Finite Group Databasebeta

← C3×#60623 #10679230

Subgroups — C3×#60623 #10679230

Order 768.

2042 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3503C211yesmaximal, characteristic, complemented
2C3×#3499C211yesmaximal, characteristic
3C3×#3141C213yesmaximal, complemented
4C3×#3139C213yesmaximal, complemented
5C3×#3137C2112yesmaximal, complemented
6C3×#3131C213yesmaximal, complemented
7C2×C3×#487C213yesmaximal, complemented
8C3×#2898C214yesmaximal, complemented
9C3×C4×#131C211yesmaximal, characteristic, complemented
10#60623C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×2−1+4C2212yes—
12C2×C3×#131C2212yes—
13C3×Q82—21no—
14C3×#557—21no—
15C3×#554—23no—
16C3×D4×Q8—21no—
17C3×D4×Q8—21no—
18C3×D4×Q8—23no—
19C3×#547—23no—
20C3×#547—23no—
21C3×#544C2216yes—
22C3×#536C2216yes—
23C3×#533C2213yes—
24C3×#532C2213yes—
25C2×C3×C4⋊Q8C2213yes—
26C2×C3×C4⋊Q8C2213yes—
27C3×#520C2212yes—
28C3×C4×#41C2211yescharacteristic
29C3×#491C22112yes—
30C3×#489C22112yes—
31C3×#488C22112yes—
32C3×#487C2213yes—
33C3×#487C22112yes—
34C3×#487C2216yes—
35C3×#487C2213yes—
36C3×#486C2211yescharacteristic
37C3×#486C2213yes—
38C3×#419C22112yes—
39C3×#417C22112yes—
40C3×#416C22112yes—
41C2×C3×#103C22112yes—
42C3×#405C2216yes—
43C3×C4×M16C2213yes—
44C2×C3×C4×C8C2213yes—
45#3503C2×C311yescharacteristic, complemented
46#3499C2×C311yescharacteristic
47#3141C2×C313yescomplemented
48#3139C2×C313yescomplemented
49#3137C2×C3112yescomplemented
50#3131C2×C313yescomplemented
51C2×#487C2×C313yescomplemented
52#2898C2×C314yescomplemented
53C4×#131C2×C311yescharacteristic, complemented
54C3×2−1+4—216no—
55C2×C3×#41—26no—
56C2×C3×#41C2316yes—
57C2×C3×#41C2311yescharacteristic
58C22×C3×Q8—22no—
59C22×C3×Q8—23no—
60C22×C3×Q8C2316yes—
61C3×#131—28no—
62C2×C3×M16C2316yes—
63C22×C3×C8C2316yes—
64C3×C4⋊Q8—26no—
65C3×C4⋊Q8—23no—
66C3×C4⋊Q8—23no—
67C3×C4⋊Q8—23no—
68C3×C4⋊Q8—23no—
69C3×C4⋊Q8C2313yes—
70C3×C4⋊Q8C2316yes—
71C3×C4⋊Q8C2313yes—
72C3×C4⋊Q8C2316yes—
73C3×C4⋊Q8C2311yescharacteristic
74C3×#127C2311yescharacteristic
75C3×#125—26no—
76C3×#125C2316yes—
77C3×#124C2316yes—
78C3×#123—212no—
79C3×C22⋊Q8—23no—
80C3×C22⋊Q8—23no—
81C3×C22⋊Q8—26no—
82C3×C22⋊Q8—26no—
83C3×C22⋊Q8—212no—
84C3×#121—23no—
85C3×#121—23no—
86C3×C4×Q8—22no—
87C3×C4×Q8—23no—
88C3×C4×Q8—23no—
89C3×C4×Q8D411yescharacteristic
90C3×C4×Q8D411yescharacteristic
91C3×C4×D4—26no—
92C3×C4×D4—23no—
93C3×C4×D4—23no—
94C3×C4×D4D413yes—
95C3×C4×D4D413yes—
96C3×#117C2316yes—
97C3×#117C2313yes—
98C2×C3×C4⋊C4—23no—
99C2×C3×C4⋊C4—23no—
100C2×C3×C4⋊C4—23no—
101C2×C3×C4⋊C4C2316yes—
102C2×C3×C42C2313yes—
103C3×#103C23112yes—
104C3×#103C23124yes—
105C3×#103C23112yes—
106C3×#102C2314yes—
107C3×#102C23112yes—
108C3×#97C2316yes—
109C3×C4×C8C2311yescharacteristic
110C3×C4×C8C2313yes—
111C3×C4×C8C2316yes—
112C2×2−1+4C22×C312yes—
113C2×#131C22×C312yes—
114Q82—21no—
115#557—21no—
116#554—23no—
117D4×Q8—21no—
118D4×Q8—21no—
119D4×Q8—23no—
120#547—23no—
121#547—23no—
122#544C22×C316yes—
123#536C22×C316yes—
124#533C22×C313yes—
125#532C22×C313yes—
126C2×C4⋊Q8C22×C313yes—
127C2×C4⋊Q8C22×C313yes—
128#520C22×C312yes—
129C4×#41C22×C311yescharacteristic
130#491C22×C3112yes—
131#489C22×C3112yes—
132#488C22×C3112yes—
133#487C22×C313yes—
134#487C22×C3112yes—
135#487C22×C316yes—
136#487C22×C313yes—
137#486C22×C311yescharacteristic
138#486C22×C313yes—
139#419C22×C3112yes—
140#417C22×C3112yes—
141#416C22×C3112yes—
142C2×#103C22×C3112yes—
143#405C22×C316yes—
144C4×M16C22×C313yes—
145C2×C4×C8C22×C313yes—
146C3×#41—224no—
147C3×#41—424no—
148C3×#41#4118yes—
149C2×C3×Q8—21no—
150C2×C3×Q8—23no—
151C2×C3×Q8—26no—
152C2×C3×Q8—26no—
153C2×C3×Q8—224no—
154C2×C3×Q8—48no—
155C2×C3×Q8—86no—
156C2×C3×Q8C2×D411yescharacteristic
157C2×C3×Q8C2416yes—
158C2×C3×D4—26no—
159C2×C3×D4—21no—
160C2×C3×D4—21no—
161C2×C3×D4C2×D413yes—
162C2×C3×D4C2412yes—
163C22×C3×C4—26no—
164C22×C3×C4—26no—
165C22×C3×C4—26no—
166C22×C3×C4C2×D413yes—
167C22×C3×C4C2×D413yes—
168C22×C3×C4C2413yes—
169C3×M16—212no—
170C2×C3×C8—212no—
171C2×C3×C8C2412yes—
172C2×C3×C8C2416yes—
173C3×C4⋊C4—23no—
174C3×C4⋊C4—23no—
175C3×C4⋊C4—23no—
176C3×C4⋊C4—26no—
177C3×C4⋊C4—26no—
178C3×C4⋊C4—26no—
179C3×C4⋊C4—26no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—26no—
182C3×C4⋊C4—26no—
183C3×C4⋊C4C2×D413yes—
184C3×C4⋊C4C2×D413yes—
185C3×C4⋊C4C2416yes—
186C3×C4⋊C4C2412yes—
187C3×C22⋊C4—26no—
188C3×C22⋊C4—26no—
189C3×C22⋊C4—26no—
190C3×C22⋊C4C2×D416yes—
191C3×C42—26no—
192C3×C42C2×D413yes—
193C3×C42C2×D413yes—
194C3×C42C2411yescharacteristic
195C3×C42C2413yes—
1962−1+4—216no—
197C2×#41—26no—
198C2×#41C23×C316yes—
199C2×#41C23×C311yescharacteristic
200C22×Q8—22no—
201C22×Q8—23no—
202C22×Q8C23×C316yes—
203#131—28no—
204C2×M16C23×C316yes—
205C22×C8C23×C316yes—
206C4⋊Q8—26no—
207C4⋊Q8—23no—
208C4⋊Q8—23no—
209C4⋊Q8—23no—
210C4⋊Q8—23no—
211C4⋊Q8C23×C313yes—
212C4⋊Q8C23×C316yes—
213C4⋊Q8C23×C313yes—
214C4⋊Q8C23×C316yes—
215C4⋊Q8C23×C311yescharacteristic
216#127C23×C311yescharacteristic
217#125—26no—
218#125C23×C316yes—
219#124C23×C316yes—
220#123—212no—
221C22⋊Q8—23no—
222C22⋊Q8—23no—
223C22⋊Q8—26no—
224C22⋊Q8—26no—
225C22⋊Q8—212no—
226#121—23no—
227#121—23no—
228C4×Q8—22no—
229C4×Q8—23no—
230C4×Q8—23no—
231C4×Q8C3×D411yescharacteristic
232C4×Q8C3×D411yescharacteristic
233C4×D4—26no—
234C4×D4—23no—
235C4×D4—23no—
236C4×D4C3×D413yes—
237C4×D4C3×D413yes—
238#117C23×C316yes—
239#117C23×C313yes—
240C2×C4⋊C4—23no—
241C2×C4⋊C4—23no—
242C2×C4⋊C4—23no—
243C2×C4⋊C4C23×C316yes—
244C2×C42C23×C313yes—
245#103C23×C3112yes—
246#103C23×C3124yes—
247#103C23×C3112yes—
248#102C23×C314yes—
249#102C23×C3112yes—
250#97C23×C316yes—
251C4×C8C23×C311yescharacteristic
252C4×C8C23×C313yes—
253C4×C8C23×C316yes—
254C23×C3—22no—
255C23×C3C22×D413yes—
256C3×Q8—212no—
257C3×Q8—412no—
258C3×Q8—46no—
259C3×Q8—42no—
260C3×Q8—86no—
261C3×Q8#12414yes—
262C3×D4—24no—
263C3×D4—412no—
264C3×D4—42no—
265C3×D4—42no—
266C3×D4#124112yes—
267C2×C3×C4—26no—
268C2×C3×C4—26no—
269C2×C3×C4—22no—
270C2×C3×C4—26no—
271C2×C3×C4—26no—
272C2×C3×C4—412no—
273C2×C3×C4—412no—
274C2×C3×C4—86no—
275C2×C3×C4C22×D413yes—
276C2×C3×C4C22×D413yes—
277C2×C3×C4C22×D413yes—
278C2×C3×C4C22×D411yescharacteristic
279C2×C3×C4C22×D411yescharacteristic
280C2×C3×C4C2×#41112yes—
281C2×C3×C4C2511yescharacteristic
282C3×C8—22no—
283C3×C8—26no—
284#41—224no—
285#41—424no—
286#41C3×#4118yes—
287C2×Q8—21no—
288C2×Q8—23no—
289C2×Q8—26no—
290C2×Q8—26no—
291C2×Q8—224no—
292C2×Q8—48no—
293C2×Q8—86no—
294C2×Q8C2×C3×D411yescharacteristic
295C2×Q8C24×C316yes—
296C2×D4—26no—
297C2×D4—21no—
298C2×D4—21no—
299C2×D4C2×C3×D413yes—
300C2×D4C24×C312yes—
301C22×C4—26no—
302C22×C4—26no—
303C22×C4—26no—
304C22×C4C2×C3×D413yes—
305C22×C4C2×C3×D413yes—
306C22×C4C24×C313yes—
307M16—212no—
308C2×C8—212no—
309C2×C8C24×C312yes—
310C2×C8C24×C316yes—
311C4⋊C4—23no—
312C4⋊C4—23no—
313C4⋊C4—23no—
314C4⋊C4—26no—
315C4⋊C4—26no—
316C4⋊C4—26no—
317C4⋊C4—26no—
318C4⋊C4—21no—
319C4⋊C4—26no—
320C4⋊C4—26no—
321C4⋊C4C2×C3×D413yes—
322C4⋊C4C2×C3×D413yes—
323C4⋊C4C24×C316yes—
324C4⋊C4C24×C312yes—
325C22⋊C4—26no—
326C22⋊C4—26no—
327C22⋊C4—26no—
328C22⋊C4C2×C3×D416yes—
329C42—26no—
330C42C2×C3×D413yes—
331C42C2×C3×D413yes—
332C42C24×C311yescharacteristic
333C42C24×C313yes—
334C22×C3—23no—
335C22×C3—23no—
336C22×C3—42no—
337C22×C3—42no—
338C22×C3—42no—
339C22×C3C2×#12416yes—
340C22×C3C23×D411yescharacteristic
341C3×C4—23no—
342C3×C4—23no—
343C3×C4—46no—
344C3×C4—46no—
345C3×C4—42no—
346C3×C4C2×#12416yes—
347C3×C4#57712yes—
348C3×C4#57812yes—
349C3×C4C22×#4112yes—
350C23—22no—
351C23C22×C3×D413yes—
352Q8—212no—
353Q8—46no—
354Q8—42no—
355Q8—412no—
356Q8—86no—
357Q8C3×#12414yes—
358D4—24no—
359D4—412no—
360D4—42no—
361D4—42no—
362D4C3×#124112yes—
363C2×C4—26no—
364C2×C4—26no—
365C2×C4—22no—
366C2×C4—26no—
367C2×C4—26no—
368C2×C4—412no—
369C2×C4—412no—
370C2×C4—86no—
371C2×C4C22×C3×D413yes—
372C2×C4C22×C3×D413yes—
373C2×C4C22×C3×D411yescharacteristic
374C2×C4C22×C3×D411yescharacteristic
375C2×C4C22×C3×D413yes—
376C2×C4C2×C3×#41112yes—
377C2×C4C25×C311yescharacteristic
378C8—22no—
379C8—26no—
380C2×C3—26no—
381C2×C3—82no—
382C2×C3C22×#12411yescharacteristic
383C2×C3C2×#57711yescharacteristic
384C2×C3C2×#57811yescharacteristic
385C22—23no—
386C22—23no—
387C22—42no—
388C22—42no—
389C22—42no—
390C22C2×C3×#12416yes—
391C22C23×C3×D411yescharacteristic
392C4—23no—
393C4—23no—
394C4—46no—
395C4—46no—
396C4—42no—
397C4C2×C3×#12416yes—
398C4C3×#57712yes—
399C4C3×#57812yes—
400C4C22×C3×#4112yes—
401C3#6062311yescharacteristic, direct product factor, Sylow
402C2—26no—
403C2—82no—
404C2C22×C3×#12411yescharacteristic
405C2C2×C3×#57711yescharacteristic
406C2C2×C3×#57811yescharacteristic
subgroupG/H
center, socleC22×C3C23×D4
commutator, Frattini, cosocleC2×C4C25×C3
prime radicalC3×#3499C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C3×C4×#41 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×#3503C2semidirect
3. C3×#3141C2semidirect
4. C3×#3139C2semidirect
5. C3×#3137C2semidirect
6. C3×#3131C2semidirect
7. C2×C3×#487C2semidirect
8. C3×#2898C2semidirect
9. C3×C4×#131C2semidirect
10. #60623C3direct (N × H)
45. #3503C2×C3semidirect
47. #3141C2×C3semidirect
48. #3139C2×C3semidirect
49. #3137C2×C3semidirect
50. #3131C2×C3semidirect
51. C2×#487C2×C3semidirect
52. #2898C2×C3semidirect
53. C4×#131C2×C3semidirect
401. C3#60623direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35032
C3×#34992
C3×#31412
C3×#31392
C3×#31372
C3×#31312
C2×C3×#4872
C3×#28982
C3×C4×#1312
#606233