Finite Group Databasebeta

← C3×#68108 #11778216

Subgroups — C3×#68108 #11778216

Order 1152.

665 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 665 conjugacy classes of subgroups — too many to draw. Only its 106 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5493C211yesmaximal, characteristic
2C3×#4947C211yesmaximal, characteristic, complemented
3C3×#4738C212yesmaximal, complemented
4C32×#323C211yesmaximal, characteristic, complemented
5C3×#4470C212yesmaximal, complemented
6C3×#1601—31nomaximal
7#68108C311yesmaximal, characteristic, direct product factor
8C22×C3×#67—21no—
9C22×C3×Dic6C2211yescharacteristic, complemented
10C2×C32×C22⋊C4C2211yescharacteristic
11C2×C3×#217—21no—
12C3×#930—21no—
13C3×#930C2212yes—
14C2×C3×#204—22no—
15C3×#886—22no—
16C3×#882—22no—
17C32×C22⋊C8C2212yescomplemented
18C3×C3⋊#95—21no—
19C3×C3⋊#95C2211yescharacteristic, complemented
20C3×#816—22no—
21C3×#534—31no—
22#5493C2×C311yescharacteristic
23C3×#448—32no—
24C3×#393—31no—
25#4947C2×C311yescharacteristic, complemented
26#4738C2×C312yescomplemented
27C3×#327—32no—
28C3×#323—21no—
29C3×#323D311yescharacteristic, complemented
30C3×#323C2×C311yescharacteristic, complemented
31#4470C2×C312yescomplemented
32C24×C32D411yescharacteristic
33C23×C3×D3—21no—
34C22×C32×C4D411yescharacteristic
35C22×C32×C4C2311yescharacteristic
36C2×C3×#67—22no—
37C2×C3×#67—22no—
38C2×C3×#67—42no—
39C2×C3×#67—42no—
40C22×C3×Dic3—21no—
41C22×C3×Dic3—21no—
42C22×C3×Dic3—21no—
43C2×C3×D12—21no—
44C2×C3×D12D412yes—
45C2×C3×Dic6—22no—
46C2×C3×Dic6—42no—
47C2×C32×C8—22no—
48C32×C22⋊C4—22no—
49C3×#217—42no—
50C3×(C2×D3)⋊C4—22no—
51C3×(C2×D3)⋊C4—22no—
52C3×C4⋊Dic3—21no—
53C3×C4⋊Dic3D412yes—
54C3×#204—44no—
55#1601—31noSylow
56C22×C3×Q8—31no—
57C22×C3×D4—61no—
58C22×#67—21no—
59C22×Dic6C22×C311yescharacteristic, complemented
60C2×C3×SD16—62no—
61C3×#123—62no—
62C3×#121—32no—
63C3×#121—61no—
64C3×#120—62no—
65C2×C3×C22⋊C4—21no—
66C2×C3×C22⋊C4—61no—
67C2×C3×C22⋊C4C2×D311yescharacteristic
68C2×C3×C22⋊C4C22×C311yescharacteristic
69C2×#217—21no—
70#930—21no—
71#930C22×C312yes—
72C2×#204—22no—
73#886—22no—
74#882—22no—
75C3×#103—62no—
76C3×C22⋊C8—22no—
77C3×C22⋊C8C2×D312yescomplemented
78C3×C22⋊C8C22×C312yescomplemented
79C3×#95—31no—
80C3×#95—61no—
81C3⋊#95—21no—
82C3⋊#95C22×C311yescharacteristic, complemented
83#816—22no—
84C23×C32—21no—
85C23×C32—21no—
86C23×C32—41no—
87C23×C32—41no—
88C23×C32SD1612yescomplemented
89C23×C32C2×D411yescharacteristic
90C22×C3×D3—22no—
91C22×C3×D3—42no—
92C22×C3×D3—81no—
93C22×C3×D3—81no—
94C2×C32×C4—21no—
95C2×C32×C4—41no—
96C2×C32×C4—41no—
97C2×C32×C4C2×D412yes—
98C3×#67—44no—
99C3×#67—42no—
100C3×#67—42no—
101C2×C3×Dic3—22no—
102C2×C3×Dic3—22no—
103C2×C3×Dic3—41no—
104C2×C3×Dic3—42no—
105C2×C3×Dic3—41no—
106C2×C3×Dic3—41no—
107C2×C3×Dic3—41no—
108C2×C3×Dic3—41no—
109C3×D12—42no—
110C3×D12—81no—
111C3×Dic6—24no—
112C3×Dic6—42no—
113C32×C8—42no—
114#534—31no—
115#448—32no—
116#393—31no—
117#327—32no—
118#323C3×D311yescharacteristic, complemented
119C24×C3—21no—
120C24×C3—61no—
121C24×C3D1211yescharacteristic
122C24×C3C3×D411yescharacteristic
123C23×D3—21no—
124C2×C3×Q8—62no—
125C2×C3×Q8—122no—
126C2×C3×D4—32no—
127C2×C3×D4—61no—
128C2×C3×D4—62no—
129C2×C3×D4—62no—
130C2×C3×D4—122no—
131C2×C3×D4—122no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—61no—
135C22×C3×C4—61no—
136C22×C3×C4—61no—
137C22×C3×C4D1211yescharacteristic
138C22×C3×C4C3×D411yescharacteristic
139C22×C3×C4C22×D311yescharacteristic
140C22×C3×C4C23×C311yescharacteristic
141C2×#67—22no—
142C2×#67—22no—
143C2×#67—42no—
144C2×#67—42no—
145C22×Dic3—21no—
146C22×Dic3—21no—
147C22×Dic3—21no—
148C2×D12—21no—
149C2×D12C3×D412yes—
150C2×Dic6—22no—
151C2×Dic6—42no—
152C3×SD16—124no—
153C2×C3×C8—22no—
154C2×C3×C8—22no—
155C2×C3×C8—42no—
156C3×C4⋊C4—32no—
157C3×C4⋊C4—61no—
158C3×C22⋊C4—22no—
159C3×C22⋊C4—22no—
160C3×C22⋊C4—42no—
161C3×C22⋊C4—62no—
162C3×C22⋊C4—62no—
163C3×C22⋊C4—122no—
164#217—42no—
165(C2×D3)⋊C4—22no—
166(C2×D3)⋊C4—22no—
167C4⋊Dic3—21no—
168C4⋊Dic3C3×D412yes—
169#204—44no—
170C22×C32—21no—
171C22×C32—21no—
172C22×C32—22no—
173C22×C32—42no—
174C22×C32—41no—
175C22×C32—41no—
176C22×C32—81no—
177C22×C32#12011yescharacteristic
178C22×C32C2×SD1611yescharacteristic
179C22×C32AGL(1,ℤ/8)11yescharacteristic
180C2×C3×D3—42no—
181C2×C3×D3—42no—
182C2×C3×D3—42no—
183C2×C3×D3—84no—
184C32×C4—22no—
185C32×C4—41no—
186C3×Dic3—42no—
187C3×Dic3—42no—
188C3×Dic3—81no—
189C22×Q8—31no—
190C22×D4—61no—
191C2×SD16—62no—
192#123—62no—
193#121—32no—
194#121—61no—
195#120—62no—
196C2×C22⋊C4—61no—
197C2×C22⋊C4C2×C3×D311yescharacteristic
198#103—62no—
199C22⋊C8C2×C3×D312yescomplemented
200#95—31no—
201#95—61no—
202C23×C3—21no—
203C23×C3—21no—
204C23×C3—21no—
205C23×C3—21no—
206C23×C3—22no—
207C23×C3—21no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3—41no—
211C23×C3—41no—
212C23×C3—41no—
213C23×C3—41no—
214C23×C3—62no—
215C23×C3—81no—
216C23×C3—81no—
217C23×C3—122no—
218C23×C3—241no—
219C23×C3—241no—
220C23×C3#20412yescomplemented
221C23×C3C3×SD1612yescomplemented
222C23×C3C2×D1211yescharacteristic
223C23×C3C2×C3×D411yescharacteristic
224C22×D3—22no—
225C22×D3—42no—
226C22×D3—81no—
227C22×D3—81no—
228C3×Q8—64no—
229C3×Q8—122no—
230C3×D4—122no—
231C3×D4—124no—
232C3×D4—122no—
233C3×D4—122no—
234C3×D4—241no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—22no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—62no—
244C2×C3×C4—62no—
245C2×C3×C4—81no—
246C2×C3×C4—81no—
247C2×C3×C4—122no—
248C2×C3×C4—121no—
249C2×C3×C4—121no—
250C2×C3×C4—121no—
251C2×C3×C4—121no—
252C2×C3×C4—121no—
253C2×C3×C4D3×D412yes—
254C2×C3×C4C2×C3×D412yes—
255#67—44no—
256#67—42no—
257#67—42no—
258C2×Dic3—22no—
259C2×Dic3—22no—
260C2×Dic3—41no—
261C2×Dic3—42no—
262C2×Dic3—41no—
263C2×Dic3—41no—
264C2×Dic3—41no—
265C2×Dic3—41no—
266D12—42no—
267D12—81no—
268Dic6—24no—
269Dic6—42no—
270C3×C8—42no—
271C3×C8—42no—
272C3×C8—82no—
273C2×C32—21no—
274C2×C32—21no—
275C2×C32—42no—
276C2×C32#45011yescharacteristic
277C2×C32#45511yescharacteristic
278C2×C32B(4,𝔽2)11yescharacteristic
279C3×D3—82no—
280C24—61no—
281C24C3×D1211yescharacteristic
282C2×Q8—62no—
283C2×Q8—122no—
284C2×D4—32no—
285C2×D4—61no—
286C2×D4—62no—
287C2×D4—62no—
288C2×D4—122no—
289C2×D4—122no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—61no—
293C22×C4C3×D1211yescharacteristic
294C22×C4C22×C3×D311yescharacteristic
295SD16—124no—
296C2×C8—22no—
297C4⋊C4—32no—
298C4⋊C4—61no—
299C22⋊C4—22no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C22⋊C4—122no—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—21no—
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—42no—
315C22×C3—42no—
316C22×C3—42no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—81no—
322C22×C3—81no—
323C22×C3—82no—
324C22×C3—81no—
325C22×C3—81no—
326C22×C3—82no—
327C22×C3—122no—
328C22×C3—122no—
329C22×C3—122no—
330C22×C3—244no—
331C22×C3#88211yescharacteristic
332C22×C3C2×#20411yescharacteristic
333C22×C3#90811yescharacteristic
334C22×C3C3×#12011yescharacteristic
335C22×C3C2×C3×SD1611yescharacteristic
336C22×C3C3×AGL(1,ℤ/8)11yescharacteristic
337C2×D3—42no—
338C2×D3—42no—
339C2×D3—42no—
340C2×D3—84no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—42no—
344C3×C4—41no—
345C3×C4—41no—
346C3×C4—81no—
347C3×C4—122no—
348C3×C4—122no—
349C3×C4—241no—
350Dic3—42no—
351Dic3—42no—
352Dic3—81no—
353C32#160111yescharacteristic, complemented, Sylow
354C23—21no—
355C23—21no—
356C23—41no—
357C23—41no—
358C23—62no—
359C23—122no—
360C23—241no—
361C23—241no—
362C23C3×#20412yescomplemented
363C23C2×C3×D1211yescharacteristic
364Q8—64no—
365Q8—122no—
366D4—122no—
367D4—124no—
368D4—122no—
369D4—122no—
370D4—241no—
371C2×C4—21no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—122no—
377C2×C4—121no—
378C2×C4—121no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4C3×D3×D412yes—
383C8—42no—
384C2×C3—21no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—21no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—41no—
392C2×C3—41no—
393C2×C3—42no—
394C2×C3—42no—
395C2×C3—82no—
396C2×C3—242no—
397C2×C3#473111yescharacteristic
398C2×C3#474111yescharacteristic
399C2×C3#474611yescharacteristic
400C2×C3C3×#45011yescharacteristic
401C2×C3C3×#45511yescharacteristic
402C2×C3C3×B(4,𝔽2)11yescharacteristic
403D3—82no—
404C22—21no—
405C22—21no—
406C22—22no—
407C22—42no—
408C22—41no—
409C22—41no—
410C22—81no—
411C22—122no—
412C22—122no—
413C22—122no—
414C22—244no—
415C22C3×#88211yescharacteristic
416C22C2×C3×#20411yescharacteristic
417C22C3×#90811yescharacteristic
418C4—22no—
419C4—41no—
420C4—122no—
421C4—122no—
422C4—241no—
423C3—21no—
424C3#6810811yescharacteristic, direct product factor
425C3C3×#160111yescharacteristic, complemented
426C2—21no—
427C2—21no—
428C2—42no—
429C2—242no—
430C2C3×#473111yescharacteristic
431C2C3×#474111yescharacteristic
432C2C3×#474611yescharacteristic
subgroupG/H
centerC22×C3#882
commutator, cosocleC22×C3×C4C23×C3
FrattiniC22×C4C22×C3×D3
FittingC32×#323C2
socleC22×C32#120
prime radicalC3×#5493C2
hypercenterC3×#323D3
hypocenterC3C3×#1601
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#323

Complements

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

NH ≅ G/Ntype
2. C3×#4947C2semidirect
3. C3×#4738C2semidirect
4. C32×#323C2semidirect
5. C3×#4470C2semidirect
7. #68108C3direct (N × H)
9. C22×C3×Dic6C22semidirect
17. C32×C22⋊C8C22semidirect
19. C3×C3⋊#95C22semidirect
25. #4947C2×C3semidirect
26. #4738C2×C3semidirect
29. C3×#323D3semidirect
30. C3×#323C2×C3semidirect
31. #4470C2×C3semidirect
59. C22×Dic6C22×C3semidirect
77. C3×C22⋊C8C2×D3semidirect
78. C3×C22⋊C8C22×C3semidirect
82. C3⋊#95C22×C3semidirect
88. C23×C32SD16semidirect
118. #323C3×D3semidirect
199. C22⋊C8C2×C3×D3semidirect
220. C23×C3#204semidirect
221. C23×C3C3×SD16semidirect
353. C32#1601semidirect
362. C23C3×#204semidirect
424. C3#68108direct (N × H)
425. C3C3×#1601semidirect

Maximal subgroups

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

subgroupindex
C3×#54932
C3×#49472
C3×#47382
C32×#3232
C3×#44702
C3×#16013
#681083