Finite Group Databasebeta

← C3×#13394 #10632001

Subgroups — C3×#13394 #10632001

Order 768.

392 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#473C211yesmaximal, characteristic
2C2×C3×#368C211yesmaximal, characteristic, complemented
3C2×C3×#362C211yesmaximal, characteristic, complemented
4C2×C3×#358C211yesmaximal, characteristic, complemented
5C3×#2113C211yesmaximal, characteristic
6C3×#1816C211yesmaximal, characteristic
7C3×#1387C211yesmaximal, characteristic, complemented
8#13394C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#137—21no—
10C22×C3×Q16C2211yescharacteristic
11C2×C3×#131C2211yescharacteristic
12C2×C3×M32C2211yescharacteristic
13C22×C3×C16C2211yescharacteristic
14C3×#473—22no—
15C3×#473—22no—
16C2×C3×#108C2211yescharacteristic
17C2×C3×#108C2211yescharacteristic
18C2×C3×#107C2211yescharacteristic
19C3×#417—21no—
20C2×C3×#101—21no—
21C3×#368—22no—
22C3×#362—22no—
23C3×#358—22no—
24C3×C22⋊C16—22no—
25C3×#343—21no—
26C2×#473C2×C311yescharacteristic
27C2×#368C2×C311yescharacteristic, complemented
28C2×#362C2×C311yescharacteristic, complemented
29C2×#358C2×C311yescharacteristic, complemented
30#2113C2×C311yescharacteristic
31#1816C2×C311yescharacteristic
32#1387C2×C311yescharacteristic, complemented
33C2×C3×#41D411yescharacteristic
34C22×C3×Q8—21no—
35C3×#137—24no—
36C3×#137—42no—
37C2×C3×Q16—21no—
38C2×C3×Q16—21no—
39C2×C3×Q16—21no—
40C2×C3×Q16—21no—
41C2×C3×Q16—21no—
42C2×C3×Q16—41no—
43C2×C3×Q16D412yes—
44C2×C3×SD16—41no—
45C3×#131—21no—
46C3×#131—21no—
47C3×#131—21no—
48C3×#131—21no—
49C2×C3×M16—21no—
50C2×C3×M16—21no—
51C2×C3×M16—21no—
52C2×C3×M16D411yescharacteristic
53C22×C3×C8—21no—
54C22×C3×C8C2311yescharacteristic
55C2×C3×C4⋊C4—21no—
56C3×M32—21no—
57C3×M32—21no—
58C2×C3×C16—21no—
59C2×C3×C16—21no—
60C2×C3×C16—21no—
61C2×C3×C16D412yes—
62C3×#108—21no—
63C3×#108—21no—
64C3×#108—21no—
65C3×#108D412yes—
66C3×#107—21no—
67C3×#107—21no—
68C3×#103—41no—
69C3×#102—41no—
70C3×#101—24no—
71C2×#137—21no—
72C22×Q16C22×C311yescharacteristic
73C2×#131C22×C311yescharacteristic
74C2×M32C22×C311yescharacteristic
75C22×C16C22×C311yescharacteristic
76#473—22no—
77#473—22no—
78C2×#108C22×C311yescharacteristic
79C2×#108C22×C311yescharacteristic
80C2×#107C22×C311yescharacteristic
81#417—21no—
82C2×#101—21no—
83#368—22no—
84#362—22no—
85#358—22no—
86C22⋊C16—22no—
87#343—21no—
88C3×#41—21no—
89C3×#41—21no—
90C3×#41—21no—
91C3×#41—21no—
92C2×C3×Q8—22no—
93C2×C3×Q8—41no—
94C2×C3×Q8—41no—
95C2×C3×Q8—41no—
96C2×C3×Q8—41no—
97C2×C3×Q8—41no—
98C2×C3×Q8D811yescharacteristic
99C2×C3×D4—21no—
100C2×C3×D4D811yescharacteristic
101C22×C3×C4—21no—
102C22×C3×C4—41no—
103C22×C3×C4—41no—
104C22×C3×C4C2×D411yescharacteristic
105C3×Q16—22no—
106C3×Q16—22no—
107C3×Q16—44no—
108C3×Q16—41no—
109C3×Q16—41no—
110C3×SD16—44no—
111C3×M16—21no—
112C3×M16—22no—
113C3×M16—21no—
114C3×M16—41no—
115C3×M16—41no—
116C3×M16—41no—
117C3×M16—41no—
118C3×M16—41no—
119C2×C3×C8—21no—
120C2×C3×C8—41no—
121C2×C3×C8—41no—
122C2×C3×C8—41no—
123C2×C3×C8—41no—
124C2×C3×C8C2×D411yescharacteristic
125C2×C3×C8C2×D411yescharacteristic
126C2×C3×C8C2×D411yescharacteristic
127C2×C3×C8#4111yescharacteristic
128C2×C3×C8#4111yescharacteristic
129C2×C3×C8#4111yescharacteristic
130C3×C4⋊C4—41no—
131C3×C4⋊C4—41no—
132C3×C16—22no—
133C3×C16—41no—
134C2×#41C3×D411yescharacteristic
135C22×Q8—21no—
136#137—24no—
137#137—42no—
138C2×Q16—21no—
139C2×Q16—21no—
140C2×Q16—21no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×Q16—41no—
144C2×Q16C3×D412yes—
145C2×SD16—41no—
146#131—21no—
147#131—21no—
148#131—21no—
149#131—21no—
150C2×M16—21no—
151C2×M16—21no—
152C2×M16—21no—
153C2×M16C3×D411yescharacteristic
154C22×C8—21no—
155C22×C8C23×C311yescharacteristic
156C2×C4⋊C4—21no—
157M32—21no—
158M32—21no—
159C2×C16—21no—
160C2×C16—21no—
161C2×C16—21no—
162C2×C16C3×D412yes—
163#108—21no—
164#108—21no—
165#108—21no—
166#108C3×D412yes—
167#107—21no—
168#107—21no—
169#103—41no—
170#102—41no—
171#101—24no—
172C23×C3—21no—
173C23×C3#13511yescharacteristic
174C3×Q8—21no—
175C3×Q8—21no—
176C3×Q8—42no—
177C3×Q8—42no—
178C3×Q8—81no—
179C3×Q8—81no—
180C3×D4—21no—
181C3×D4—21no—
182C3×D4—41no—
183C3×D4—41no—
184C2×C3×C4—21no—
185C2×C3×C4—41no—
186C2×C3×C4—41no—
187C2×C3×C4—42no—
188C2×C3×C4—81no—
189C2×C3×C4—81no—
190C2×C3×C4—81no—
191C2×C3×C4—161no—
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12111yescharacteristic
194C2×C3×C4#12311yescharacteristic
195C2×C3×C4C2×D811yescharacteristic
196C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
197C2×C3×C4#13711yescharacteristic
198C3×C8—41no—
199C3×C8—42no—
200C3×C8—81no—
201C3×C8#12111yescharacteristic
202C3×C8#12111yescharacteristic
203C3×C8#12311yescharacteristic
204C3×C8#12411yescharacteristic
205#41—21no—
206#41—21no—
207#41—21no—
208#41—21no—
209C2×Q8—22no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—41no—
214C2×Q8—41no—
215C2×Q8C3×D811yescharacteristic
216C2×D4—21no—
217C2×D4C3×D811yescharacteristic
218C22×C4—21no—
219C22×C4—41no—
220C22×C4—41no—
221C22×C4C2×C3×D411yescharacteristic
222Q16—22no—
223Q16—22no—
224Q16—44no—
225Q16—41no—
226Q16—41no—
227SD16—44no—
228M16—21no—
229M16—22no—
230M16—21no—
231M16—41no—
232M16—41no—
233M16—41no—
234M16—41no—
235M16—41no—
236C2×C8—21no—
237C2×C8—41no—
238C2×C8—41no—
239C2×C8—41no—
240C2×C8—41no—
241C2×C8C2×C3×D411yescharacteristic
242C2×C8C2×C3×D411yescharacteristic
243C2×C8C2×C3×D411yescharacteristic
244C2×C8C3×#4111yescharacteristic
245C2×C8C3×#4111yescharacteristic
246C2×C8C3×#4111yescharacteristic
247C4⋊C4—41no—
248C4⋊C4—41no—
249C16—22no—
250C16—41no—
251C22×C3—21no—
252C22×C3—21no—
253C22×C3—41no—
254C22×C3—41no—
255C22×C3—41no—
256C22×C3#45211yescharacteristic
257C22×C3#46611yescharacteristic
258C22×C3#48211yescharacteristic
259C3×C4—41no—
260C3×C4—82no—
261C3×C4—161no—
262C3×C4#39411yescharacteristic
263C3×C4#44711yescharacteristic
264C3×C4#46811yescharacteristic
265C3×C4#48011yescharacteristic
266C23—21no—
267C23C3×#13511yescharacteristic
268Q8—21no—
269Q8—21no—
270Q8—42no—
271Q8—42no—
272Q8—81no—
273Q8—81no—
274D4—21no—
275D4—21no—
276D4—41no—
277D4—41no—
278C2×C4—21no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—42no—
282C2×C4—81no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—161no—
286C2×C4C3×#12011yescharacteristic
287C2×C4C3×#12111yescharacteristic
288C2×C4C3×#12311yescharacteristic
289C2×C4C2×C3×D811yescharacteristic
290C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
291C2×C4C3×#13711yescharacteristic
292C8—41no—
293C8—42no—
294C8—81no—
295C8C3×#12111yescharacteristic
296C8C3×#12111yescharacteristic
297C8C3×#12311yescharacteristic
298C8C3×#12411yescharacteristic
299C2×C3—21no—
300C2×C3—21no—
301C2×C3—81no—
302C2×C3#203311yescharacteristic
303C2×C3#222211yescharacteristic
304C2×C3#222311yescharacteristic
305C22—21no—
306C22—21no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22C3×#45211yescharacteristic
311C22C3×#46611yescharacteristic
312C22C3×#48211yescharacteristic
313C4—41no—
314C4—82no—
315C4—161no—
316C4C3×#39411yescharacteristic
317C4C3×#44711yescharacteristic
318C4C3×#46811yescharacteristic
319C4C3×#48011yescharacteristic
320C3#1339411yescharacteristic, direct product factor, Sylow
321C2—21no—
322C2—21no—
323C2—81no—
324C2C3×#203311yescharacteristic
325C2C3×#222211yescharacteristic
326C2C3×#222311yescharacteristic
subgroupG/H
center, socleC22×C3#466
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC2×C3×#41D4
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C2×C3×#131 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#368C2semidirect
3. C2×C3×#362C2semidirect
4. C2×C3×#358C2semidirect
7. C3×#1387C2semidirect
8. #13394C3direct (N × H)
27. C2×#368C2×C3semidirect
28. C2×#362C2×C3semidirect
29. C2×#358C2×C3semidirect
32. #1387C2×C3semidirect
320. C3#13394direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4732
C2×C3×#3682
C2×C3×#3622
C2×C3×#3582
C3×#21132
C3×#18162
C3×#13872
#133943