Finite Group Databasebeta

← C3×#13388 #10631995

Subgroups — C3×#13388 #10631995

Order 768.

396 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3059C211yesmaximal, characteristic
2C3×#2159C211yesmaximal, characteristic, complemented
3C2×C3×#362C211yesmaximal, characteristic, complemented
4C2×C3×#358C211yesmaximal, characteristic, complemented
5C2×C3×#350C211yesmaximal, characteristic
6C3×#1816C211yesmaximal, characteristic
7C3×#1382C211yesmaximal, characteristic, complemented
8#13388C311yesmaximal, characteristic, direct product factor, Sylow
9C22×C3×Q16C2211yescharacteristic
10C2×C3×#131C2211yescharacteristic
11C3×#536—21no—
12C2×C3×M32C2211yescharacteristic
13C22×C3×C16C2211yescharacteristic
14C3×#470—22no—
15C3×#467—21no—
16C3×#467—21no—
17C2×C3×#108C2211yescharacteristic
18C3×#427C2211yescharacteristic
19C2×C3×#107C2211yescharacteristic
20C3×#419—21no—
21C3×#417—21no—
22C3×#367—21no—
23C3×#366—21no—
24C3×#362—22no—
25C3×#358—22no—
26C3×#350—22no—
27C3×#343—21no—
28#3059C2×C311yescharacteristic
29#2159C2×C311yescharacteristic, complemented
30C2×#362C2×C311yescharacteristic, complemented
31C2×#358C2×C311yescharacteristic, complemented
32C2×#350C2×C311yescharacteristic
33#1816C2×C311yescharacteristic
34#1382C2×C311yescharacteristic, complemented
35C2×C3×#41D411yescharacteristic
36C22×C3×Q8—21no—
37C2×C3×Q16—21no—
38C2×C3×Q16—21no—
39C2×C3×Q16—21no—
40C2×C3×Q16—21no—
41C2×C3×Q16—21no—
42C2×C3×Q16D412yes—
43C3×#131—21no—
44C3×#131—21no—
45C3×#131—21no—
46C3×#131—21no—
47C2×C3×M16—21no—
48C2×C3×M16—21no—
49C2×C3×M16D411yescharacteristic
50C22×C3×C8—21no—
51C22×C3×C8C2311yescharacteristic
52C3×C4⋊Q8—41no—
53C3×#124—41no—
54C3×#123—42no—
55C3×C22⋊Q8—22no—
56C3×C22⋊Q8—22no—
57C3×#117—21no—
58C2×C3×C4⋊C4—21no—
59C3×M32—21no—
60C3×M32D412yes—
61C2×C3×C16—21no—
62C2×C3×C16—21no—
63C2×C3×C16—21no—
64C2×C3×C16—21no—
65C3×#108—21no—
66C3×#108—21no—
67C3×#107—21no—
68C3×#107—21no—
69C3×#107D411yescharacteristic
70C3×#107D411yescharacteristic
71C3×#106—21no—
72C3×#103—22no—
73C3×#103—22no—
74C3×#103—41no—
75C3×#102—41no—
76C22×Q16C22×C311yescharacteristic
77C2×#131C22×C311yescharacteristic
78#536—21no—
79C2×M32C22×C311yescharacteristic
80C22×C16C22×C311yescharacteristic
81#470—22no—
82#467—21no—
83#467—21no—
84C2×#108C22×C311yescharacteristic
85#427C22×C311yescharacteristic
86C2×#107C22×C311yescharacteristic
87#419—21no—
88#417—21no—
89#367—21no—
90#366—21no—
91#362—22no—
92#358—22no—
93#350—22no—
94#343—21no—
95C3×#41—21no—
96C3×#41—21no—
97C3×#41—21no—
98C3×#41—21no—
99C2×C3×Q8—22no—
100C2×C3×Q8—41no—
101C2×C3×Q8—41no—
102C2×C3×Q8—41no—
103C2×C3×Q8—41no—
104C2×C3×Q8—41no—
105C2×C3×Q8D811yescharacteristic
106C2×C3×D4—21no—
107C2×C3×D4D811yescharacteristic
108C22×C3×C4—21no—
109C22×C3×C4—41no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C3×Q16—22no—
113C3×Q16—22no—
114C3×Q16—41no—
115C3×Q16—41no—
116C3×M16—21no—
117C3×M16—21no—
118C3×M16—41no—
119C3×M16—41no—
120C3×M16—41no—
121C3×M16—41no—
122C2×C3×C8—21no—
123C2×C3×C8—41no—
124C2×C3×C8—41no—
125C2×C3×C8—41no—
126C2×C3×C8C2×D411yescharacteristic
127C2×C3×C8C2×D411yescharacteristic
128C2×C3×C8C2×D411yescharacteristic
129C2×C3×C8#4111yescharacteristic
130C2×C3×C8#4111yescharacteristic
131C2×C3×C8#4111yescharacteristic
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—42no—
135C3×C4⋊C4—42no—
136C3×C4⋊C4—41no—
137C3×C4⋊C4—41no—
138C3×C22⋊C4—42no—
139C3×C22⋊C4—42no—
140C3×C22⋊C4—41no—
141C3×C42—41no—
142C3×C16—22no—
143C3×C16—41no—
144C2×#41C3×D411yescharacteristic
145C22×Q8—21no—
146C2×Q16—21no—
147C2×Q16—21no—
148C2×Q16—21no—
149C2×Q16—21no—
150C2×Q16—21no—
151C2×Q16C3×D412yes—
152#131—21no—
153#131—21no—
154#131—21no—
155#131—21no—
156C2×M16—21no—
157C2×M16—21no—
158C2×M16C3×D411yescharacteristic
159C22×C8—21no—
160C22×C8C23×C311yescharacteristic
161C4⋊Q8—41no—
162#124—41no—
163#123—42no—
164C22⋊Q8—22no—
165C22⋊Q8—22no—
166#117—21no—
167C2×C4⋊C4—21no—
168M32—21no—
169M32C3×D412yes—
170C2×C16—21no—
171C2×C16—21no—
172C2×C16—21no—
173C2×C16—21no—
174#108—21no—
175#108—21no—
176#107—21no—
177#107—21no—
178#107C3×D411yescharacteristic
179#107C3×D411yescharacteristic
180#106—21no—
181#103—22no—
182#103—22no—
183#103—41no—
184#102—41no—
185C23×C3—21no—
186C23×C3AGL(1,ℤ/8)11yescharacteristic
187C3×Q8—21no—
188C3×Q8—21no—
189C3×Q8—42no—
190C3×Q8—42no—
191C3×Q8—81no—
192C3×Q8—81no—
193C3×D4—21no—
194C3×D4—21no—
195C3×D4—41no—
196C3×D4—41no—
197C2×C3×C4—21no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4—41no—
201C2×C3×C4—41no—
202C2×C3×C4—42no—
203C2×C3×C4—81no—
204C2×C3×C4—81no—
205C2×C3×C4—81no—
206C2×C3×C4—161no—
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12311yescharacteristic
210C2×C3×C4C2×D811yescharacteristic
211C2×C3×C4#13511yescharacteristic
212C2×C3×C4#13711yescharacteristic
213C3×C8—41no—
214C3×C8—81no—
215C3×C8#12111yescharacteristic
216C3×C8#12111yescharacteristic
217C3×C8#12311yescharacteristic
218C3×C8#12411yescharacteristic
219#41—21no—
220#41—21no—
221#41—21no—
222#41—21no—
223C2×Q8—22no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×Q8—41no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8C3×D811yescharacteristic
230C2×D4—21no—
231C2×D4C3×D811yescharacteristic
232C22×C4—21no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4C2×C3×D411yescharacteristic
236Q16—22no—
237Q16—22no—
238Q16—41no—
239Q16—41no—
240M16—21no—
241M16—21no—
242M16—41no—
243M16—41no—
244M16—41no—
245M16—41no—
246C2×C8—21no—
247C2×C8—41no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8C2×C3×D411yescharacteristic
251C2×C8C2×C3×D411yescharacteristic
252C2×C8C2×C3×D411yescharacteristic
253C2×C8C3×#4111yescharacteristic
254C2×C8C3×#4111yescharacteristic
255C2×C8C3×#4111yescharacteristic
256C4⋊C4—21no—
257C4⋊C4—21no—
258C4⋊C4—42no—
259C4⋊C4—42no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C22⋊C4—42no—
263C22⋊C4—42no—
264C22⋊C4—41no—
265C42—41no—
266C16—22no—
267C16—41no—
268C22×C3—21no—
269C22×C3—21no—
270C22×C3—41no—
271C22×C3—41no—
272C22×C3—41no—
273C22×C3#44711yescharacteristic
274C22×C3#46811yescharacteristic
275C22×C3#48211yescharacteristic
276C3×C4—41no—
277C3×C4—81no—
278C3×C4—81no—
279C3×C4—82no—
280C3×C4—161no—
281C3×C4#39411yescharacteristic
282C3×C4#45211yescharacteristic
283C3×C4#46611yescharacteristic
284C3×C4#48011yescharacteristic
285C23—21no—
286C23C3×AGL(1,ℤ/8)11yescharacteristic
287Q8—21no—
288Q8—21no—
289Q8—42no—
290Q8—42no—
291Q8—81no—
292Q8—81no—
293D4—21no—
294D4—21no—
295D4—41no—
296D4—41no—
297C2×C4—21no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—42no—
303C2×C4—81no—
304C2×C4—81no—
305C2×C4—81no—
306C2×C4—161no—
307C2×C4C3×#12011yescharacteristic
308C2×C4C3×#12111yescharacteristic
309C2×C4C3×#12311yescharacteristic
310C2×C4C2×C3×D811yescharacteristic
311C2×C4C3×#13511yescharacteristic
312C2×C4C3×#13711yescharacteristic
313C8—41no—
314C8—81no—
315C8C3×#12111yescharacteristic
316C8C3×#12111yescharacteristic
317C8C3×#12311yescharacteristic
318C8C3×#12411yescharacteristic
319C2×C3—21no—
320C2×C3—21no—
321C2×C3—81no—
322C2×C3#203311yescharacteristic
323C2×C3#219311yescharacteristic
324C2×C3#219511yescharacteristic
325C22—21no—
326C22—21no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22C3×#44711yescharacteristic
331C22C3×#46811yescharacteristic
332C22C3×#48211yescharacteristic
333C4—41no—
334C4—82no—
335C4—81no—
336C4—81no—
337C4—161no—
338C4C3×#39411yescharacteristic
339C4C3×#45211yescharacteristic
340C4C3×#46611yescharacteristic
341C4C3×#48011yescharacteristic
342C3#1338811yescharacteristic, direct product factor, Sylow
343C2—21no—
344C2—21no—
345C2—81no—
346C2C3×#203311yescharacteristic
347C2C3×#219311yescharacteristic
348C2C3×#219511yescharacteristic
subgroupG/H
center, socleC22×C3#447
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. C3×#2159C2semidirect
3. C2×C3×#362C2semidirect
4. C2×C3×#358C2semidirect
7. C3×#1382C2semidirect
8. #13388C3direct (N × H)
29. #2159C2×C3semidirect
30. C2×#362C2×C3semidirect
31. C2×#358C2×C3semidirect
34. #1382C2×C3semidirect
342. C3#13388direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30592
C3×#21592
C2×C3×#3622
C2×C3×#3582
C2×C3×#3502
C3×#18162
C3×#13822
#133883