Finite Group Databasebeta

← C3×#13385 #10631992

Subgroups — C3×#13385 #10631992

Order 768.

408 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3061C211yesmaximal, characteristic
2C2×C3×#367C211yesmaximal, characteristic, complemented
3C2×C3×#359C211yesmaximal, characteristic, complemented
4C3×#2139C211yesmaximal, characteristic, complemented
5C2×C3×#350C211yesmaximal, characteristic, complemented
6C3×#1816C211yesmaximal, characteristic, complemented
7C3×#1387C211yesmaximal, characteristic, complemented
8#13385C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#135C2211yescharacteristic
10C2×C3×#131C2211yescharacteristic
11C3×#537—21no—
12C2×C3×M32C2211yescharacteristic
13C22×C3×C16C2211yescharacteristic
14C3×#468—22no—
15C3×#467—21no—
16C3×#466—21no—
17C2×C3×#108C2211yescharacteristic
18C2×C3×#107C2211yescharacteristic
19C2×C3×#107C2211yescharacteristic
20C3×#417—21no—
21C3×#417—21no—
22C3×#367—22no—
23C3×#359—22no—
24C3×#358—21no—
25C3×#357—21no—
26C3×#350—22no—
27C3×#343—21no—
28#3061C2×C311yescharacteristic
29C2×#367C2×C311yescharacteristic, complemented
30C2×#359C2×C311yescharacteristic, complemented
31#2139C2×C311yescharacteristic, complemented
32C2×#350C2×C311yescharacteristic, complemented
33#1816C2×C311yescharacteristic, complemented
34#1387C2×C311yescharacteristic, complemented
35C2×C3×#41—21no—
36C2×C3×#41D411yescharacteristic
37C3×#135—21no—
38C3×#135—21no—
39C3×#135—21no—
40C3×#135—21no—
41C2×C3×Q16D411yescharacteristic
42C2×C3×SD16—21no—
43C2×C3×D8D411yescharacteristic
44C3×#131—21no—
45C3×#131—21no—
46C3×#131—21no—
47C3×#131—21no—
48C2×C3×M16—21no—
49C2×C3×M16—21no—
50C2×C3×M16D411yescharacteristic
51C22×C3×C8—21no—
52C22×C3×C8C2311yescharacteristic
53C3×C22⋊Q8—22no—
54C3×C22⋊Q8—41no—
55C3×#121—22no—
56C3×#121—42no—
57C3×#121—41no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C3×M32—21no—
61C3×M32—21no—
62C2×C3×C16—21no—
63C2×C3×C16—21no—
64C2×C3×C16—21no—
65C2×C3×C16D412yescomplemented
66C3×#108—21no—
67C3×#108—21no—
68C3×#107—21no—
69C3×#107—21no—
70C3×#107—21no—
71C3×#107D412yes—
72C3×#103—22no—
73C3×#103—41no—
74C3×#102—22no—
75C3×#102—41no—
76C2×#135C22×C311yescharacteristic
77C2×#131C22×C311yescharacteristic
78#537—21no—
79C2×M32C22×C311yescharacteristic
80C22×C16C22×C311yescharacteristic
81#468—22no—
82#467—21no—
83#466—21no—
84C2×#108C22×C311yescharacteristic
85C2×#107C22×C311yescharacteristic
86C2×#107C22×C311yescharacteristic
87#417—21no—
88#417—21no—
89#367—22no—
90#359—22no—
91#358—21no—
92#357—21no—
93#350—22no—
94#343—21no—
95C3×#41—21no—
96C3×#41—21no—
97C3×#41—21no—
98C3×#41—21no—
99C3×#41—41no—
100C3×#41—41no—
101C3×#41—41no—
102C3×#41—41no—
103C2×C3×Q8—21no—
104C2×C3×Q8D811yescharacteristic
105C2×C3×D4—21no—
106C2×C3×D4—21no—
107C2×C3×D4—41no—
108C2×C3×D4—42no—
109C2×C3×D4D811yescharacteristic
110C22×C3×C4—21no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C3×Q16—21no—
116C3×Q16—21no—
117C3×SD16—41no—
118C3×SD16—41no—
119C3×D8—21no—
120C3×D8—21no—
121C3×M16—21no—
122C3×M16—21no—
123C3×M16—41no—
124C3×M16—41no—
125C3×M16—41no—
126C3×M16—41no—
127C2×C3×C8—21no—
128C2×C3×C8—41no—
129C2×C3×C8—41no—
130C2×C3×C8—41no—
131C2×C3×C8C2×D411yescharacteristic
132C2×C3×C8C2×D411yescharacteristic
133C2×C3×C8C2×D411yescharacteristic
134C2×C3×C8#4111yescharacteristic
135C2×C3×C8#4111yescharacteristic
136C2×C3×C8#4111yescharacteristic
137C3×C4⋊C4—22no—
138C3×C4⋊C4—42no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C22⋊C4—42no—
143C3×C22⋊C4—42no—
144C3×C16—22no—
145C3×C16—41no—
146C2×#41—21no—
147C2×#41C3×D411yescharacteristic
148#135—21no—
149#135—21no—
150#135—21no—
151#135—21no—
152C2×Q16C3×D411yescharacteristic
153C2×SD16—21no—
154C2×D8C3×D411yescharacteristic
155#131—21no—
156#131—21no—
157#131—21no—
158#131—21no—
159C2×M16—21no—
160C2×M16—21no—
161C2×M16C3×D411yescharacteristic
162C22×C8—21no—
163C22×C8C23×C311yescharacteristic
164C22⋊Q8—22no—
165C22⋊Q8—41no—
166#121—22no—
167#121—42no—
168#121—41no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—21no—
171M32—21no—
172M32—21no—
173C2×C16—21no—
174C2×C16—21no—
175C2×C16—21no—
176C2×C16C3×D412yescomplemented
177#108—21no—
178#108—21no—
179#107—21no—
180#107—21no—
181#107—21no—
182#107C3×D412yes—
183#103—22no—
184#103—41no—
185#102—22no—
186#102—41no—
187C23×C3—21no—
188C23×C3—41no—
189C23×C3AGL(1,ℤ/8)11yescharacteristic
190C3×Q8—21no—
191C3×Q8—21no—
192C3×Q8—41no—
193C3×Q8—41no—
194C3×D4—21no—
195C3×D4—21no—
196C3×D4—41no—
197C3×D4—41no—
198C3×D4—41no—
199C3×D4—41no—
200C3×D4—81no—
201C3×D4—81no—
202C3×D4—162no—
203C2×C3×C4—21no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—42no—
207C2×C3×C4—41no—
208C2×C3×C4—81no—
209C2×C3×C4—81no—
210C2×C3×C4—81no—
211C2×C3×C4—161no—
212C2×C3×C4—161no—
213C2×C3×C4#12011yescharacteristic
214C2×C3×C4#12111yescharacteristic
215C2×C3×C4#12311yescharacteristic
216C2×C3×C4C2×D811yescharacteristic
217C2×C3×C4#13511yescharacteristic
218C2×C3×C4#13711yescharacteristic
219C3×C8—41no—
220C3×C8—81no—
221C3×C8#12111yescharacteristic
222C3×C8#12111yescharacteristic
223C3×C8#12311yescharacteristic
224C3×C8#12411yescharacteristic
225#41—21no—
226#41—21no—
227#41—21no—
228#41—21no—
229#41—41no—
230#41—41no—
231#41—41no—
232#41—41no—
233C2×Q8—21no—
234C2×Q8C3×D811yescharacteristic
235C2×D4—21no—
236C2×D4—21no—
237C2×D4—41no—
238C2×D4—42no—
239C2×D4C3×D811yescharacteristic
240C22×C4—21no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4C2×C3×D411yescharacteristic
245Q16—21no—
246Q16—21no—
247SD16—41no—
248SD16—41no—
249D8—21no—
250D8—21no—
251M16—21no—
252M16—21no—
253M16—41no—
254M16—41no—
255M16—41no—
256M16—41no—
257C2×C8—21no—
258C2×C8—41no—
259C2×C8—41no—
260C2×C8—41no—
261C2×C8C2×C3×D411yescharacteristic
262C2×C8C2×C3×D411yescharacteristic
263C2×C8C2×C3×D411yescharacteristic
264C2×C8C3×#4111yescharacteristic
265C2×C8C3×#4111yescharacteristic
266C2×C8C3×#4111yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—42no—
269C4⋊C4—41no—
270C4⋊C4—41no—
271C4⋊C4—41no—
272C22⋊C4—42no—
273C22⋊C4—42no—
274C16—22no—
275C16—41no—
276C22×C3—21no—
277C22×C3—21no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—81no—
282C22×C3—81no—
283C22×C3—81no—
284C22×C3#44711yescharacteristic
285C22×C3#46811yescharacteristic
286C22×C3#48211yescharacteristic
287C3×C4—41no—
288C3×C4—82no—
289C3×C4—81no—
290C3×C4—161no—
291C3×C4#39411yescharacteristic
292C3×C4#45211yescharacteristic
293C3×C4#46611yescharacteristic
294C3×C4#48011yescharacteristic
295C23—21no—
296C23—41no—
297C23C3×AGL(1,ℤ/8)11yescharacteristic
298Q8—21no—
299Q8—21no—
300Q8—41no—
301Q8—41no—
302D4—21no—
303D4—21no—
304D4—41no—
305D4—41no—
306D4—41no—
307D4—41no—
308D4—81no—
309D4—81no—
310D4—162no—
311C2×C4—21no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—42no—
315C2×C4—41no—
316C2×C4—81no—
317C2×C4—81no—
318C2×C4—81no—
319C2×C4—161no—
320C2×C4—161no—
321C2×C4C3×#12011yescharacteristic
322C2×C4C3×#12111yescharacteristic
323C2×C4C3×#12311yescharacteristic
324C2×C4C2×C3×D811yescharacteristic
325C2×C4C3×#13511yescharacteristic
326C2×C4C3×#13711yescharacteristic
327C8—41no—
328C8—81no—
329C8C3×#12111yescharacteristic
330C8C3×#12111yescharacteristic
331C8C3×#12311yescharacteristic
332C8C3×#12411yescharacteristic
333C2×C3—21no—
334C2×C3—21no—
335C2×C3—81no—
336C2×C3—161no—
337C2×C3#203311yescharacteristic
338C2×C3#219311yescharacteristic
339C2×C3#219411yescharacteristic
340C22—21no—
341C22—21no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—81no—
346C22—81no—
347C22—81no—
348C22C3×#44711yescharacteristic
349C22C3×#46811yescharacteristic
350C22C3×#48211yescharacteristic
351C4—41no—
352C4—81no—
353C4—82no—
354C4—161no—
355C4C3×#39411yescharacteristic
356C4C3×#45211yescharacteristic
357C4C3×#46611yescharacteristic
358C4C3×#48011yescharacteristic
359C3#1338511yescharacteristic, direct product factor, Sylow
360C2—21no—
361C2—21no—
362C2—81no—
363C2—161no—
364C2C3×#203311yescharacteristic
365C2C3×#219311yescharacteristic
366C2C3×#219411yescharacteristic
subgroupG/H
center, socleC22×C3#447
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC3×#3061C2
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×#367C2semidirect
3. C2×C3×#359C2semidirect
4. C3×#2139C2semidirect
5. C2×C3×#350C2semidirect
6. C3×#1816C2semidirect
7. C3×#1387C2semidirect
8. #13385C3direct (N × H)
29. C2×#367C2×C3semidirect
30. C2×#359C2×C3semidirect
31. #2139C2×C3semidirect
32. C2×#350C2×C3semidirect
33. #1816C2×C3semidirect
34. #1387C2×C3semidirect
65. C2×C3×C16D4semidirect
176. C2×C16C3×D4semidirect
359. C3#13385direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30612
C2×C3×#3672
C2×C3×#3592
C3×#21392
C2×C3×#3502
C3×#18162
C3×#13872
#133853