Finite Group Databasebeta

← C3×#13400 #10632007

Subgroups — C3×#13400 #10632007

Order 768.

402 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3060C211yesmaximal, characteristic
2C2×C3×#365C211yesmaximal, characteristic, complemented
3C3×#2148C211yesmaximal, characteristic, complemented
4C3×#2145C211yesmaximal, characteristic, complemented
5C3×#2117C211yesmaximal, characteristic, complemented
6C3×#1814C211yesmaximal, characteristic, complemented
7C3×#1386C211yesmaximal, characteristic, complemented
8#13400C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×#135C2211yescharacteristic
10C2×C3×#131C2211yescharacteristic
11C3×#537—21no—
12C2×C3×M32C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14C3×#470—21no—
15C3×#469—21no—
16C3×#465—22no—
17C2×C3×#108C2211yescharacteristic
18C3×#427C2211yescharacteristic
19C2×C3×#106C2211yescharacteristic
20C3×#421—21no—
21C3×#417—21no—
22C3×#365—22no—
23C3×#362—21no—
24C3×#361—21no—
25C3×#360—22no—
26C3×#350—22no—
27C3×#344—21no—
28#3060C2×C311yescharacteristic
29C2×#365C2×C311yescharacteristic, complemented
30#2148C2×C311yescharacteristic, complemented
31#2145C2×C311yescharacteristic, complemented
32#2117C2×C311yescharacteristic, complemented
33#1814C2×C311yescharacteristic, complemented
34#1386C2×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—
58C3×#117—21no—
59C2×C3×C4⋊C4—21no—
60C3×M32—21no—
61C3×M32—21no—
62C3×M32—21no—
63C3×M32D412yescomplemented
64C2×C3×C16—21no—
65C2×C3×C16—21no—
66C3×#108—21no—
67C3×#108—21no—
68C3×#107—21no—
69C3×#106—21no—
70C3×#106—21no—
71C3×#106D412yes—
72C3×#104—42no—
73C3×#103—22no—
74C3×#102—22no—
75C2×#135C22×C311yescharacteristic
76C2×#131C22×C311yescharacteristic
77#537—21no—
78C2×M32C22×C311yescharacteristic
79C2×M32C22×C311yescharacteristic
80#470—21no—
81#469—21no—
82#465—22no—
83C2×#108C22×C311yescharacteristic
84#427C22×C311yescharacteristic
85C2×#106C22×C311yescharacteristic
86#421—21no—
87#417—21no—
88#365—22no—
89#362—21no—
90#361—21no—
91#360—22no—
92#350—22no—
93#344—21no—
94C3×#41—21no—
95C3×#41—21no—
96C3×#41—21no—
97C3×#41—21no—
98C3×#41—41no—
99C3×#41—41no—
100C3×#41—41no—
101C3×#41—41no—
102C2×C3×Q8—21no—
103C2×C3×Q8SD1611yescharacteristic
104C2×C3×D4—21no—
105C2×C3×D4—21no—
106C2×C3×D4—41no—
107C2×C3×D4—42no—
108C2×C3×D4SD1611yescharacteristic
109C22×C3×C4—21no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4C2×D411yescharacteristic
113C3×Q16—21no—
114C3×Q16—21no—
115C3×SD16—41no—
116C3×SD16—41no—
117C3×D8—21no—
118C3×D8—21no—
119C3×M16—21no—
120C3×M16—21no—
121C3×M16—41no—
122C3×M16—41no—
123C3×M16—41no—
124C3×M16—41no—
125C2×C3×C8—21no—
126C2×C3×C8—41no—
127C2×C3×C8—41no—
128C2×C3×C8—41no—
129C2×C3×C8C2×D411yescharacteristic
130C2×C3×C8C2×D411yescharacteristic
131C2×C3×C8C2×D411yescharacteristic
132C2×C3×C8#4111yescharacteristic
133C2×C3×C8#4111yescharacteristic
134C2×C3×C8#4111yescharacteristic
135C3×C4⋊C4—22no—
136C3×C4⋊C4—42no—
137C3×C4⋊C4—41no—
138C3×C4⋊C4—41no—
139C3×C22⋊C4—42no—
140C3×C22⋊C4—42no—
141C3×C22⋊C4—41no—
142C3×C42—41no—
143C3×C16—22no—
144C3×C16—41no—
145C2×#41—21no—
146C2×#41C3×D411yescharacteristic
147#135—21no—
148#135—21no—
149#135—21no—
150#135—21no—
151C2×Q16C3×D411yescharacteristic
152C2×SD16—21no—
153C2×D8C3×D411yescharacteristic
154#131—21no—
155#131—21no—
156#131—21no—
157#131—21no—
158C2×M16—21no—
159C2×M16—21no—
160C2×M16C3×D411yescharacteristic
161C22×C8—21no—
162C22×C8C23×C311yescharacteristic
163C22⋊Q8—22no—
164C22⋊Q8—41no—
165#121—22no—
166#121—42no—
167#121—41no—
168#117—21no—
169C2×C4⋊C4—21no—
170M32—21no—
171M32—21no—
172M32—21no—
173M32C3×D412yescomplemented
174C2×C16—21no—
175C2×C16—21no—
176#108—21no—
177#108—21no—
178#107—21no—
179#106—21no—
180#106—21no—
181#106C3×D412yes—
182#104—42no—
183#103—22no—
184#102—22no—
185C23×C3—21no—
186C23×C3—41no—
187C23×C3AGL(1,ℤ/8)11yescharacteristic
188C3×Q8—21no—
189C3×Q8—21no—
190C3×Q8—41no—
191C3×Q8—41no—
192C3×D4—21no—
193C3×D4—21no—
194C3×D4—41no—
195C3×D4—41no—
196C3×D4—41no—
197C3×D4—41no—
198C3×D4—81no—
199C3×D4—81no—
200C3×D4—162no—
201C2×C3×C4—21no—
202C2×C3×C4—41no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—41no—
206C2×C3×C4—81no—
207C2×C3×C4—81no—
208C2×C3×C4—81no—
209C2×C3×C4—161no—
210C2×C3×C4#12011yescharacteristic
211C2×C3×C4#12111yescharacteristic
212C2×C3×C4#12311yescharacteristic
213C2×C3×C4C2×SD1611yescharacteristic
214C2×C3×C4#13511yescharacteristic
215C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
216C3×C8—41no—
217C3×C8—81no—
218C3×C8#12111yescharacteristic
219C3×C8#12111yescharacteristic
220C3×C8#12311yescharacteristic
221C3×C8#12411yescharacteristic
222#41—21no—
223#41—21no—
224#41—21no—
225#41—21no—
226#41—41no—
227#41—41no—
228#41—41no—
229#41—41no—
230C2×Q8—21no—
231C2×Q8C3×SD1611yescharacteristic
232C2×D4—21no—
233C2×D4—21no—
234C2×D4—41no—
235C2×D4—42no—
236C2×D4C3×SD1611yescharacteristic
237C22×C4—21no—
238C22×C4—41no—
239C22×C4—41no—
240C22×C4C2×C3×D411yescharacteristic
241Q16—21no—
242Q16—21no—
243SD16—41no—
244SD16—41no—
245D8—21no—
246D8—21no—
247M16—21no—
248M16—21no—
249M16—41no—
250M16—41no—
251M16—41no—
252M16—41no—
253C2×C8—21no—
254C2×C8—41no—
255C2×C8—41no—
256C2×C8—41no—
257C2×C8C2×C3×D411yescharacteristic
258C2×C8C2×C3×D411yescharacteristic
259C2×C8C2×C3×D411yescharacteristic
260C2×C8C3×#4111yescharacteristic
261C2×C8C3×#4111yescharacteristic
262C2×C8C3×#4111yescharacteristic
263C4⋊C4—22no—
264C4⋊C4—42no—
265C4⋊C4—41no—
266C4⋊C4—41no—
267C22⋊C4—42no—
268C22⋊C4—42no—
269C22⋊C4—41no—
270C42—41no—
271C16—22no—
272C16—41no—
273C22×C3—41no—
274C22×C3—42no—
275C22×C3—41no—
276C22×C3—81no—
277C22×C3—81no—
278C22×C3—81no—
279C22×C3#45011yescharacteristic
280C22×C3#46911yescharacteristic
281C22×C3#48211yescharacteristic
282C3×C4—41no—
283C3×C4—82no—
284C3×C4—81no—
285C3×C4—161no—
286C3×C4#39411yescharacteristic
287C3×C4#44911yescharacteristic
288C3×C4#46511yescharacteristic
289C3×C4#48111yescharacteristic
290C23—21no—
291C23—41no—
292C23C3×AGL(1,ℤ/8)11yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—41no—
296Q8—41no—
297D4—21no—
298D4—21no—
299D4—41no—
300D4—41no—
301D4—41no—
302D4—41no—
303D4—81no—
304D4—81no—
305D4—162no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—81no—
312C2×C4—81no—
313C2×C4—81no—
314C2×C4—161no—
315C2×C4C3×#12011yescharacteristic
316C2×C4C3×#12111yescharacteristic
317C2×C4C3×#12311yescharacteristic
318C2×C4C2×C3×SD1611yescharacteristic
319C2×C4C3×#13511yescharacteristic
320C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
321C8—41no—
322C8—81no—
323C8C3×#12111yescharacteristic
324C8C3×#12111yescharacteristic
325C8C3×#12311yescharacteristic
326C8C3×#12411yescharacteristic
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—21no—
330C2×C3—81no—
331C2×C3—161no—
332C2×C3#203411yescharacteristic
333C22—41no—
334C22—42no—
335C22—41no—
336C22—81no—
337C22—81no—
338C22—81no—
339C22C3×#45011yescharacteristic
340C22C3×#46911yescharacteristic
341C22C3×#48211yescharacteristic
342C4—41no—
343C4—82no—
344C4—81no—
345C4—161no—
346C4C3×#39411yescharacteristic
347C4C3×#44911yescharacteristic
348C4C3×#46511yescharacteristic
349C4C3×#48111yescharacteristic
350C3#1340011yescharacteristic, direct product factor, Sylow
351C2—21no—
352C2—21no—
353C2—21no—
354C2—81no—
355C2—161no—
356C2C3×#203411yescharacteristic
subgroupG/H
center, socleC2×C3#2034
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC3×#3060C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×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×#365C2semidirect
3. C3×#2148C2semidirect
4. C3×#2145C2semidirect
5. C3×#2117C2semidirect
6. C3×#1814C2semidirect
7. C3×#1386C2semidirect
8. #13400C3direct (N × H)
29. C2×#365C2×C3semidirect
30. #2148C2×C3semidirect
31. #2145C2×C3semidirect
32. #2117C2×C3semidirect
33. #1814C2×C3semidirect
34. #1386C2×C3semidirect
63. C3×M32D4semidirect
173. M32C3×D4semidirect
350. C3#13400direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#30602
C2×C3×#3652
C3×#21482
C3×#21452
C3×#21172
C3×#18142
C3×#13862
#134003