Finite Group Databasebeta

← C3×#29700 #10648307

Subgroups — C3×#29700 #10648307

Order 768.

1068 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#540C211yesmaximal, characteristic
2C2×C3×#478C212yesmaximal, complemented
3C3×#2629C211yesmaximal, characteristic, complemented
4C3×#2059C212yesmaximal, complemented
5C3×#2041C212yesmaximal, complemented
6C3×#2034C212yesmaximal, complemented
7C3×#2033C212yesmaximal, complemented
8C3×#1802C212yesmaximal, complemented
9C3×#1798C211yesmaximal, characteristic, complemented
10#29700C311yesmaximal, characteristic, direct product factor, Sylow
11C3×#540—28no—
12C2×C3×#127C2211yescharacteristic
13C2×C3×#125C2211yescharacteristic
14C2×C3×#125C2211yescharacteristic
15C2×C3×#123—22no—
16C2×C3×#121—22no—
17C2×C3×#121C2212yes—
18C2×C3×C4×D4C2212yes—
19C2×C3×#117C2211yescharacteristic
20C3×#478—28no—
21C2×C3×#107C2212yes—
22C2×C3×#106C2212yes—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#102C2212yes—
25C2×C3×#102C2212yes—
26C2×C3×#102C2212yes—
27C2×C3×#102C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#398—22no—
31C3×#396—22no—
32C3×#389C2212yes—
33C3×#384C2211yescharacteristic
34C3×#380C2212yes—
35C3×#340C2212yes—
36C3×#340C2212yes—
37C3×#336C2212yes—
38C2×#540C2×C311yescharacteristic
39C2×#478C2×C312yescomplemented
40#2629C2×C311yescharacteristic, complemented
41#2059C2×C312yescomplemented
42#2041C2×C312yescomplemented
43#2034C2×C312yescomplemented
44#2033C2×C312yescomplemented
45#1802C2×C312yescomplemented
46#1798C2×C311yescharacteristic, complemented
47C22×C3×D4—22no—
48C22×C3×D4—22no—
49C22×C3×D4C2312yes—
50C23×C3×C4—22no—
51C23×C3×C4C2311yescharacteristic
52C22×C3×C8C2312yes—
53C22×C3×C8C2312yes—
54C3×#127—24no—
55C3×#125—24no—
56C3×#125—24no—
57C3×#123—216no—
58C3×#121—216no—
59C3×#121—24no—
60C3×#121—24no—
61C3×C4×D4—24no—
62C3×C4×D4D418yes—
63C3×#117—24no—
64C2×C3×C4⋊C4—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4D412yes—
69C2×C3×C4⋊C4C2312yes—
70C2×C3×C4⋊C4C2312yes—
71C2×C3×C4⋊C4C2312yes—
72C2×C3×C22⋊C4—22no—
73C2×C3×C22⋊C4—22no—
74C2×C3×C22⋊C4—22no—
75C2×C3×C22⋊C4D412yes—
76C2×C3×C42C2311yescharacteristic
77C2×C3×C42C2311yescharacteristic
78C3×#107—24no—
79C3×#106—24no—
80C3×C4⋊C8—24no—
81C3×#102—24no—
82C3×#102—24no—
83C3×#102—42no—
84C3×#102—42no—
85C3×C22⋊C8—41no—
86C3×C22⋊C8—41no—
87C3×#95—22no—
88C3×#95—22no—
89#540—28no—
90C2×#127C22×C311yescharacteristic
91C2×#125C22×C311yescharacteristic
92C2×#125C22×C311yescharacteristic
93C2×#123—22no—
94C2×#121—22no—
95C2×#121C22×C312yes—
96C2×C4×D4C22×C312yes—
97C2×#117C22×C311yescharacteristic
98#478—28no—
99C2×#107C22×C312yes—
100C2×#106C22×C312yes—
101C2×C4⋊C8C22×C312yes—
102C2×#102C22×C312yes—
103C2×#102C22×C312yes—
104C2×#102C22×C312yes—
105C2×#102C22×C312yes—
106C2×C22⋊C8C22×C311yescharacteristic
107C2×C22⋊C8C22×C311yescharacteristic
108#398—22no—
109#396—22no—
110#389C22×C312yes—
111#384C22×C311yescharacteristic
112#380C22×C312yes—
113#340C22×C312yes—
114#340C22×C312yes—
115#336C22×C312yes—
116C24×C3—22no—
117C24×C3C2×D411yescharacteristic
118C2×C3×D4—28no—
119C2×C3×D4—28no—
120C2×C3×D4—22no—
121C2×C3×D4—22no—
122C2×C3×D4—22no—
123C2×C3×D4—22no—
124C2×C3×D4—22no—
125C2×C3×D4—82no—
126C2×C3×D4—82no—
127C2×C3×D4—82no—
128C2×C3×D4—82no—
129C2×C3×D4C2×D414yes—
130C22×C3×C4—22no—
131C22×C3×C4—22no—
132C22×C3×C4—21no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—28no—
136C22×C3×C4—22no—
137C22×C3×C4—22no—
138C22×C3×C4—82no—
139C22×C3×C4—82no—
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D412yes—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2411yescharacteristic
144C2×C3×C8—22no—
145C2×C3×C8—22no—
146C2×C3×C8—22no—
147C2×C3×C8—22no—
148C2×C3×C8—22no—
149C2×C3×C8—22no—
150C3×C4⋊C4—28no—
151C3×C4⋊C4—24no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—22no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4—44no—
158C3×C4⋊C4—82no—
159C3×C4⋊C4—82no—
160C3×C4⋊C4C2×D414yes—
161C3×C22⋊C4—28no—
162C3×C22⋊C4—28no—
163C3×C22⋊C4—24no—
164C3×C22⋊C4—82no—
165C3×C42—22no—
166C3×C42C2×D414yes—
167C22×D4—22no—
168C22×D4—22no—
169C22×D4C23×C312yes—
170C23×C4—22no—
171C23×C4C23×C311yescharacteristic
172C22×C8C23×C312yes—
173C22×C8C23×C312yes—
174#127—24no—
175#125—24no—
176#125—24no—
177#123—216no—
178#121—216no—
179#121—24no—
180#121—24no—
181C4×D4—24no—
182C4×D4C3×D418yes—
183#117—24no—
184C2×C4⋊C4—22no—
185C2×C4⋊C4—22no—
186C2×C4⋊C4—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4C3×D412yes—
189C2×C4⋊C4C23×C312yes—
190C2×C4⋊C4C23×C312yes—
191C2×C4⋊C4C23×C312yes—
192C2×C22⋊C4—22no—
193C2×C22⋊C4—22no—
194C2×C22⋊C4—22no—
195C2×C22⋊C4C3×D412yes—
196C2×C42C23×C311yescharacteristic
197C2×C42C23×C311yescharacteristic
198#107—24no—
199#106—24no—
200C4⋊C8—24no—
201#102—24no—
202#102—24no—
203#102—42no—
204#102—42no—
205C22⋊C8—41no—
206C22⋊C8—41no—
207#95—22no—
208#95—22no—
209C23×C3—21no—
210C23×C3—24no—
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—22no—
214C23×C3—22no—
215C23×C3—42no—
216C23×C3—42no—
217C23×C3—42no—
218C23×C3—42no—
219C23×C3—42no—
220C23×C3—42no—
221C23×C3C22×D411yescharacteristic
222C3×D4—24no—
223C3×D4—24no—
224C3×D4—42no—
225C3×D4—42no—
226C3×D4—88no—
227C3×D4—88no—
228C2×C3×C4—22no—
229C2×C3×C4—22no—
230C2×C3×C4—22no—
231C2×C3×C4—21no—
232C2×C3×C4—24no—
233C2×C3×C4—21no—
234C2×C3×C4—21no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—42no—
238C2×C3×C4—42no—
239C2×C3×C4—42no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—88no—
248C2×C3×C4#12014yes—
249C2×C3×C4C22×D412yes—
250C2×C3×C4H(5,𝔽2)11yescharacteristic
251C2×C3×C42−1+411yescharacteristic
252C2×C3×C42−1+412yes—
253C3×C8—42no—
254C3×C8—42no—
255C24—22no—
256C24C2×C3×D411yescharacteristic
257C2×D4—28no—
258C2×D4—28no—
259C2×D4—22no—
260C2×D4—22no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—82no—
265C2×D4—82no—
266C2×D4—82no—
267C2×D4—82no—
268C2×D4C2×C3×D414yes—
269C22×C4—22no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—21no—
274C22×C4—28no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—82no—
278C22×C4—82no—
279C22×C4C2×C3×D411yescharacteristic
280C22×C4C2×C3×D412yes—
281C22×C4C2×C3×D412yes—
282C22×C4C24×C311yescharacteristic
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—22no—
287C2×C8—22no—
288C2×C8—22no—
289C4⋊C4—28no—
290C4⋊C4—24no—
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—44no—
297C4⋊C4—82no—
298C4⋊C4—82no—
299C4⋊C4C2×C3×D414yes—
300C22⋊C4—28no—
301C22⋊C4—28no—
302C22⋊C4—24no—
303C22⋊C4—82no—
304C42—22no—
305C42C2×C3×D414yes—
306C22×C3—41no—
307C22×C3—44no—
308C22×C3—42no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—44no—
312C22×C3—42no—
313C22×C3—44no—
314C22×C3—82no—
315C22×C3—82no—
316C22×C3—82no—
317C22×C3—82no—
318C22×C3C2×#12011yescharacteristic
319C22×C3#53611yescharacteristic
320C22×C3#53711yescharacteristic
321C22×C3#57612yes—
322C22×C3#57712yes—
323C3×C4—24no—
324C3×C4—41no—
325C3×C4—42no—
326C3×C4—44no—
327C3×C4—82no—
328C3×C4—82no—
329C3×C4#53612yes—
330C3×C4#53712yes—
331C23—22no—
332C23—22no—
333C23—21no—
334C23—24no—
335C23—21no—
336C23—21no—
337C23—42no—
338C23—42no—
339C23—42no—
340C23—42no—
341C23—42no—
342C23—42no—
343C23C22×C3×D411yescharacteristic
344D4—24no—
345D4—24no—
346D4—42no—
347D4—42no—
348D4—88no—
349D4—88no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—21no—
356C2×C4—24no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—42no—
360C2×C4—42no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—42no—
369C2×C4—88no—
370C2×C4C3×#12014yes—
371C2×C4C22×C3×D412yes—
372C2×C4C3×H(5,𝔽2)11yescharacteristic
373C2×C4C3×2−1+411yescharacteristic
374C2×C4C3×2−1+412yes—
375C8—42no—
376C8—42no—
377C2×C3—81no—
378C2×C3—84no—
379C2×C3#261611yescharacteristic
380C2×C3#301212yes—
381C2×C3#324112yes—
382C2×C3#325911yescharacteristic
383C2×C3#326011yescharacteristic
384C22—41no—
385C22—44no—
386C22—42no—
387C22—41no—
388C22—41no—
389C22—44no—
390C22—42no—
391C22—44no—
392C22—82no—
393C22—82no—
394C22—82no—
395C22—82no—
396C22C2×C3×#12011yescharacteristic
397C22C3×#53611yescharacteristic
398C22C3×#53711yescharacteristic
399C22C3×#57612yes—
400C22C3×#57712yes—
401C4—24no—
402C4—41no—
403C4—42no—
404C4—44no—
405C4—82no—
406C4—82no—
407C4C3×#53612yes—
408C4C3×#53712yes—
409C3#2970011yescharacteristic, direct product factor, Sylow
410C2—81no—
411C2—84no—
412C2C3×#261611yescharacteristic
413C2C3×#301212yes—
414C2C3×#324112yes—
415C2C3×#325911yescharacteristic
416C2C3×#326011yescharacteristic
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC2×C3×#540C2
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C2×C3×#117 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C3×#478C2semidirect
3. C3×#2629C2semidirect
4. C3×#2059C2semidirect
5. C3×#2041C2semidirect
6. C3×#2034C2semidirect
7. C3×#2033C2semidirect
8. C3×#1802C2semidirect
9. C3×#1798C2semidirect
10. #29700C3direct (N × H)
39. C2×#478C2×C3semidirect
40. #2629C2×C3semidirect
41. #2059C2×C3semidirect
42. #2041C2×C3semidirect
43. #2034C2×C3semidirect
44. #2033C2×C3semidirect
45. #1802C2×C3semidirect
46. #1798C2×C3semidirect
409. C3#29700direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#5402
C2×C3×#4782
C3×#26292
C3×#20592
C3×#20412
C3×#20342
C3×#20332
C3×#18022
C3×#17982
#297003