Finite Group Databasebeta

← C3×#60627 #10679234

Subgroups — C3×#60627 #10679234

Order 768.

1998 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3504C211yesmaximal, characteristic, complemented
2C3×#3499C211yesmaximal, characteristic
3C3×#3142C213yesmaximal, complemented
4C3×#3139C213yesmaximal, complemented
5C3×#3137C2112yesmaximal, complemented
6C3×#3135C213yesmaximal, complemented
7C2×C3×#491C213yesmaximal, complemented
8C3×#2898C214yesmaximal, complemented
9C3×#2875C211yesmaximal, characteristic, complemented
10#60627C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×2−1+4C2212yes—
12C2×C3×#131C2212yes—
13C3×#557—22no—
14C3×#554—26no—
15C3×D4×Q8—22no—
16C3×#547—26no—
17C3×#544C2216yes—
18C3×#536C2216yes—
19C3×#533C2213yes—
20C3×#532C2213yes—
21C2×C3×C4⋊Q8C2213yes—
22C2×C3×#125C2213yes—
23C3×#520C2212yes—
24C3×C4×#41C2211yescharacteristic
25C3×#491C2213yes—
26C3×#491C22112yes—
27C3×#491C2216yes—
28C3×#491C2213yes—
29C3×#490C2211yescharacteristic
30C3×#490C2213yes—
31C3×#489C22112yes—
32C3×#488C22112yes—
33C3×#487C22112yes—
34C3×#419C22112yes—
35C3×#417C22112yes—
36C3×#416C22112yes—
37C2×C3×#103C22112yes—
38C3×#405C2216yes—
39C3×C4×M16C2213yes—
40C2×C3×#97C2213yes—
41#3504C2×C311yescharacteristic, complemented
42#3499C2×C311yescharacteristic
43#3142C2×C313yescomplemented
44#3139C2×C313yescomplemented
45#3137C2×C3112yescomplemented
46#3135C2×C313yescomplemented
47C2×#491C2×C313yescomplemented
48#2898C2×C314yescomplemented
49#2875C2×C311yescharacteristic, complemented
50C3×2−1+4—216no—
51C2×C3×#41—26no—
52C2×C3×#41C2316yes—
53C2×C3×#41C2311yescharacteristic
54C22×C3×Q8—22no—
55C22×C3×Q8C2316yes—
56C3×#131—28no—
57C2×C3×M16C2316yes—
58C22×C3×C8C2316yes—
59C3×C4⋊Q8—26no—
60C3×C4⋊Q8—26no—
61C3×C4⋊Q8C2313yes—
62C3×C4⋊Q8C2316yes—
63C3×C4⋊Q8C2316yes—
64C3×#127C2311yescharacteristic
65C3×#125—26no—
66C3×#125—26no—
67C3×#125C2313yes—
68C3×#125C2316yes—
69C3×#125C2311yescharacteristic
70C3×#124C2316yes—
71C3×#123—212no—
72C3×C22⋊Q8—26no—
73C3×C22⋊Q8—212no—
74C3×C22⋊Q8—212no—
75C3×#121—26no—
76C3×C4×Q8—22no—
77C3×C4×Q8—26no—
78C3×C4×Q8D412yes—
79C3×C4×D4—26no—
80C3×C4×D4—26no—
81C3×C4×D4D416yes—
82C3×#117C2316yes—
83C3×#117C2313yes—
84C2×C3×C4⋊C4—26no—
85C2×C3×C4⋊C4—26no—
86C2×C3×C4⋊C4C2316yes—
87C2×C3×C42C2313yes—
88C3×#103C23112yes—
89C3×#103C23124yes—
90C3×#103C23112yes—
91C3×#102C2314yes—
92C3×#102C23112yes—
93C3×#97C2311yescharacteristic
94C3×#97C2313yes—
95C3×#97C2316yes—
96C3×C4×C8C2316yes—
97C2×2−1+4C22×C312yes—
98C2×#131C22×C312yes—
99#557—22no—
100#554—26no—
101D4×Q8—22no—
102#547—26no—
103#544C22×C316yes—
104#536C22×C316yes—
105#533C22×C313yes—
106#532C22×C313yes—
107C2×C4⋊Q8C22×C313yes—
108C2×#125C22×C313yes—
109#520C22×C312yes—
110C4×#41C22×C311yescharacteristic
111#491C22×C313yes—
112#491C22×C3112yes—
113#491C22×C316yes—
114#491C22×C313yes—
115#490C22×C311yescharacteristic
116#490C22×C313yes—
117#489C22×C3112yes—
118#488C22×C3112yes—
119#487C22×C3112yes—
120#419C22×C3112yes—
121#417C22×C3112yes—
122#416C22×C3112yes—
123C2×#103C22×C3112yes—
124#405C22×C316yes—
125C4×M16C22×C313yes—
126C2×#97C22×C313yes—
127C3×#41—224no—
128C3×#41—424no—
129C3×#41#4118yes—
130C2×C3×Q8—26no—
131C2×C3×Q8—26no—
132C2×C3×Q8—224no—
133C2×C3×Q8—48no—
134C2×C3×Q8C2×D411yescharacteristic
135C2×C3×Q8C2416yes—
136C2×C3×D4—26no—
137C2×C3×D4—22no—
138C2×C3×D4C2×D413yes—
139C2×C3×D4C2412yes—
140C22×C3×C4—26no—
141C22×C3×C4—26no—
142C22×C3×C4—26no—
143C22×C3×C4C2×D416yes—
144C22×C3×C4C2413yes—
145C3×M16—212no—
146C2×C3×C8—212no—
147C2×C3×C8C2416yes—
148C2×C3×C8C2412yes—
149C3×C4⋊C4—26no—
150C3×C4⋊C4—26no—
151C3×C4⋊C4—212no—
152C3×C4⋊C4—212no—
153C3×C4⋊C4—26no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—26no—
156C3×C4⋊C4C2×D416yes—
157C3×C4⋊C4C2416yes—
158C3×C4⋊C4C2412yes—
159C3×C22⋊C4—26no—
160C3×C22⋊C4—212no—
161C3×C22⋊C4C2×D416yes—
162C3×C42—26no—
163C3×C42C2×D416yes—
164C3×C42C2413yes—
165C3×C42C2411yescharacteristic
1662−1+4—216no—
167C2×#41—26no—
168C2×#41C23×C316yes—
169C2×#41C23×C311yescharacteristic
170C22×Q8—22no—
171C22×Q8C23×C316yes—
172#131—28no—
173C2×M16C23×C316yes—
174C22×C8C23×C316yes—
175C4⋊Q8—26no—
176C4⋊Q8—26no—
177C4⋊Q8C23×C313yes—
178C4⋊Q8C23×C316yes—
179C4⋊Q8C23×C316yes—
180#127C23×C311yescharacteristic
181#125—26no—
182#125—26no—
183#125C23×C313yes—
184#125C23×C316yes—
185#125C23×C311yescharacteristic
186#124C23×C316yes—
187#123—212no—
188C22⋊Q8—26no—
189C22⋊Q8—212no—
190C22⋊Q8—212no—
191#121—26no—
192C4×Q8—22no—
193C4×Q8—26no—
194C4×Q8C3×D412yes—
195C4×D4—26no—
196C4×D4—26no—
197C4×D4C3×D416yes—
198#117C23×C316yes—
199#117C23×C313yes—
200C2×C4⋊C4—26no—
201C2×C4⋊C4—26no—
202C2×C4⋊C4C23×C316yes—
203C2×C42C23×C313yes—
204#103C23×C3112yes—
205#103C23×C3124yes—
206#103C23×C3112yes—
207#102C23×C314yes—
208#102C23×C3112yes—
209#97C23×C311yescharacteristic
210#97C23×C313yes—
211#97C23×C316yes—
212C4×C8C23×C316yes—
213C23×C3—22no—
214C23×C3C22×D413yes—
215C3×Q8—212no—
216C3×Q8—412no—
217C3×Q8—86no—
218C3×Q8#12414yes—
219C3×D4—24no—
220C3×D4—44no—
221C3×D4—412no—
222C3×D4#124112yes—
223C2×C3×C4—22no—
224C2×C3×C4—26no—
225C2×C3×C4—26no—
226C2×C3×C4—46no—
227C2×C3×C4—412no—
228C2×C3×C4—412no—
229C2×C3×C4—86no—
230C2×C3×C4C22×D412yes—
231C2×C3×C4C22×D416yes—
232C2×C3×C4C22×D413yes—
233C2×C3×C4C2×#41112yes—
234C2×C3×C4C2511yescharacteristic
235C3×C8—22no—
236C3×C8—26no—
237#41—224no—
238#41—424no—
239#41C3×#4118yes—
240C2×Q8—26no—
241C2×Q8—26no—
242C2×Q8—224no—
243C2×Q8—48no—
244C2×Q8C2×C3×D411yescharacteristic
245C2×Q8C24×C316yes—
246C2×D4—26no—
247C2×D4—22no—
248C2×D4C2×C3×D413yes—
249C2×D4C24×C312yes—
250C22×C4—26no—
251C22×C4—26no—
252C22×C4—26no—
253C22×C4C2×C3×D416yes—
254C22×C4C24×C313yes—
255M16—212no—
256C2×C8—212no—
257C2×C8C24×C316yes—
258C2×C8C24×C312yes—
259C4⋊C4—26no—
260C4⋊C4—26no—
261C4⋊C4—212no—
262C4⋊C4—212no—
263C4⋊C4—26no—
264C4⋊C4—22no—
265C4⋊C4—26no—
266C4⋊C4C2×C3×D416yes—
267C4⋊C4C24×C316yes—
268C4⋊C4C24×C312yes—
269C22⋊C4—26no—
270C22⋊C4—212no—
271C22⋊C4C2×C3×D416yes—
272C42—26no—
273C42C2×C3×D416yes—
274C42C24×C313yes—
275C42C24×C311yescharacteristic
276C22×C3—26no—
277C22×C3—44no—
278C22×C3—42no—
279C22×C3C2×#12416yes—
280C22×C3C23×D411yescharacteristic
281C3×C4—22no—
282C3×C4—26no—
283C3×C4—46no—
284C3×C4—42no—
285C3×C4—46no—
286C3×C4C2×#12416yes—
287C3×C4C22×#4112yes—
288C23—22no—
289C23C22×C3×D413yes—
290Q8—212no—
291Q8—412no—
292Q8—86no—
293Q8C3×#12414yes—
294D4—24no—
295D4—44no—
296D4—412no—
297D4C3×#124112yes—
298C2×C4—22no—
299C2×C4—26no—
300C2×C4—26no—
301C2×C4—46no—
302C2×C4—412no—
303C2×C4—412no—
304C2×C4—86no—
305C2×C4C22×C3×D412yes—
306C2×C4C22×C3×D416yes—
307C2×C4C22×C3×D413yes—
308C2×C4C2×C3×#41112yes—
309C2×C4C25×C311yescharacteristic
310C8—22no—
311C8—26no—
312C2×C3—26no—
313C2×C3—82no—
314C2×C3C22×#12411yescharacteristic
315C2×C3#358612yes—
316C22—26no—
317C22—42no—
318C22—44no—
319C22C2×C3×#12416yes—
320C22C23×C3×D411yescharacteristic
321C4—22no—
322C4—26no—
323C4—46no—
324C4—46no—
325C4—42no—
326C4C2×C3×#12416yes—
327C4C22×C3×#4112yes—
328C3#6062711yescharacteristic, direct product factor, Sylow
329C2—26no—
330C2—82no—
331C2C22×C3×#12411yescharacteristic
332C2C3×#358612yes—
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×#3504C2semidirect
3. C3×#3142C2semidirect
4. C3×#3139C2semidirect
5. C3×#3137C2semidirect
6. C3×#3135C2semidirect
7. C2×C3×#491C2semidirect
8. C3×#2898C2semidirect
9. C3×#2875C2semidirect
10. #60627C3direct (N × H)
41. #3504C2×C3semidirect
43. #3142C2×C3semidirect
44. #3139C2×C3semidirect
45. #3137C2×C3semidirect
46. #3135C2×C3semidirect
47. C2×#491C2×C3semidirect
48. #2898C2×C3semidirect
49. #2875C2×C3semidirect
328. C3#60627direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35042
C3×#34992
C3×#31422
C3×#31392
C3×#31372
C3×#31352
C2×C3×#4912
C3×#28982
C3×#28752
#606273