Finite Group Databasebeta

← C3×#13417 #10632024

Subgroups — C3×#13417 #10632024

Order 768.

454 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×#3155C211yesmaximal, characteristic
2C3×#2148C211yesmaximal, characteristic, complemented
3C2×C3×#361C211yesmaximal, characteristic, complemented
4C2×C3×#361C211yesmaximal, characteristic, complemented
5C3×#2143C211yesmaximal, characteristic, complemented
6C3×#1953C211yesmaximal, characteristic, complemented
7C3×#1378C211yesmaximal, characteristic, complemented
8#13417C311yesmaximal, characteristic, direct product factor, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C2×C3×#135C2211yescharacteristic
11C22×C3×D8C2211yescharacteristic
12C3×#535—21no—
13C2×C3×M32C2211yescharacteristic
14C2×C3×M32C2211yescharacteristic
15C3×#496—22no—
16C3×#495—21no—
17C3×#493—21no—
18C3×#430—21no—
19C2×C3×#108C2211yescharacteristic
20C2×C3×#108C2211yescharacteristic
21C3×#405C2211yescharacteristic
22C3×#362—21no—
23C3×#361—22no—
24C3×#361—21no—
25C3×#361—22no—
26C3×#359—22no—
27C3×#344—21no—
28#3155C2×C311yescharacteristic
29#2148C2×C311yescharacteristic, complemented
30C2×#361C2×C311yescharacteristic, complemented
31C2×#361C2×C311yescharacteristic, complemented
32#2143C2×C311yescharacteristic, complemented
33#1953C2×C311yescharacteristic, complemented
34#1378C2×C311yescharacteristic, complemented
35C2×C3×#41—21no—
36C22×C3×D4—21no—
37C3×AGL(1,ℤ/8)—42no—
38C3×AGL(1,ℤ/8)—42no—
39C3×#135—21no—
40C3×#135—21no—
41C3×#135—21no—
42C3×#135—21no—
43C2×C3×Q16D411yescharacteristic
44C2×C3×SD16—22no—
45C2×C3×SD16—21no—
46C2×C3×D8—22no—
47C2×C3×D8—21no—
48C2×C3×D8—21no—
49C2×C3×D8—21no—
50C2×C3×D8—21no—
51C2×C3×D8—21no—
52C2×C3×D8D412yes—
53C2×C3×D8D411yescharacteristic
54C2×C3×M16—21no—
55C2×C3×M16—21no—
56C2×C3×M16D411yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C3×#127—22no—
59C3×#124—22no—
60C3×#121—42no—
61C3×#120—42no—
62C3×#117D411yescharacteristic
63C3×M32—21no—
64C3×M32—21no—
65C3×M32—21no—
66C3×M32—21no—
67C2×C3×C16—21no—
68C2×C3×C16—21no—
69C3×#108—21no—
70C3×#108—21no—
71C3×#108—21no—
72C3×#108—42no—
73C3×#108D412yescomplemented
74C3×#97—21no—
75C3×C4×C8—21no—
76C2×AGL(1,ℤ/8)—21no—
77C2×#135C22×C311yescharacteristic
78C22×D8C22×C311yescharacteristic
79#535—21no—
80C2×M32C22×C311yescharacteristic
81C2×M32C22×C311yescharacteristic
82#496—22no—
83#495—21no—
84#493—21no—
85#430—21no—
86C2×#108C22×C311yescharacteristic
87C2×#108C22×C311yescharacteristic
88#405C22×C311yescharacteristic
89#362—21no—
90#361—22no—
91#361—21no—
92#361—22no—
93#359—22no—
94#344—21no—
95C24×C3—41no—
96C3×#41—41no—
97C3×#41—41no—
98C3×#41—41no—
99C3×#41—41no—
100C2×C3×Q8—21no—
101C2×C3×D4—22no—
102C2×C3×D4—21no—
103C2×C3×D4—44no—
104C2×C3×D4—41no—
105C2×C3×D4—41no—
106C2×C3×D4—41no—
107C2×C3×D4—41no—
108C2×C3×D4—41no—
109C2×C3×D4—41no—
110C22×C3×C4—41no—
111C22×C3×C4C2×D411yescharacteristic
112C3×Q16—21no—
113C3×Q16—21no—
114C3×SD16—44no—
115C3×SD16—41no—
116C3×SD16—41no—
117C3×D8—22no—
118C3×D8—21no—
119C3×D8—22no—
120C3×D8—21no—
121C3×D8—44no—
122C3×D8—41no—
123C3×D8—41no—
124C3×M16—21no—
125C3×M16—22no—
126C3×M16—41no—
127C3×M16—41no—
128C3×M16—41no—
129C3×M16D812yes—
130C2×C3×C8—21no—
131C2×C3×C8—41no—
132C2×C3×C8—41no—
133C2×C3×C8C2×D411yescharacteristic
134C2×C3×C8C2×D411yescharacteristic
135C2×C3×C8C2×D411yescharacteristic
136C2×C3×C8#4111yescharacteristic
137C2×C3×C8#4111yescharacteristic
138C2×C3×C8#4111yescharacteristic
139C3×C4⋊C4—21no—
140C3×C22⋊C4—44no—
141C3×C22⋊C4SD1612yes—
142C3×C42—21no—
143C3×C16—41no—
144C3×C16—41no—
145C2×#41—21no—
146C22×D4—21no—
147AGL(1,ℤ/8)—42no—
148AGL(1,ℤ/8)—42no—
149#135—21no—
150#135—21no—
151#135—21no—
152#135—21no—
153C2×Q16C3×D411yescharacteristic
154C2×SD16—22no—
155C2×SD16—21no—
156C2×D8—22no—
157C2×D8—21no—
158C2×D8—21no—
159C2×D8—21no—
160C2×D8—21no—
161C2×D8—21no—
162C2×D8C3×D412yes—
163C2×D8C3×D411yescharacteristic
164C2×M16—21no—
165C2×M16—21no—
166C2×M16C3×D411yescharacteristic
167C22×C8C23×C311yescharacteristic
168#127—22no—
169#124—22no—
170#121—42no—
171#120—42no—
172#117C3×D411yescharacteristic
173M32—21no—
174M32—21no—
175M32—21no—
176M32—21no—
177C2×C16—21no—
178C2×C16—21no—
179#108—21no—
180#108—21no—
181#108—21no—
182#108—42no—
183#108C3×D412yescomplemented
184#97—21no—
185C4×C8—21no—
186C23×C3—42no—
187C23×C3—41no—
188C23×C3—81no—
189C23×C3—81no—
190C23×C3—162no—
191C23×C3C2×SD1611yescharacteristic
192C3×Q8—41no—
193C3×Q8—41no—
194C3×D4—42no—
195C3×D4—41no—
196C3×D4—41no—
197C3×D4—42no—
198C3×D4—81no—
199C3×D4—81no—
200C3×D4—81no—
201C3×D4—81no—
202C3×D4—164no—
203C2×C3×C4—22no—
204C2×C3×C4—41no—
205C2×C3×C4—81no—
206C2×C3×C4—81no—
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12411yescharacteristic
210C2×C3×C4C2×D811yescharacteristic
211C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
212C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
213C3×C8—22no—
214C3×C8—42no—
215C3×C8—81no—
216C3×C8#12111yescharacteristic
217C3×C8#12111yescharacteristic
218C3×C8#12311yescharacteristic
219C3×C8#12311yescharacteristic
220C24—41no—
221#41—41no—
222#41—41no—
223#41—41no—
224#41—41no—
225C2×Q8—21no—
226C2×D4—22no—
227C2×D4—21no—
228C2×D4—44no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C22×C4—41no—
236C22×C4C2×C3×D411yescharacteristic
237Q16—21no—
238Q16—21no—
239SD16—44no—
240SD16—41no—
241SD16—41no—
242D8—22no—
243D8—21no—
244D8—22no—
245D8—21no—
246D8—44no—
247D8—41no—
248D8—41no—
249M16—21no—
250M16—22no—
251M16—41no—
252M16—41no—
253M16—41no—
254M16C3×D812yes—
255C2×C8—21no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8C2×C3×D411yescharacteristic
259C2×C8C2×C3×D411yescharacteristic
260C2×C8C2×C3×D411yescharacteristic
261C2×C8C3×#4111yescharacteristic
262C2×C8C3×#4111yescharacteristic
263C2×C8C3×#4111yescharacteristic
264C4⋊C4—21no—
265C22⋊C4—44no—
266C22⋊C4C3×SD1612yes—
267C42—21no—
268C16—41no—
269C16—41no—
270C22×C3—41no—
271C22×C3—82no—
272C22×C3—84no—
273C22×C3—81no—
274C22×C3—82no—
275C22×C3—162no—
276C22×C3—162no—
277C22×C3#45011yescharacteristic
278C22×C3#46011yescharacteristic
279C22×C3#48611yescharacteristic
280C3×C4—42no—
281C3×C4—81no—
282C3×C4#39411yescharacteristic
283C3×C4#44711yescharacteristic
284C3×C4#45911yescharacteristic
285C3×C4#49011yescharacteristic
286C23—42no—
287C23—41no—
288C23—81no—
289C23—81no—
290C23—162no—
291C23C2×C3×SD1611yescharacteristic
292Q8—41no—
293Q8—41no—
294D4—42no—
295D4—41no—
296D4—41no—
297D4—42no—
298D4—81no—
299D4—81no—
300D4—81no—
301D4—81no—
302D4—164no—
303C2×C4—22no—
304C2×C4—41no—
305C2×C4—81no—
306C2×C4—81no—
307C2×C4C3×#12011yescharacteristic
308C2×C4C3×#12111yescharacteristic
309C2×C4C3×#12411yescharacteristic
310C2×C4C2×C3×D811yescharacteristic
311C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
312C2×C4C3×AGL(1,ℤ/8)11yescharacteristic
313C8—22no—
314C8—42no—
315C8—81no—
316C8C3×#12111yescharacteristic
317C8C3×#12111yescharacteristic
318C8C3×#12311yescharacteristic
319C8C3×#12311yescharacteristic
320C2×C3—21no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—161no—
324C2×C3—162no—
325C2×C3#204111yescharacteristic
326C22—41no—
327C22—82no—
328C22—84no—
329C22—81no—
330C22—82no—
331C22—162no—
332C22—162no—
333C22C3×#45011yescharacteristic
334C22C3×#46011yescharacteristic
335C22C3×#48611yescharacteristic
336C4—42no—
337C4—81no—
338C4C3×#39411yescharacteristic
339C4C3×#44711yescharacteristic
340C4C3×#45911yescharacteristic
341C4C3×#49011yescharacteristic
342C3#1341711yescharacteristic, direct product factor, Sylow
343C2—21no—
344C2—21no—
345C2—21no—
346C2—162no—
347C2—161no—
348C2C3×#204111yescharacteristic
subgroupG/H
center, socleC2×C3#2041
commutator, Frattini, cosocleC22×C8C23×C3
prime radicalC3×#3155C2
LCSG ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3 ◁ C22×C3×C4 ◁ C3×#405 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#2148C2semidirect
3. C2×C3×#361C2semidirect
4. C2×C3×#361C2semidirect
5. C3×#2143C2semidirect
6. C3×#1953C2semidirect
7. C3×#1378C2semidirect
8. #13417C3direct (N × H)
29. #2148C2×C3semidirect
30. C2×#361C2×C3semidirect
31. C2×#361C2×C3semidirect
32. #2143C2×C3semidirect
33. #1953C2×C3semidirect
34. #1378C2×C3semidirect
73. C3×#108D4semidirect
183. #108C3×D4semidirect
342. C3#13417direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#31552
C3×#21482
C2×C3×#3612
C3×#21432
C3×#19532
C3×#13782
#134173