Finite Group Databasebeta

← C3×#29836 #10648443

Subgroups — C3×#29836 #10648443

Order 768.

760 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#478C211yesmaximal, characteristic
2C2×C3×#461C211yesmaximal, characteristic
3C3×#2642C211yesmaximal, characteristic, complemented
4C3×#2319C211yesmaximal, characteristic
5C3×#2068C212yesmaximal, complemented
6C3×#2066C212yesmaximal, complemented
7C3×#2055C212yesmaximal
8C3×#2055C212yesmaximal
9C3×#1989C211yesmaximal, characteristic, complemented
10C3×#1981C211yesmaximal, characteristic, complemented
11C3×#1843C211yesmaximal, characteristic, complemented
12#29836C311yesmaximal, characteristic, direct product factor, Sylow
13C22×C3×SD16C2212yes—
14C2×C3×C4⋊Q8C2211yescharacteristic
15C2×C3×#125C2211yescharacteristic
16C2×C3×C4×D4C2211yescharacteristic
17C3×#478—24no—
18C3×#461—24no—
19C2×C3×#107C2211yescharacteristic
20C2×C3×#106C2211yescharacteristic
21C2×C3×C4⋊C8C2211yescharacteristic
22C2×C3×C4⋊C8C2211yescharacteristic
23C2×C3×C4⋊C8C2212yes—
24C2×C3×C4⋊C8C2212yes—
25C2×C3×#103C2212yes—
26C2×C3×#103C2212yes—
27C2×C3×#103C2212yes—
28C2×C3×#102C2211yescharacteristic
29C2×C3×#102C2211yescharacteristic
30C3×#400—21no—
31C3×#400—21no—
32C3×#400—22no—
33C3×#391C2212yes—
34C3×#391C2212yes—
35C3×#390C2212yes—
36C3×#389C2212yes—
37C3×#388—22no—
38C3×#381—21no—
39C3×#380—21no—
40C3×C4×C4⋊C4C2211yescharacteristic
41C3×#376C2211yescharacteristic
42C3×#340C2212yes—
43C3×#340C2212yes—
44C2×#478C2×C311yescharacteristic
45C2×#461C2×C311yescharacteristic
46#2642C2×C311yescharacteristic, complemented
47#2319C2×C311yescharacteristic
48#2068C2×C312yescomplemented
49#2066C2×C312yescomplemented
50#2055C2×C312yes—
51#2055C2×C312yes—
52#1989C2×C311yescharacteristic, complemented
53#1981C2×C311yescharacteristic, complemented
54#1843C2×C311yescharacteristic, complemented
55C22×C3×Q8C2312yes—
56C22×C3×D4C2311yescharacteristic
57C23×C3×C4—21no—
58C2×C3×SD16—24no—
59C2×C3×SD16—42no—
60C2×C3×SD16—42no—
61C22×C3×C8C2312yes—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×C4⋊Q8—22no—
65C3×C4⋊Q8—22no—
66C3×#125—22no—
67C3×#125—22no—
68C3×C4×D4—22no—
69C3×C4×D4D414yes—
70C2×C3×C4⋊C4—21no—
71C2×C3×C4⋊C4—21no—
72C2×C3×C4⋊C4—21no—
73C2×C3×C4⋊C4—21no—
74C2×C3×C4⋊C4—22no—
75C2×C3×C4⋊C4C2312yes—
76C2×C3×C4⋊C4C2311yescharacteristic
77C2×C3×C4⋊C4C2311yescharacteristic
78C2×C3×C4⋊C4C2311yescharacteristic
79C2×C3×C22⋊C4—21no—
80C2×C3×C22⋊C4—22no—
81C2×C3×C22⋊C4—22no—
82C2×C3×C42—21no—
83C2×C3×C42C2312yes—
84C2×C3×C42C2311yescharacteristic
85C3×#107—22no—
86C3×#106—22no—
87C3×C4⋊C8—22no—
88C3×C4⋊C8—22no—
89C3×C4⋊C8—42no—
90C3×C4⋊C8—42no—
91C3×#103—24no—
92C3×#103—42no—
93C3×#103—42no—
94C3×#102—22no—
95C3×#102—22no—
96C3×#95—21no—
97C3×#95—22no—
98C3×#95—22no—
99C3×#95—22no—
100C3×#95—22no—
101C3×#95—22no—
102C3×#95D411yescharacteristic
103C3×#95D411yescharacteristic
104C3×#95D412yes—
105C22×SD16C22×C312yes—
106C2×C4⋊Q8C22×C311yescharacteristic
107C2×#125C22×C311yescharacteristic
108C2×C4×D4C22×C311yescharacteristic
109#478—24no—
110#461—24no—
111C2×#107C22×C311yescharacteristic
112C2×#106C22×C311yescharacteristic
113C2×C4⋊C8C22×C311yescharacteristic
114C2×C4⋊C8C22×C311yescharacteristic
115C2×C4⋊C8C22×C312yes—
116C2×C4⋊C8C22×C312yes—
117C2×#103C22×C312yes—
118C2×#103C22×C312yes—
119C2×#103C22×C312yes—
120C2×#102C22×C311yescharacteristic
121C2×#102C22×C311yescharacteristic
122#400—21no—
123#400—21no—
124#400—22no—
125#391C22×C312yes—
126#391C22×C312yes—
127#390C22×C312yes—
128#389C22×C312yes—
129#388—22no—
130#381—21no—
131#380—21no—
132C4×C4⋊C4C22×C311yescharacteristic
133#376C22×C311yescharacteristic
134#340C22×C312yes—
135#340C22×C312yes—
136C24×C3—21no—
137C2×C3×Q8—22no—
138C2×C3×Q8—22no—
139C2×C3×Q8—22no—
140C2×C3×Q8—22no—
141C2×C3×Q8—22no—
142C2×C3×Q8—22no—
143C2×C3×D4—21no—
144C2×C3×D4—22no—
145C2×C3×D4—22no—
146C2×C3×D4C2×D412yes—
147C22×C3×C4—21no—
148C22×C3×C4—24no—
149C22×C3×C4—21no—
150C22×C3×C4—21no—
151C22×C3×C4—22no—
152C22×C3×C4—22no—
153C22×C3×C4—82no—
154C22×C3×C4C2×D412yes—
155C22×C3×C4C2×D412yes—
156C22×C3×C4C2×D411yescharacteristic
157C22×C3×C4C2×D411yescharacteristic
158C22×C3×C4C2411yescharacteristic
159C3×SD16—48no—
160C2×C3×C8—21no—
161C2×C3×C8—21no—
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—21no—
171C3×C4⋊C4—21no—
172C3×C4⋊C4—22no—
173C3×C4⋊C4—22no—
174C3×C4⋊C4—42no—
175C3×C4⋊C4—42no—
176C3×C4⋊C4—42no—
177C3×C4⋊C4—42no—
178C3×C4⋊C4—82no—
179C3×C4⋊C4C2×D412yes—
180C3×C22⋊C4—24no—
181C3×C22⋊C4—82no—
182C3×C22⋊C4—82no—
183C3×C42—22no—
184C3×C42—22no—
185C3×C42—81no—
186C3×C42C2×D412yes—
187C3×C42#4112yes—
188C22×Q8C23×C312yes—
189C22×D4C23×C311yescharacteristic
190C23×C4—21no—
191C2×SD16—24no—
192C2×SD16—42no—
193C2×SD16—42no—
194C22×C8C23×C312yes—
195C22×C8C23×C311yescharacteristic
196C22×C8C23×C311yescharacteristic
197C4⋊Q8—22no—
198C4⋊Q8—22no—
199#125—22no—
200#125—22no—
201C4×D4—22no—
202C4×D4C3×D414yes—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—21no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4C23×C311yescharacteristic
209C2×C4⋊C4C23×C311yescharacteristic
210C2×C4⋊C4C23×C312yes—
211C2×C4⋊C4C23×C311yescharacteristic
212C2×C22⋊C4—21no—
213C2×C22⋊C4—22no—
214C2×C22⋊C4—22no—
215C2×C42—21no—
216C2×C42C23×C312yes—
217C2×C42C23×C311yescharacteristic
218#107—22no—
219#106—22no—
220C4⋊C8—22no—
221C4⋊C8—22no—
222C4⋊C8—42no—
223C4⋊C8—42no—
224#103—24no—
225#103—42no—
226#103—42no—
227#102—22no—
228#102—22no—
229#95—21no—
230#95—22no—
231#95—22no—
232#95—22no—
233#95—22no—
234#95—22no—
235#95C3×D411yescharacteristic
236#95C3×D411yescharacteristic
237#95C3×D412yes—
238C23×C3—22no—
239C23×C3—42no—
240C23×C3—41no—
241C23×C3—42no—
242C23×C3—41no—
243C23×C3C22×D411yescharacteristic
244C3×Q8—42no—
245C3×Q8—42no—
246C3×Q8—42no—
247C3×Q8—42no—
248C3×D4—22no—
249C3×D4—22no—
250C3×D4—42no—
251C2×C3×C4—21no—
252C2×C3×C4—22no—
253C2×C3×C4—21no—
254C2×C3×C4—22no—
255C2×C3×C4—22no—
256C2×C3×C4—22no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—42no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4—42no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—42no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—84no—
274C2×C3×C4#12112yes—
275C2×C3×C4#12112yes—
276C2×C3×C4C22×D411yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4H(5,𝔽2)12yes—
279C2×C3×C42−1+412yes—
280C3×C8—41no—
281C3×C8—41no—
282C3×C8—42no—
283C24—21no—
284C2×Q8—22no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—22no—
288C2×Q8—22no—
289C2×Q8—22no—
290C2×D4—21no—
291C2×D4—22no—
292C2×D4—22no—
293C2×D4C2×C3×D412yes—
294C22×C4—21no—
295C22×C4—24no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—22no—
299C22×C4—22no—
300C22×C4—82no—
301C22×C4C2×C3×D412yes—
302C22×C4C2×C3×D412yes—
303C22×C4C2×C3×D411yescharacteristic
304C22×C4C2×C3×D411yescharacteristic
305C22×C4C24×C311yescharacteristic
306SD16—48no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—22no—
310C2×C8—22no—
311C2×C8—22no—
312C2×C8—22no—
313C2×C8—22no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—22no—
320C4⋊C4—22no—
321C4⋊C4—42no—
322C4⋊C4—42no—
323C4⋊C4—42no—
324C4⋊C4—42no—
325C4⋊C4—82no—
326C4⋊C4C2×C3×D412yes—
327C22⋊C4—24no—
328C22⋊C4—82no—
329C22⋊C4—82no—
330C42—22no—
331C42—22no—
332C42—81no—
333C42C2×C3×D412yes—
334C42C3×#4112yes—
335C22×C3—42no—
336C22×C3—42no—
337C22×C3—42no—
338C22×C3—82no—
339C22×C3—82no—
340C22×C3C2×#12111yescharacteristic
341C22×C3#53712yes—
342C22×C3#57712yes—
343C22×C3#57711yescharacteristic
344C22×C3#57811yescharacteristic
345C3×C4—22no—
346C3×C4—22no—
347C3×C4—42no—
348C3×C4—42no—
349C3×C4—42no—
350C3×C4—82no—
351C3×C4—82no—
352C3×C4—81no—
353C3×C4—81no—
354C3×C4#53511yescharacteristic
355C3×C4#53611yescharacteristic
356C3×C4#54212yes—
357C23—22no—
358C23—41no—
359C23—42no—
360C23—41no—
361C23—42no—
362C23C22×C3×D411yescharacteristic
363Q8—42no—
364Q8—42no—
365Q8—42no—
366Q8—42no—
367D4—22no—
368D4—22no—
369D4—42no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—22no—
375C2×C4—22no—
376C2×C4—22no—
377C2×C4—22no—
378C2×C4—22no—
379C2×C4—22no—
380C2×C4—42no—
381C2×C4—42no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—84no—
393C2×C4C3×#12112yes—
394C2×C4C3×#12112yes—
395C2×C4C22×C3×D411yescharacteristic
396C2×C4C2×C3×#4111yescharacteristic
397C2×C4C3×H(5,𝔽2)12yes—
398C2×C4C3×2−1+412yes—
399C8—42no—
400C8—41no—
401C8—41no—
402C2×C3—82no—
403C2×C3#262011yescharacteristic
404C2×C3#304211yescharacteristic
405C2×C3#304611yescharacteristic
406C2×C3#325812yes—
407C2×C3#326012yes—
408C22—42no—
409C22—42no—
410C22—42no—
411C22—82no—
412C22—82no—
413C22C2×C3×#12111yescharacteristic
414C22C3×#53712yes—
415C22C3×#57712yes—
416C22C3×#57711yescharacteristic
417C22C3×#57811yescharacteristic
418C4—22no—
419C4—22no—
420C4—42no—
421C4—42no—
422C4—42no—
423C4—82no—
424C4—82no—
425C4—81no—
426C4—81no—
427C4C3×#53511yescharacteristic
428C4C3×#53611yescharacteristic
429C4C3×#54212yes—
430C3#2983611yescharacteristic, direct product factor, Sylow
431C2—82no—
432C2C3×#262011yescharacteristic
433C2C3×#304211yescharacteristic
434C2C3×#304611yescharacteristic
435C2C3×#325812yes—
436C2C3×#326012yes—
subgroupG/H
center, socleC23×C3C22×D4
commutator, Frattini, cosocleC22×C4C24×C3
prime radicalC22×C3×D4C23
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23×C3 ◁ C3×#376 ◁ G

Complements

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

NH ≅ G/Ntype
3. C3×#2642C2semidirect
5. C3×#2068C2semidirect
6. C3×#2066C2semidirect
9. C3×#1989C2semidirect
10. C3×#1981C2semidirect
11. C3×#1843C2semidirect
12. #29836C3direct (N × H)
46. #2642C2×C3semidirect
48. #2068C2×C3semidirect
49. #2066C2×C3semidirect
52. #1989C2×C3semidirect
53. #1981C2×C3semidirect
54. #1843C2×C3semidirect
430. C3#29836direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×#4782
C2×C3×#4612
C3×#26422
C3×#23192
C3×#20682
C3×#20662
C3×#20552
C3×#19892
C3×#19812
C3×#18432
#298363