Finite Group Databasebeta

← C3×#25691 #10644298

Subgroups — C3×#25691 #10644298

Order 768.

920 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3509C211yesmaximal, characteristic
2C3×#3166C211yesmaximal, characteristic, complemented
3C3×#3080C211yesmaximal, characteristic
4C3×#2642C211yesmaximal, characteristic, complemented
5C3×#2038C214yesmaximal
6C3×#1995C211yesmaximal, characteristic, complemented
7C3×#1966C211yesmaximal, characteristic, complemented
8C3×#1857C214yesmaximal, complemented
9C3×#1759C211yesmaximal, characteristic
10#25691C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#557—22no—
12C3×#556—22no—
13C3×#552—24no—
14C3×#541—22no—
15C3×#536C2212yes—
16C3×#533—22no—
17C3×#533C2211yescharacteristic
18C2×C3×C4⋊Q8C2211yescharacteristic
19C2×C3×#125C2211yescharacteristic
20C3×#519C2212yes—
21C3×C4×#41C2211yescharacteristic
22C3×#501—22no—
23C3×#499—22no—
24C3×#479—22no—
25C3×#478—22no—
26C3×#428C2214yes—
27C3×#428C2212yes—
28C3×#423C2211yescharacteristic
29C2×C3×#104C2212yes—
30C3×#417C2212yes—
31C2×C3×#101C2214yes—
32C3×C4×M16C2211yescharacteristic
33C3×#400—24no—
34C3×#391C2214yes—
35C3×#376C2211yescharacteristic
36C3×#339C2214yes—
37C3×#339C2212yes—
38#3509C2×C311yescharacteristic
39#3166C2×C311yescharacteristic, complemented
40#3080C2×C311yescharacteristic
41#2642C2×C311yescharacteristic, complemented
42#2038C2×C314yes—
43#1995C2×C311yescharacteristic, complemented
44#1966C2×C311yescharacteristic, complemented
45#1857C2×C314yescomplemented
46#1759C2×C311yescharacteristic
47C2×C3×#41C2311yescharacteristic
48C22×C3×Q8C2312yes—
49C2×C3×M16C2312yes—
50C2×C3×M16C2312yes—
51C3×C4⋊Q8—22no—
52C3×C4⋊Q8—21no—
53C3×C4⋊Q8—21no—
54C3×C4⋊Q8—22no—
55C3×C4⋊Q8D412yes—
56C3×#126—28no—
57C3×#125—24no—
58C3×#125—28no—
59C3×#125—22no—
60C3×#125—21no—
61C3×#125—21no—
62C3×#125D412yes—
63C3×#124—22no—
64C3×#123—24no—
65C3×C22⋊Q8—24no—
66C3×C22⋊Q8—22no—
67C3×C22⋊Q8—24no—
68C3×C4×Q8—24no—
69C3×C4×Q8—24no—
70C3×C4×Q8—24no—
71C3×C4×Q8—22no—
72C3×C4×Q8D412yes—
73C3×C4×D4—22no—
74C3×C4×D4D412yes—
75C3×#117—24no—
76C3×#117—24no—
77C3×#117—21no—
78C3×#117C2311yescharacteristic
79C3×#117C2312yes—
80C2×C3×C4⋊C4—22no—
81C2×C3×C4⋊C4D414yes—
82C2×C3×C4⋊C4C2312yes—
83C2×C3×C42—21no—
84C2×C3×C42C2312yes—
85C2×C3×C42C2311yescharacteristic
86C3×#107—22no—
87C3×#107—24no—
88C3×#106—22no—
89C3×#106—24no—
90C3×C4⋊C8—22no—
91C3×#104—24no—
92C3×#103—22no—
93C3×#102—22no—
94C3×#101—28no—
95C3×#97—21no—
96C3×C4×C8—21no—
97C3×#95—22no—
98C3×#95—28no—
99#557—22no—
100#556—22no—
101#552—24no—
102#541—22no—
103#536C22×C312yes—
104#533—22no—
105#533C22×C311yescharacteristic
106C2×C4⋊Q8C22×C311yescharacteristic
107C2×#125C22×C311yescharacteristic
108#519C22×C312yes—
109C4×#41C22×C311yescharacteristic
110#501—22no—
111#499—22no—
112#479—22no—
113#478—22no—
114#428C22×C312yes—
115#428C22×C314yes—
116#423C22×C311yescharacteristic
117C2×#104C22×C312yes—
118#417C22×C312yes—
119C2×#101C22×C314yes—
120C4×M16C22×C311yescharacteristic
121#400—24no—
122#391C22×C314yes—
123#376C22×C311yescharacteristic
124#339C22×C312yes—
125#339C22×C314yes—
126C3×#41—22no—
127C3×#41—22no—
128C2×C3×Q8—22no—
129C2×C3×Q8—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—24no—
132C2×C3×Q8—24no—
133C2×C3×Q8C2×D411yescharacteristic
134C2×C3×D4—21no—
135C2×C3×D4C2×D411yescharacteristic
136C22×C3×C4—22no—
137C22×C3×C4—21no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4C2×D414yes—
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2411yescharacteristic
143C3×M16—22no—
144C3×M16—24no—
145C3×M16#4114yes—
146C2×C3×C8—22no—
147C2×C3×C8—22no—
148C3×C4⋊C4—28no—
149C3×C4⋊C4—28no—
150C3×C4⋊C4—24no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—24no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—42no—
158C3×C4⋊C4—42no—
159C3×C4⋊C4C2×D412yes—
160C3×C4⋊C4C2×D414yes—
161C3×C22⋊C4—28no—
162C3×C22⋊C4—21no—
163C3×C22⋊C4—22no—
164C3×C22⋊C4—22no—
165C3×C42—28no—
166C3×C42—22no—
167C3×C42—21no—
168C3×C42—22no—
169C3×C42—22no—
170C3×C42—22no—
171C3×C42C2×D411yescharacteristic
172C3×C42C2×D411yescharacteristic
173C3×C42C2×D412yes—
174C2×#41C23×C311yescharacteristic
175C22×Q8C23×C312yes—
176C2×M16C23×C312yes—
177C2×M16C23×C312yes—
178C4⋊Q8—22no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
181C4⋊Q8—22no—
182C4⋊Q8C3×D412yes—
183#126—28no—
184#125—24no—
185#125—28no—
186#125—22no—
187#125—21no—
188#125—21no—
189#125C3×D412yes—
190#124—22no—
191#123—24no—
192C22⋊Q8—24no—
193C22⋊Q8—22no—
194C22⋊Q8—24no—
195C4×Q8—24no—
196C4×Q8—24no—
197C4×Q8—24no—
198C4×Q8—22no—
199C4×Q8C3×D412yes—
200C4×D4—22no—
201C4×D4C3×D412yes—
202#117—24no—
203#117—24no—
204#117—21no—
205#117C23×C311yescharacteristic
206#117C23×C312yes—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C3×D414yes—
209C2×C4⋊C4C23×C312yes—
210C2×C42—21no—
211C2×C42C23×C312yes—
212C2×C42C23×C311yescharacteristic
213#107—22no—
214#107—24no—
215#106—22no—
216#106—24no—
217C4⋊C8—22no—
218#104—24no—
219#103—22no—
220#102—22no—
221#101—28no—
222#97—21no—
223C4×C8—21no—
224#95—22no—
225#95—28no—
226C23×C3—21no—
227C23×C3C22×D411yescharacteristic
228C3×Q8—21no—
229C3×Q8—21no—
230C3×Q8—44no—
231C3×Q8—42no—
232C3×Q8—42no—
233C3×Q8—44no—
234C3×D4—21no—
235C3×D4—21no—
236C3×D4—42no—
237C2×C3×C4—22no—
238C2×C3×C4—24no—
239C2×C3×C4—24no—
240C2×C3×C4—21no—
241C2×C3×C4—28no—
242C2×C3×C4—42no—
243C2×C3×C4—44no—
244C2×C3×C4—42no—
245C2×C3×C4—44no—
246C2×C3×C4—44no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4#12012yes—
250C2×C3×C4#12012yes—
251C2×C3×C4C22×D411yescharacteristic
252C2×C3×C4C22×D411yescharacteristic
253C2×C3×C4C2×#4112yes—
254C2×C3×C42−1+412yes—
255C3×C8—24no—
256C3×C8—42no—
257#41—22no—
258#41—22no—
259C2×Q8—22no—
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—24no—
263C2×Q8—24no—
264C2×Q8C2×C3×D411yescharacteristic
265C2×D4—21no—
266C2×D4C2×C3×D411yescharacteristic
267C22×C4—22no—
268C22×C4—21no—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4C2×C3×D414yes—
272C22×C4C2×C3×D412yes—
273C22×C4C24×C311yescharacteristic
274M16—22no—
275M16—24no—
276M16C3×#4114yes—
277C2×C8—22no—
278C2×C8—22no—
279C4⋊C4—28no—
280C4⋊C4—28no—
281C4⋊C4—24no—
282C4⋊C4—24no—
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—42no—
289C4⋊C4—42no—
290C4⋊C4C2×C3×D412yes—
291C4⋊C4C2×C3×D414yes—
292C22⋊C4—28no—
293C22⋊C4—21no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C42—28no—
297C42—22no—
298C42—21no—
299C42—22no—
300C42—22no—
301C42—22no—
302C42C2×C3×D411yescharacteristic
303C42C2×C3×D411yescharacteristic
304C42C2×C3×D412yes—
305C22×C3—22no—
306C22×C3—41no—
307C22×C3—42no—
308C22×C3C2×#12011yescharacteristic
309C22×C3#54712yes—
310C3×C4—22no—
311C3×C4—22no—
312C3×C4—44no—
313C3×C4—41no—
314C3×C4—42no—
315C3×C4—44no—
316C3×C4—82no—
317C3×C4—82no—
318C3×C4#53211yescharacteristic
319C3×C4#53611yescharacteristic
320C3×C4#54812yes—
321C23—21no—
322C23C22×C3×D411yescharacteristic
323Q8—21no—
324Q8—21no—
325Q8—44no—
326Q8—42no—
327Q8—42no—
328Q8—44no—
329D4—21no—
330D4—21no—
331D4—42no—
332C2×C4—22no—
333C2×C4—24no—
334C2×C4—24no—
335C2×C4—21no—
336C2×C4—28no—
337C2×C4—42no—
338C2×C4—44no—
339C2×C4—42no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4C3×#12012yes—
345C2×C4C3×#12012yes—
346C2×C4C22×C3×D411yescharacteristic
347C2×C4C22×C3×D411yescharacteristic
348C2×C4C2×C3×#4112yes—
349C2×C4C3×2−1+412yes—
350C8—24no—
351C8—42no—
352C2×C3—22no—
353C2×C3—81no—
354C2×C3#243211yescharacteristic
355C2×C3#302012yes—
356C22—22no—
357C22—41no—
358C22—42no—
359C22C2×C3×#12011yescharacteristic
360C22C3×#54712yes—
361C4—22no—
362C4—22no—
363C4—44no—
364C4—41no—
365C4—42no—
366C4—44no—
367C4—82no—
368C4—82no—
369C4C3×#53211yescharacteristic
370C4C3×#53611yescharacteristic
371C4C3×#54812yes—
372C3#2569111yescharacteristic, direct product factor, Sylow
373C2—22no—
374C2—81no—
375C2C3×#243211yescharacteristic
376C2C3×#302012yes—
subgroupG/H
center, socleC22×C3C2×#120
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#41C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C3 ◁ C2×C3×C42 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×#3166C2semidirect
4. C3×#2642C2semidirect
6. C3×#1995C2semidirect
7. C3×#1966C2semidirect
8. C3×#1857C2semidirect
10. #25691C3direct (N × H)
39. #3166C2×C3semidirect
41. #2642C2×C3semidirect
43. #1995C2×C3semidirect
44. #1966C2×C3semidirect
45. #1857C2×C3semidirect
372. C3#25691direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#35092
C3×#31662
C3×#30802
C3×#26422
C3×#20382
C3×#19952
C3×#19662
C3×#18572
C3×#17592
#256913