Finite Group Databasebeta

← C3×#29766 #10648373

Subgroups — C3×#29766 #10648373

Order 768.

766 conjugacy classes of subgroups.

The subgroup lattice has 436 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#485C212yesmaximal
2C3×#2621C211yesmaximal, characteristic
3C3×#2317C211yesmaximal, characteristic
4C3×#2085C212yesmaximal, complemented
5C3×#2068C212yesmaximal, complemented
6C3×#2066C212yesmaximal, complemented
7C3×#2043C212yesmaximal, complemented
8C3×#1872C212yesmaximal
9C3×#1798C211yesmaximal, characteristic
10#29766C311yesmaximal, characteristic, direct product factor, Sylow
11C2×C3×C4⋊Q8C2211yescharacteristic
12C2×C3×C22⋊Q8C2212yes—
13C2×C3×#117C2211yescharacteristic
14C2×C3×#117C2211yescharacteristic
15C22×C3×C4⋊C4C2211yescharacteristic
16C3×#485—28no—
17C2×C3×#107C2212yes—
18C2×C3×#106C2212yes—
19C2×C3×C4⋊C8C2211yescharacteristic
20C2×C3×C4⋊C8C2211yescharacteristic
21C2×C3×#103C2212yes—
22C2×C3×#103C2212yes—
23C2×C3×#103C2212yes—
24C2×C3×#103C2212yes—
25C2×C3×C22⋊C8C2212yes—
26C3×#398—21no—
27C3×#398—21no—
28C3×#397—21no—
29C3×#397—21no—
30C3×#391C2212yes—
31C3×#389C2212yes—
32C3×#389C2212yes—
33C3×#385—21no—
34C3×#385—21no—
35C3×#384—21no—
36C3×#382—21no—
37C3×#382C2211yescharacteristic
38C3×#340C2212yes—
39C3×#340C2212yes—
40C3×#336C2212yes—
41C2×#485C2×C312yes—
42#2621C2×C311yescharacteristic
43#2317C2×C311yescharacteristic
44#2085C2×C312yescomplemented
45#2068C2×C312yescomplemented
46#2066C2×C312yescomplemented
47#2043C2×C312yescomplemented
48#1872C2×C312yes—
49#1798C2×C311yescharacteristic
50C22×C3×Q8C2312yes—
51C23×C3×C4—21no—
52C23×C3×C4C2311yescharacteristic
53C22×C3×C8C2312yes—
54C22×C3×C8C2312yes—
55C3×C4⋊Q8—41no—
56C3×C4⋊Q8—41no—
57C3×C22⋊Q8—24no—
58C3×C22⋊Q8—24no—
59C3×#117—22no—
60C3×#117—41no—
61C3×#117—41no—
62C3×#117D414yes—
63C2×C3×C4⋊C4—21no—
64C2×C3×C4⋊C4—21no—
65C2×C3×C4⋊C4—21no—
66C2×C3×C4⋊C4—22no—
67C2×C3×C4⋊C4—22no—
68C2×C3×C4⋊C4—22no—
69C2×C3×C4⋊C4—41no—
70C2×C3×C4⋊C4—41no—
71C2×C3×C4⋊C4D411yescharacteristic
72C2×C3×C4⋊C4D411yescharacteristic
73C2×C3×C4⋊C4C2312yes—
74C2×C3×C4⋊C4C2312yes—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C22⋊C4—21no—
77C2×C3×C22⋊C4—22no—
78C2×C3×C22⋊C4D411yescharacteristic
79C2×C3×C22⋊C4D411yescharacteristic
80C2×C3×C42—21no—
81C2×C3×C42C2311yescharacteristic
82C2×C3×C42C2311yescharacteristic
83C3×#107—24no—
84C3×#106—24no—
85C3×C4⋊C8—41no—
86C3×C4⋊C8—41no—
87C3×#103—24no—
88C3×#103—24no—
89C3×#103—42no—
90C3×#103—42no—
91C3×C22⋊C8—24no—
92C3×#95—22no—
93C3×#95—22no—
94C3×#95—21no—
95C3×#95—21no—
96C3×#95—22no—
97C3×#95—22no—
98C2×C4⋊Q8C22×C311yescharacteristic
99C2×C22⋊Q8C22×C312yes—
100C2×#117C22×C311yescharacteristic
101C2×#117C22×C311yescharacteristic
102C22×C4⋊C4C22×C311yescharacteristic
103#485—28no—
104C2×#107C22×C312yes—
105C2×#106C22×C312yes—
106C2×C4⋊C8C22×C311yescharacteristic
107C2×C4⋊C8C22×C311yescharacteristic
108C2×#103C22×C312yes—
109C2×#103C22×C312yes—
110C2×#103C22×C312yes—
111C2×#103C22×C312yes—
112C2×C22⋊C8C22×C312yes—
113#398—21no—
114#398—21no—
115#397—21no—
116#397—21no—
117#391C22×C312yes—
118#389C22×C312yes—
119#389C22×C312yes—
120#385—21no—
121#385—21no—
122#384—21no—
123#382—21no—
124#382C22×C311yescharacteristic
125#340C22×C312yes—
126#340C22×C312yes—
127#336C22×C312yes—
128C24×C3C2×D411yescharacteristic
129C2×C3×Q8—22no—
130C2×C3×Q8—22no—
131C2×C3×Q8—22no—
132C2×C3×Q8—22no—
133C2×C3×Q8—22no—
134C2×C3×Q8—22no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—22no—
140C22×C3×C4—22no—
141C22×C3×C4—22no—
142C22×C3×C4—22no—
143C22×C3×C4—42no—
144C22×C3×C4—82no—
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4C2×D412yes—
151C22×C3×C4#4112yes—
152C22×C3×C4C2411yescharacteristic
153C2×C3×C8—22no—
154C2×C3×C8—22no—
155C2×C3×C8—22no—
156C2×C3×C8—22no—
157C2×C3×C8—22no—
158C2×C3×C8—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—22no—
161C3×C4⋊C4—22no—
162C3×C4⋊C4—22no—
163C3×C4⋊C4—22no—
164C3×C4⋊C4—41no—
165C3×C4⋊C4—41no—
166C3×C4⋊C4—42no—
167C3×C4⋊C4—42no—
168C3×C4⋊C4—42no—
169C3×C4⋊C4—44no—
170C3×C4⋊C4—82no—
171C3×C4⋊C4—82no—
172C3×C4⋊C4C2×D414yes—
173C3×C22⋊C4—24no—
174C3×C22⋊C4—41no—
175C3×C22⋊C4—41no—
176C3×C22⋊C4—44no—
177C3×C42—24no—
178C3×C42—21no—
179C3×C42—21no—
180C3×C42—21no—
181C3×C42—21no—
182C22×Q8C23×C312yes—
183C23×C4—21no—
184C23×C4C23×C311yescharacteristic
185C22×C8C23×C312yes—
186C22×C8C23×C312yes—
187C4⋊Q8—41no—
188C4⋊Q8—41no—
189C22⋊Q8—24no—
190C22⋊Q8—24no—
191#117—22no—
192#117—41no—
193#117—41no—
194#117C3×D414yes—
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—41no—
202C2×C4⋊C4—41no—
203C2×C4⋊C4C3×D411yescharacteristic
204C2×C4⋊C4C3×D411yescharacteristic
205C2×C4⋊C4C23×C312yes—
206C2×C4⋊C4C23×C312yes—
207C2×C4⋊C4C23×C312yes—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—22no—
210C2×C22⋊C4C3×D411yescharacteristic
211C2×C22⋊C4C3×D411yescharacteristic
212C2×C42—21no—
213C2×C42C23×C311yescharacteristic
214C2×C42C23×C311yescharacteristic
215#107—24no—
216#106—24no—
217C4⋊C8—41no—
218C4⋊C8—41no—
219#103—24no—
220#103—24no—
221#103—42no—
222#103—42no—
223C22⋊C8—24no—
224#95—22no—
225#95—22no—
226#95—21no—
227#95—21no—
228#95—22no—
229#95—22no—
230C23×C3—21no—
231C23×C3—21no—
232C23×C3—22no—
233C23×C3—22no—
234C23×C3#12112yes—
235C23×C3C22×D411yescharacteristic
236C3×Q8—42no—
237C3×Q8—42no—
238C3×Q8—42no—
239C3×Q8—42no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—22no—
243C2×C3×C4—22no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—24no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—42no—
256C2×C3×C4—42no—
257C2×C3×C4—42no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—84no—
267C2×C3×C4#12112yes—
268C2×C3×C4C22×D411yescharacteristic
269C2×C3×C4C2×#4111yescharacteristic
270C2×C3×C4H(5,𝔽2)11yescharacteristic
271C2×C3×C4H(5,𝔽2)11yescharacteristic
272C2×C3×C42−1+411yescharacteristic
273C2×C3×C42−1+411yescharacteristic
274C3×C8—42no—
275C3×C8—42no—
276C24C2×C3×D411yescharacteristic
277C2×Q8—22no—
278C2×Q8—22no—
279C2×Q8—22no—
280C2×Q8—22no—
281C2×Q8—22no—
282C2×Q8—22no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—22no—
288C22×C4—22no—
289C22×C4—22no—
290C22×C4—22no—
291C22×C4—42no—
292C22×C4—82no—
293C22×C4C2×C3×D411yescharacteristic
294C22×C4C2×C3×D411yescharacteristic
295C22×C4C2×C3×D411yescharacteristic
296C22×C4C2×C3×D411yescharacteristic
297C22×C4C2×C3×D411yescharacteristic
298C22×C4C2×C3×D412yes—
299C22×C4C3×#4112yes—
300C22×C4C24×C311yescharacteristic
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—22no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—41no—
313C4⋊C4—41no—
314C4⋊C4—42no—
315C4⋊C4—42no—
316C4⋊C4—42no—
317C4⋊C4—44no—
318C4⋊C4—82no—
319C4⋊C4—82no—
320C4⋊C4C2×C3×D414yes—
321C22⋊C4—24no—
322C22⋊C4—41no—
323C22⋊C4—41no—
324C22⋊C4—44no—
325C42—24no—
326C42—21no—
327C42—21no—
328C42—21no—
329C42—21no—
330C22×C3—22no—
331C22×C3—24no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—42no—
335C22×C3C2×#12111yescharacteristic
336C22×C3#53411yescharacteristic
337C22×C3#53611yescharacteristic
338C22×C3#57712yes—
339C22×C3#57812yes—
340C3×C4—22no—
341C3×C4—41no—
342C3×C4—41no—
343C3×C4—41no—
344C3×C4—41no—
345C3×C4—44no—
346C3×C4—82no—
347C3×C4—82no—
348C3×C4—82no—
349C3×C4#53711yescharacteristic
350C3×C4#53711yescharacteristic
351C3×C4#53911yescharacteristic
352C3×C4#53911yescharacteristic
353C23—22no—
354C23—22no—
355C23—21no—
356C23—21no—
357C23C3×#12112yes—
358C23C22×C3×D411yescharacteristic
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—42no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—22no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—24no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—42no—
379C2×C4—42no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—84no—
390C2×C4C3×#12112yes—
391C2×C4C22×C3×D411yescharacteristic
392C2×C4C2×C3×#4111yescharacteristic
393C2×C4C3×H(5,𝔽2)11yescharacteristic
394C2×C4C3×H(5,𝔽2)11yescharacteristic
395C2×C4C3×2−1+411yescharacteristic
396C2×C4C3×2−1+411yescharacteristic
397C8—42no—
398C8—42no—
399C2×C3—42no—
400C2×C3#261811yescharacteristic
401C2×C3#304612yes—
402C2×C3#320011yescharacteristic
403C2×C3#320211yescharacteristic
404C2×C3#324212yes—
405C22—22no—
406C22—24no—
407C22—41no—
408C22—41no—
409C22—42no—
410C22C2×C3×#12111yescharacteristic
411C22C3×#53411yescharacteristic
412C22C3×#53611yescharacteristic
413C22C3×#57712yes—
414C22C3×#57812yes—
415C4—22no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—44no—
421C4—82no—
422C4—82no—
423C4—82no—
424C4C3×#53711yescharacteristic
425C4C3×#53711yescharacteristic
426C4C3×#53911yescharacteristic
427C4C3×#53911yescharacteristic
428C3#2976611yescharacteristic, direct product factor, Sylow
429C2—42no—
430C2C3×#261811yescharacteristic
431C2C3×#304612yes—
432C2C3×#320011yescharacteristic
433C2C3×#320211yescharacteristic
434C2C3×#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
4. C3×#2085C2semidirect
5. C3×#2068C2semidirect
6. C3×#2066C2semidirect
7. C3×#2043C2semidirect
10. #29766C3direct (N × H)
44. #2085C2×C3semidirect
45. #2068C2×C3semidirect
46. #2066C2×C3semidirect
47. #2043C2×C3semidirect
428. C3#29766direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4852
C3×#26212
C3×#23172
C3×#20852
C3×#20682
C3×#20662
C3×#20432
C3×#18722
C3×#17982
#297663