Finite Group Databasebeta

← C3×#13395 #10632002

Subgroups — C3×#13395 #10632002

Order 768.

422 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3050C211yesmaximal, characteristic
2C3×#2155C211yesmaximal, characteristic, complemented
3C3×#2148C211yesmaximal, characteristic, complemented
4C3×#2143C211yesmaximal, characteristic, complemented
5C3×#2112C211yesmaximal, characteristic, complemented
6C3×#1810C211yesmaximal, characteristic, complemented
7C3×#1389C211yesmaximal, characteristic, complemented
8#13395C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#135C2211yescharacteristic
10C23×C3×C8C2211yescharacteristic
11C3×#525—21no—
12C2×C3×M32C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14C3×#467—21no—
15C3×#466—21no—
16C3×#465—22no—
17C2×C3×#108C2211yescharacteristic
18C2×C3×#108C2211yescharacteristic
19C3×#427C2211yescharacteristic
20C3×#416—21no—
21C3×#407—21no—
22C3×#365—22no—
23C3×#362—21no—
24C3×#361—21no—
25C3×#359—22no—
26C3×C22⋊C16—22no—
27C3×#341—21no—
28#3050C2×C311yescharacteristic
29#2155C2×C311yescharacteristic, complemented
30#2148C2×C311yescharacteristic, complemented
31#2143C2×C311yescharacteristic, complemented
32#2112C2×C311yescharacteristic, complemented
33#1810C2×C311yescharacteristic, complemented
34#1389C2×C311yescharacteristic, complemented
35C2×C3×#41—21no—
36C23×C3×C4D411yescharacteristic
37C3×#135—21no—
38C3×#135—21no—
39C3×#135—21no—
40C3×#135—21no—
41C2×C3×Q16D411yescharacteristic
42C2×C3×SD16—21no—
43C2×C3×D8D411yescharacteristic
44C2×C3×M16—21no—
45C2×C3×M16—21no—
46C22×C3×C8—21no—
47C22×C3×C8—21no—
48C22×C3×C8—21no—
49C22×C3×C8—21no—
50C22×C3×C8—21no—
51C22×C3×C8—21no—
52C22×C3×C8D411yescharacteristic
53C22×C3×C8C2311yescharacteristic
54C3×#123—41no—
55C3×C22⋊Q8—22no—
56C3×#121—22no—
57C3×#120—41no—
58C3×C4×D4—42no—
59C3×#117—21no—
60C3×M32—21no—
61C3×M32—21no—
62C3×M32—21no—
63C3×M32—21no—
64C2×C3×C16—21no—
65C2×C3×C16D412yescomplemented
66C3×#108—21no—
67C3×#108—21no—
68C3×#108—21no—
69C3×#108—21no—
70C3×#107—21no—
71C3×#106D412yes—
72C3×#103—22no—
73C3×#102—22no—
74C3×C22⋊C8—42no—
75C2×#135C22×C311yescharacteristic
76C23×C8C22×C311yescharacteristic
77#525—21no—
78C2×M32C22×C311yescharacteristic
79C2×M32C22×C311yescharacteristic
80#467—21no—
81#466—21no—
82#465—22no—
83C2×#108C22×C311yescharacteristic
84C2×#108C22×C311yescharacteristic
85#427C22×C311yescharacteristic
86#416—21no—
87#407—21no—
88#365—22no—
89#362—21no—
90#361—21no—
91#359—22no—
92C22⋊C16—22no—
93#341—21no—
94C24×C3SD1611yescharacteristic
95C3×#41—41no—
96C3×#41—41no—
97C3×#41—41no—
98C3×#41—41no—
99C2×C3×Q8—21no—
100C2×C3×D4—21no—
101C2×C3×D4—42no—
102C2×C3×D4—41no—
103C22×C3×C4—21no—
104C22×C3×C4—21no—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—41no—
110C22×C3×C4SD1611yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C3×Q16—21no—
113C3×Q16—21no—
114C3×SD16—41no—
115C3×SD16—41no—
116C3×D8—21no—
117C3×D8—21no—
118C3×M16—41no—
119C3×M16—41no—
120C3×M16—41no—
121C3×M16—41no—
122C2×C3×C8—21no—
123C2×C3×C8—21no—
124C2×C3×C8—21no—
125C2×C3×C8—41no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
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×C22⋊C4—42no—
140C3×C22⋊C4—42no—
141C3×C22⋊C4—41no—
142C3×C42—41no—
143C3×C16—22no—
144C3×C16—41no—
145C2×#41—21no—
146C23×C4C3×D411yescharacteristic
147#135—21no—
148#135—21no—
149#135—21no—
150#135—21no—
151C2×Q16C3×D411yescharacteristic
152C2×SD16—21no—
153C2×D8C3×D411yescharacteristic
154C2×M16—21no—
155C2×M16—21no—
156C22×C8—21no—
157C22×C8—21no—
158C22×C8—21no—
159C22×C8—21no—
160C22×C8—21no—
161C22×C8—21no—
162C22×C8C3×D411yescharacteristic
163C22×C8C23×C311yescharacteristic
164#123—41no—
165C22⋊Q8—22no—
166#121—22no—
167#120—41no—
168C4×D4—42no—
169#117—21no—
170M32—21no—
171M32—21no—
172M32—21no—
173M32—21no—
174C2×C16—21no—
175C2×C16C3×D412yescomplemented
176#108—21no—
177#108—21no—
178#108—21no—
179#108—21no—
180#107—21no—
181#106C3×D412yes—
182#103—22no—
183#102—22no—
184C22⋊C8—42no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—42no—
189C23×C3—41no—
190C23×C3C2×SD1611yescharacteristic
191C3×Q8—41no—
192C3×Q8—41no—
193C3×D4—41no—
194C3×D4—41no—
195C3×D4—84no—
196C3×D4—81no—
197C3×D4—81no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—41no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—42no—
206C2×C3×C4—41no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4#12011yescharacteristic
210C2×C3×C4#12111yescharacteristic
211C2×C3×C4#12311yescharacteristic
212C2×C3×C4#13511yescharacteristic
213C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
214C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
215C3×C8—41no—
216C3×C8—81no—
217C3×C8—81no—
218C3×C8#12111yescharacteristic
219C3×C8#12111yescharacteristic
220C3×C8#12311yescharacteristic
221C3×C8#12411yescharacteristic
222C24C3×SD1611yescharacteristic
223#41—41no—
224#41—41no—
225#41—41no—
226#41—41no—
227C2×Q8—21no—
228C2×D4—21no—
229C2×D4—42no—
230C2×D4—41no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—41no—
238C22×C4C3×SD1611yescharacteristic
239C22×C4C2×C3×D411yescharacteristic
240Q16—21no—
241Q16—21no—
242SD16—41no—
243SD16—41no—
244D8—21no—
245D8—21no—
246M16—41no—
247M16—41no—
248M16—41no—
249M16—41no—
250C2×C8—21no—
251C2×C8—21no—
252C2×C8—21no—
253C2×C8—41no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8C2×C3×D411yescharacteristic
260C2×C8C2×C3×D411yescharacteristic
261C2×C8C2×C3×D411yescharacteristic
262C2×C8C3×#4111yescharacteristic
263C2×C8C3×#4111yescharacteristic
264C2×C8C3×#4111yescharacteristic
265C4⋊C4—22no—
266C4⋊C4—42no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C42—41no—
271C16—22no—
272C16—41no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—82no—
282C22×C3—81no—
283C22×C3#45011yescharacteristic
284C22×C3#46511yescharacteristic
285C22×C3#48111yescharacteristic
286C3×C4—41no—
287C3×C4—82no—
288C3×C4—81no—
289C3×C4#39411yescharacteristic
290C3×C4#44911yescharacteristic
291C3×C4#46911yescharacteristic
292C3×C4#48211yescharacteristic
293C23—21no—
294C23—21no—
295C23—21no—
296C23—42no—
297C23—41no—
298C23C2×C3×SD1611yescharacteristic
299Q8—41no—
300Q8—41no—
301D4—41no—
302D4—41no—
303D4—84no—
304D4—81no—
305D4—81no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—42no—
314C2×C4—41no—
315C2×C4—81no—
316C2×C4—81no—
317C2×C4C3×#12011yescharacteristic
318C2×C4C3×#12111yescharacteristic
319C2×C4C3×#12311yescharacteristic
320C2×C4C3×#13511yescharacteristic
321C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
322C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
323C8—41no—
324C8—81no—
325C8—81no—
326C8C3×#12111yescharacteristic
327C8C3×#12111yescharacteristic
328C8C3×#12311yescharacteristic
329C8C3×#12411yescharacteristic
330C2×C3—21no—
331C2×C3—21no—
332C2×C3—21no—
333C2×C3—42no—
334C2×C3—161no—
335C2×C3#203411yescharacteristic
336C22—22no—
337C22—22no—
338C22—22no—
339C22—41no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22—82no—
345C22—81no—
346C22C3×#45011yescharacteristic
347C22C3×#46511yescharacteristic
348C22C3×#48111yescharacteristic
349C4—41no—
350C4—82no—
351C4—81no—
352C4C3×#39411yescharacteristic
353C4C3×#44911yescharacteristic
354C4C3×#46911yescharacteristic
355C4C3×#48211yescharacteristic
356C3#1339511yescharacteristic, direct product factor, Sylow
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—161no—
362C2C3×#203411yescharacteristic
subgroupG/H
centerC3×C4#469
commutator, Frattini, cosocleC22×C8C23×C3
socleC2×C3#2034
prime radicalC3×#3050C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4 ◁ C22×C3×C4 ◁ C23×C3×C8 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2155C2semidirect
3. C3×#2148C2semidirect
4. C3×#2143C2semidirect
5. C3×#2112C2semidirect
6. C3×#1810C2semidirect
7. C3×#1389C2semidirect
8. #13395C3direct (N × H)
29. #2155C2×C3semidirect
30. #2148C2×C3semidirect
31. #2143C2×C3semidirect
32. #2112C2×C3semidirect
33. #1810C2×C3semidirect
34. #1389C2×C3semidirect
65. C2×C3×C16D4semidirect
175. C2×C16C3×D4semidirect
356. C3#13395direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30502
C3×#21552
C3×#21482
C3×#21432
C3×#21122
C3×#18102
C3×#13892
#133953