Finite Group Databasebeta

← C3×#85507 #11922791

Subgroups — C3×#85507 #11922791

Order 1152.

777 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 777 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C23×C3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
2C2×C3×#983C211yesmaximal, characteristic, complemented
3C3×#5421C212yesmaximal, complemented
4C3×#5420C211yesmaximal, characteristic, complemented
5C3×#5419C212yesmaximal, complemented
6#85507C311yesmaximal, characteristic, direct product factor
7C3×#3004—31nomaximal
8C22×C3×SL(2,𝔽3)—22no—
9C22×C3×SL(2,𝔽3)C2212yes—
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C3×#983—22no—
12C2×C3×GL(2,𝔽3)C2211yescharacteristic, complemented
13C2×C3×#226C2211yescharacteristic, complemented
14C3×#953—41nomaximal
15C3×#859C2212yescomplemented
16C23×C3×Q8D311yescharacteristic, complemented
17C23×SL(2,𝔽3)—21no—
18C23×SL(2,𝔽3)C2×C311yescharacteristic, complemented
19C2×#983C2×C311yescharacteristic, complemented
20C2×C3×#137—31no—
21C2×C3×#137—31no—
22C3×#525—31no—
23#5421C2×C312yescomplemented
24#5420C2×C311yescharacteristic, complemented
25#5419C2×C312yescomplemented
26C3×#451—32no—
27C3×#451—32no—
28C3×#448—32no—
29C3×#448—32no—
30C3×#419—31no—
31C3×#419—31no—
32C3×#407—31no—
33C24×C32—41no—
34C2×C3×#67—42no—
35C2×C3×#67—41no—
36C2×C3×SL(2,𝔽3)—22no—
37C2×C3×SL(2,𝔽3)D412yescomplemented
38C2×C3×SL(2,𝔽3)C2311yescharacteristic
39C3×GL(2,𝔽3)—21no—
40C3×#226—21no—
41C3×#217—42no—
42C3×#217—41no—
43#3004—31noSylow
44C2×C3×#41—31no—
45C22×C3×Q8—22no—
46C22×C3×Q8—31no—
47C22×C3×Q8—32no—
48C22×C3×Q8—62no—
49C22×C3×Q8C2×D312yes—
50C22×C3×Q8C2×D311yescharacteristic
51C23×C3×C4—31no—
52C22×SL(2,𝔽3)—24no—
53C22×SL(2,𝔽3)—22no—
54C22×SL(2,𝔽3)—22no—
55C22×SL(2,𝔽3)—21no—
56C22×SL(2,𝔽3)C22×C312yes—
57C22×SL(2,𝔽3)C22×C311yescharacteristic
58#983—22no—
59C2×GL(2,𝔽3)C22×C311yescharacteristic, complemented
60C2×#226C22×C311yescharacteristic, complemented
61C3×#137—62no—
62C3×#137—64no—
63C3×#137—62no—
64C2×C3×Q16—32no—
65C2×C3×Q16—31no—
66C2×C3×Q16—31no—
67C2×C3×SD16—32no—
68C2×C3×SD16—31no—
69C2×C3×SD16—31no—
70C2×C3×M16—31no—
71C2×C3×M16—31no—
72C3×#123—61no—
73C3×C22⋊Q8—32no—
74C3×#121—32no—
75C3×#120—61no—
76C3×C4×D4—62no—
77C3×#117—31no—
78#953—41no—
79#859C22×C312yescomplemented
80C3×#103—32no—
81C3×#103—32no—
82C3×#103—32no—
83C3×#103—32no—
84C3×C22⋊C8—32no—
85C3×C22⋊C8—32no—
86C23×C32—42no—
87C23×C32—41no—
88C23×C32—84no—
89C23×C32—82no—
90C22×C3×D3—41no—
91C3×#67—84no—
92C3×#67—82no—
93C2×C3×Dic3—42no—
94C2×C3×Dic3—41no—
95C3×SL(2,𝔽3)C2×D411yescharacteristic
96C23×Q8C3×D311yescharacteristic, complemented
97C2×#137—31no—
98C2×#137—31no—
99#525—31no—
100#451—32no—
101#451—32no—
102#448—32no—
103#448—32no—
104#419—31no—
105#419—31no—
106#407—31no—
107C24×C3—41no—
108C24×C3—81no—
109C24×C3S411yescharacteristic
110C3×#41—62no—
111C3×#41—62no—
112C2×C3×Q8—22no—
113C2×C3×Q8—31no—
114C2×C3×Q8—32no—
115C2×C3×Q8—32no—
116C2×C3×Q8—34no—
117C2×C3×Q8—31no—
118C2×C3×Q8—61no—
119C2×C3×Q8—62no—
120C2×C3×Q8—64no—
121C2×C3×Q8—62no—
122C2×C3×Q8—64no—
123C2×C3×Q8#6712yescomplemented
124C2×C3×Q8C22×D311yescharacteristic
125C2×C3×D4—31no—
126C2×C3×D4—62no—
127C2×C3×D4—61no—
128C22×C3×C4—31no—
129C22×C3×C4—32no—
130C22×C3×C4—32no—
131C22×C3×C4—31no—
132C22×C3×C4—62no—
133C22×C3×C4—62no—
134C22×C3×C4—61no—
135C2×#67—41no—
136C2×#67—42no—
137C2×SL(2,𝔽3)—24no—
138C2×SL(2,𝔽3)—22no—
139C2×SL(2,𝔽3)—22no—
140C2×SL(2,𝔽3)—21no—
141C2×SL(2,𝔽3)C3×D412yescomplemented
142C2×SL(2,𝔽3)C23×C311yescharacteristic
143GL(2,𝔽3)—21no—
144#226—21no—
145C3×Q16—62no—
146C3×Q16—61no—
147C3×Q16—64no—
148C3×Q16—61no—
149C3×SD16—62no—
150C3×SD16—61no—
151C3×SD16—64no—
152C3×SD16—61no—
153C3×M16—62no—
154C3×M16—62no—
155C2×C3×C8—31no—
156C2×C3×C8—31no—
157C2×C3×C8—32no—
158C3×C4⋊C4—32no—
159C3×C4⋊C4—62no—
160C3×C22⋊C4—62no—
161C3×C22⋊C4—61no—
162C3×C22⋊C4—62no—
163C3×C42—61no—
164#217—42no—
165#217—41no—
166C22×C32—44no—
167C22×C32—42no—
168C22×C32—41no—
169C22×C32—82no—
170C22×C32—82no—
171C22×C32—82no—
172C22×C32—88no—
173C2×C3×D3—82no—
174C2×C3×D3—81no—
175C3×Dic3—82no—
176C3×Dic3—81no—
177C2×#41—31no—
178C22×Q8—22no—
179C22×Q8—31no—
180C22×Q8—32no—
181C22×Q8—62no—
182C22×Q8C2×C3×D312yes—
183C22×Q8C2×C3×D311yescharacteristic
184C23×C4—31no—
185#137—62no—
186#137—64no—
187#137—62no—
188C2×Q16—32no—
189C2×Q16—31no—
190C2×Q16—31no—
191C2×SD16—32no—
192C2×SD16—31no—
193C2×SD16—31no—
194C2×M16—31no—
195C2×M16—31no—
196#123—61no—
197C22⋊Q8—32no—
198#121—32no—
199#120—61no—
200C4×D4—62no—
201#117—31no—
202#103—32no—
203#103—32no—
204#103—32no—
205#103—32no—
206C22⋊C8—32no—
207C22⋊C8—32no—
208C23×C3—24no—
209C23×C3—22no—
210C23×C3—41no—
211C23×C3—42no—
212C23×C3—61no—
213C23×C3—82no—
214C23×C3—84no—
215C23×C3—84no—
216C23×C3—88no—
217C23×C3—81no—
218C23×C3—82no—
219C23×C3C2×S412yes—
220C23×C3C2×S411yescharacteristic
221C22×D3—41no—
222C3×Q8—32no—
223C3×Q8—31no—
224C3×Q8—34no—
225C3×Q8—61no—
226C3×Q8—62no—
227C3×Q8—64no—
228C3×Q8—61no—
229C3×Q8—61no—
230C3×Q8C2×#6711yescharacteristic
231C3×D4—61no—
232C3×D4—61no—
233C3×D4—124no—
234C3×D4—122no—
235C2×C3×C4—32no—
236C2×C3×C4—32no—
237C2×C3×C4—31no—
238C2×C3×C4—31no—
239C2×C3×C4—32no—
240C2×C3×C4—62no—
241C2×C3×C4—62no—
242C2×C3×C4—64no—
243C2×C3×C4—62no—
244C2×C3×C4—62no—
245C2×C3×C4—61no—
246C2×C3×C4—122no—
247#67—82no—
248#67—84no—
249C2×Dic3—42no—
250C2×Dic3—41no—
251SL(2,𝔽3)—21no—
252SL(2,𝔽3)C2×C3×D411yescharacteristic
253C3×C8—62no—
254C3×C8—61no—
255C3×C8—61no—
256C2×C32—42no—
257C2×C32—41no—
258C2×C32—84no—
259C2×C32—82no—
260C3×D3—161no—
261C24C3×S411yescharacteristic
262#41—62no—
263#41—62no—
264C2×Q8—22no—
265C2×Q8—31no—
266C2×Q8—32no—
267C2×Q8—32no—
268C2×Q8—34no—
269C2×Q8—31no—
270C2×Q8—61no—
271C2×Q8—62no—
272C2×Q8—64no—
273C2×Q8—62no—
274C2×Q8—64no—
275C2×Q8C3×#6712yescomplemented
276C2×Q8C22×C3×D311yescharacteristic
277C2×D4—31no—
278C2×D4—62no—
279C2×D4—61no—
280C22×C4—31no—
281C22×C4—32no—
282C22×C4—32no—
283C22×C4—31no—
284C22×C4—62no—
285C22×C4—62no—
286C22×C4—61no—
287Q16—62no—
288Q16—61no—
289Q16—64no—
290Q16—61no—
291SD16—62no—
292SD16—61no—
293SD16—64no—
294SD16—61no—
295M16—62no—
296M16—62no—
297C2×C8—31no—
298C2×C8—31no—
299C2×C8—32no—
300C4⋊C4—32no—
301C4⋊C4—62no—
302C22⋊C4—62no—
303C22⋊C4—61no—
304C22⋊C4—62no—
305C42—61no—
306C22×C3—28no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—42no—
311C22×C3—41no—
312C22×C3—44no—
313C22×C3—82no—
314C22×C3—82no—
315C22×C3—82no—
316C22×C3—88no—
317C22×C3—816no—
318C22×C3—84no—
319C22×C3—84no—
320C22×C3—84no—
321C22×C3—82no—
322C22×C3—81no—
323C22×C3—84no—
324C22×C3—122no—
325C22×C3—121no—
326C22×C3#98314yescomplemented
327C22×C3GL(2,ℤ/4)12yes—
328C22×C3C22×S411yescharacteristic
329C2×D3—82no—
330C2×D3—81no—
331C3×C4—31no—
332C3×C4—31no—
333C3×C4—32no—
334C3×C4—62no—
335C3×C4—122no—
336C3×C4—121no—
337Dic3—82no—
338Dic3—81no—
339C32—41noSylow
340C23—22no—
341C23—24no—
342C23—61no—
343C23C2×C3×S411yescharacteristic
344C23C2×C3×S412yes—
345Q8—32no—
346Q8—31no—
347Q8—34no—
348Q8—61no—
349Q8—62no—
350Q8—64no—
351Q8—61no—
352Q8—61no—
353Q8C2×C3×#6711yescharacteristic
354D4—61no—
355D4—61no—
356D4—122no—
357D4—124no—
358C2×C4—32no—
359C2×C4—32no—
360C2×C4—31no—
361C2×C4—31no—
362C2×C4—32no—
363C2×C4—62no—
364C2×C4—62no—
365C2×C4—64no—
366C2×C4—62no—
367C2×C4—62no—
368C2×C4—61no—
369C2×C4—122no—
370C8—62no—
371C8—61no—
372C8—61no—
373C2×C3—24no—
374C2×C3—22no—
375C2×C3—42no—
376C2×C3—41no—
377C2×C3—82no—
378C2×C3—84no—
379C2×C3—84no—
380C2×C3—88no—
381C2×C3—82no—
382C2×C3—81no—
383C2×C3—241no—
384C2×C3C2×#98312yes—
385C2×C3C2×GL(2,ℤ/4)11yescharacteristic
386D3—161no—
387C22—28no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—122no—
392C22—121no—
393C22C3×#98314yescomplemented
394C22C3×GL(2,ℤ/4)12yes—
395C22C22×C3×S411yescharacteristic
396C4—31no—
397C4—31no—
398C4—32no—
399C4—62no—
400C4—122no—
401C4—121no—
402C3—41no—
403C3—81no—
404C3#8550711yescharacteristic, direct product factor
405C2—22no—
406C2—24no—
407C2—241no—
408C2C2×C3×#98312yes—
409C2C2×C3×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, socleC22×C3C22×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C3
FrattiniC22C22×C3×S4
FittingC23×C3×Q8D3
hypercenterC24×C3S4
hypocenterSL(2,𝔽3)C2×C3×D4
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C22×C3 ◁ C24×C3

Complements

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

NH ≅ G/Ntype
1. C23×C3×SL(2,𝔽3)C2semidirect
2. C2×C3×#983C2semidirect
3. C3×#5421C2semidirect
4. C3×#5420C2semidirect
5. C3×#5419C2semidirect
6. #85507C3direct (N × H)
12. C2×C3×GL(2,𝔽3)C22semidirect
13. C2×C3×#226C22semidirect
15. C3×#859C22semidirect
16. C23×C3×Q8D3semidirect
18. C23×SL(2,𝔽3)C2×C3semidirect
19. C2×#983C2×C3semidirect
23. #5421C2×C3semidirect
24. #5420C2×C3semidirect
25. #5419C2×C3semidirect
37. C2×C3×SL(2,𝔽3)D4semidirect
59. C2×GL(2,𝔽3)C22×C3semidirect
60. C2×#226C22×C3semidirect
79. #859C22×C3semidirect
96. C23×Q8C3×D3semidirect
123. C2×C3×Q8#67semidirect
141. C2×SL(2,𝔽3)C3×D4semidirect
275. C2×Q8C3×#67semidirect
326. C22×C3#983semidirect
393. C22C3×#983semidirect
404. C3#85507direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C3×SL(2,𝔽3)2
C2×C3×#9832
C3×#54212
C3×#54202
C3×#54192
#855073
C3×#30043
C3×#9534