32112
C3×#29682
C3×#15322
C2×C3×#3292
C3×#14692
C3×#13022
C3×#12812
#92653