Finite Group Databasebeta

← C3×#29764 #10648371

Subgroups — C3×#29764 #10648371

Order 768.

766 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#484C211yesmaximal, characteristic
2C2×C3×#483C211yesmaximal, characteristic
3C3×#2621C211yesmaximal, characteristic
4C3×#2316C211yesmaximal, characteristic
5C3×#2085C212yesmaximal, complemented
6C3×#2068C212yesmaximal, complemented
7C3×#2066C212yesmaximal, complemented
8C3×#2043C212yesmaximal, complemented
9C3×#1873C211yesmaximal, characteristic
10C3×#1873C211yesmaximal, characteristic
11C3×#1798C211yesmaximal, characteristic
12#29764C311yesmaximal, characteristic, direct product factor, Sylow
13C2×C3×C4⋊Q8C2211yescharacteristic
14C2×C3×C22⋊Q8C2211yescharacteristic
15C2×C3×C22⋊Q8C2211yescharacteristic
16C2×C3×#117C2211yescharacteristic
17C2×C3×#117C2211yescharacteristic
18C22×C3×C4⋊C4C2211yescharacteristic
19C3×#484—24no—
20C3×#483—24no—
21C2×C3×#107C2212yes—
22C2×C3×#106C2212yes—
23C2×C3×C4⋊C8C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×C22⋊C8C2211yescharacteristic
29C2×C3×C22⋊C8C2211yescharacteristic
30C3×#398—22no—
31C3×#397—22no—
32C3×#391C2211yescharacteristic
33C3×#391C2211yescharacteristic
34C3×#389—22no—
35C3×#389C2212yes—
36C3×#389C2212yes—
37C3×#382C2211yescharacteristic
38C3×#380—22no—
39C3×#340C2212yes—
40C3×#340C2212yes—
41C3×#336C2212yes—
42C2×#484C2×C311yescharacteristic
43C2×#483C2×C311yescharacteristic
44#2621C2×C311yescharacteristic
45#2316C2×C311yescharacteristic
46#2085C2×C312yescomplemented
47#2068C2×C312yescomplemented
48#2066C2×C312yescomplemented
49#2043C2×C312yescomplemented
50#1873C2×C311yescharacteristic
51#1873C2×C311yescharacteristic
52#1798C2×C311yescharacteristic
53C22×C3×Q8C2311yescharacteristic
54C22×C3×Q8C2311yescharacteristic
55C23×C3×C4—21no—
56C23×C3×C4C2311yescharacteristic
57C22×C3×C8C2312yes—
58C22×C3×C8C2312yes—
59C3×C4⋊Q8—42no—
60C3×C22⋊Q8—22no—
61C3×C22⋊Q8—22no—
62C3×C22⋊Q8—22no—
63C3×C22⋊Q8—22no—
64C3×#117—42no—
65C3×#117—42no—
66C2×C3×C4⋊C4—21no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—22no—
72C2×C3×C4⋊C4—22no—
73C2×C3×C4⋊C4D414yes—
74C2×C3×C4⋊C4D412yes—
75C2×C3×C4⋊C4C2311yescharacteristic
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2312yes—
78C2×C3×C4⋊C4C2312yes—
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—21no—
81C2×C3×C22⋊C4—21no—
82C2×C3×C22⋊C4D412yes—
83C2×C3×C42—21no—
84C2×C3×C42C2311yescharacteristic
85C2×C3×C42C2311yescharacteristic
86C3×#107—24no—
87C3×#106—24no—
88C3×C4⋊C8—42no—
89C3×#103—24no—
90C3×#103—24no—
91C3×#103—42no—
92C3×#103—42no—
93C3×C22⋊C8—22no—
94C3×C22⋊C8—22no—
95C3×#95—22no—
96C3×#95—22no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—22no—
100C2×C4⋊Q8C22×C311yescharacteristic
101C2×C22⋊Q8C22×C311yescharacteristic
102C2×C22⋊Q8C22×C311yescharacteristic
103C2×#117C22×C311yescharacteristic
104C2×#117C22×C311yescharacteristic
105C22×C4⋊C4C22×C311yescharacteristic
106#484—24no—
107#483—24no—
108C2×#107C22×C312yes—
109C2×#106C22×C312yes—
110C2×C4⋊C8C22×C312yes—
111C2×#103C22×C312yes—
112C2×#103C22×C312yes—
113C2×#103C22×C312yes—
114C2×#103C22×C312yes—
115C2×C22⋊C8C22×C311yescharacteristic
116C2×C22⋊C8C22×C311yescharacteristic
117#398—22no—
118#397—22no—
119#391C22×C311yescharacteristic
120#391C22×C311yescharacteristic
121#389—22no—
122#389C22×C312yes—
123#389C22×C312yes—
124#382C22×C311yescharacteristic
125#380—22no—
126#340C22×C312yes—
127#340C22×C312yes—
128#336C22×C312yes—
129C24×C3C2×D411yescharacteristic
130C2×C3×Q8—21no—
131C2×C3×Q8—21no—
132C2×C3×Q8—21no—
133C2×C3×Q8—21no—
134C2×C3×Q8—22no—
135C2×C3×Q8—22no—
136C2×C3×Q8—22no—
137C2×C3×Q8—22no—
138C22×C3×C4—22no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—24no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4—22no—
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4—82no—
148C22×C3×C4C2×D412yes—
149C22×C3×C4C2×D412yes—
150C22×C3×C4C2×D411yescharacteristic
151C22×C3×C4C2×D412yes—
152C22×C3×C4#4112yes—
153C22×C3×C4C2411yescharacteristic
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C2×C3×C8—22no—
160C3×C4⋊C4—24no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—21no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—22no—
167C3×C4⋊C4—22no—
168C3×C4⋊C4—42no—
169C3×C4⋊C4—44no—
170C3×C4⋊C4—42no—
171C3×C4⋊C4—42no—
172C3×C4⋊C4—82no—
173C3×C4⋊C4—82no—
174C3×C4⋊C4C2×D414yes—
175C3×C22⋊C4—42no—
176C3×C22⋊C4—42no—
177C3×C22⋊C4—42no—
178C3×C22⋊C4—42no—
179C3×C42—22no—
180C3×C42—22no—
181C3×C42—42no—
182C22×Q8C23×C311yescharacteristic
183C22×Q8C23×C311yescharacteristic
184C23×C4—21no—
185C23×C4C23×C311yescharacteristic
186C22×C8C23×C312yes—
187C22×C8C23×C312yes—
188C4⋊Q8—42no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C22⋊Q8—22no—
192C22⋊Q8—22no—
193#117—42no—
194#117—42no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—22no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4C3×D414yes—
203C2×C4⋊C4C3×D412yes—
204C2×C4⋊C4C23×C312yes—
205C2×C4⋊C4C23×C311yescharacteristic
206C2×C4⋊C4C23×C311yescharacteristic
207C2×C4⋊C4C23×C312yes—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—21no—
210C2×C22⋊C4—21no—
211C2×C22⋊C4C3×D412yes—
212C2×C42—21no—
213C2×C42C23×C311yescharacteristic
214C2×C42C23×C311yescharacteristic
215#107—24no—
216#106—24no—
217C4⋊C8—42no—
218#103—24no—
219#103—24no—
220#103—42no—
221#103—42no—
222C22⋊C8—22no—
223C22⋊C8—22no—
224#95—22no—
225#95—22no—
226#95—22no—
227#95—22no—
228#95—22no—
229C23×C3—21no—
230C23×C3—21no—
231C23×C3—22no—
232C23×C3—22no—
233C23×C3#12112yes—
234C23×C3C22×D411yescharacteristic
235C3×Q8—42no—
236C3×Q8—42no—
237C3×Q8—42no—
238C3×Q8—42no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—24no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—22no—
250C2×C3×C4—42no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—41no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—84no—
266C2×C3×C4#12112yes—
267C2×C3×C4C22×D411yescharacteristic
268C2×C3×C4C2×#4111yescharacteristic
269C2×C3×C4H(5,𝔽2)12yes—
270C2×C3×C42−1+412yes—
271C3×C8—42no—
272C3×C8—42no—
273C24C2×C3×D411yescharacteristic
274C2×Q8—21no—
275C2×Q8—21no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C22×C4—22no—
283C22×C4—22no—
284C22×C4—22no—
285C22×C4—24no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—22no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—82no—
292C22×C4C2×C3×D412yes—
293C22×C4C2×C3×D412yes—
294C22×C4C2×C3×D411yescharacteristic
295C22×C4C2×C3×D412yes—
296C22×C4C3×#4112yes—
297C22×C4C24×C311yescharacteristic
298C2×C8—22no—
299C2×C8—22no—
300C2×C8—22no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C4⋊C4—24no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—21no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—42no—
313C4⋊C4—44no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—82no—
317C4⋊C4—82no—
318C4⋊C4C2×C3×D414yes—
319C22⋊C4—42no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—42no—
323C42—22no—
324C42—22no—
325C42—42no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—42no—
330C22×C3—42no—
331C22×C3C2×#12111yescharacteristic
332C22×C3#53411yescharacteristic
333C22×C3#53611yescharacteristic
334C22×C3#57712yes—
335C22×C3#57812yes—
336C3×C4—22no—
337C3×C4—42no—
338C3×C4—42no—
339C3×C4—44no—
340C3×C4—81no—
341C3×C4—81no—
342C3×C4—82no—
343C3×C4—81no—
344C3×C4—81no—
345C3×C4#53712yes—
346C3×C4#53912yes—
347C23—22no—
348C23—22no—
349C23—21no—
350C23—21no—
351C23C3×#12112yes—
352C23C22×C3×D411yescharacteristic
353Q8—42no—
354Q8—42no—
355Q8—42no—
356Q8—42no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—42no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—84no—
384C2×C4C3×#12112yes—
385C2×C4C22×C3×D411yescharacteristic
386C2×C4C2×C3×#4111yescharacteristic
387C2×C4C3×H(5,𝔽2)12yes—
388C2×C4C3×2−1+412yes—
389C8—42no—
390C8—42no—
391C2×C3—42no—
392C2×C3#261811yescharacteristic
393C2×C3#304211yescharacteristic
394C2×C3#304411yescharacteristic
395C2×C3#320112yes—
396C2×C3#324212yes—
397C22—22no—
398C22—22no—
399C22—22no—
400C22—42no—
401C22—42no—
402C22C2×C3×#12111yescharacteristic
403C22C3×#53411yescharacteristic
404C22C3×#53611yescharacteristic
405C22C3×#57712yes—
406C22C3×#57812yes—
407C4—22no—
408C4—42no—
409C4—42no—
410C4—44no—
411C4—81no—
412C4—81no—
413C4—82no—
414C4—81no—
415C4—81no—
416C4C3×#53712yes—
417C4C3×#53912yes—
418C3#2976411yescharacteristic, direct product factor, Sylow
419C2—42no—
420C2C3×#261811yescharacteristic
421C2C3×#304211yescharacteristic
422C2C3×#304411yescharacteristic
423C2C3×#320112yes—
424C2C3×#324212yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC24×C3C2×D4
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
5. C3×#2085C2semidirect
6. C3×#2068C2semidirect
7. C3×#2066C2semidirect
8. C3×#2043C2semidirect
12. #29764C3direct (N × H)
46. #2085C2×C3semidirect
47. #2068C2×C3semidirect
48. #2066C2×C3semidirect
49. #2043C2×C3semidirect
418. C3#29764direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4842
C2×C3×#4832
C3×#26212
C3×#23162
C3×#20852
C3×#20682
C3×#20662
C3×#20432
C3×#18732
C3×#17982
#297643