Finite Group Databasebeta

← #85881⋊C3 #11923214

Subgroups — #85881⋊C3 #11923214

Order 1152.

829 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 829 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C2×#1007⋊C3C211yesmaximal, characteristic, complemented
2C22×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
3C3×#519⋊C3C211yesmaximal, characteristic, complemented
4#5576⋊C3C212yesmaximal, complemented
5SL(2,𝔽3)×D12C212yesmaximal, complemented
6#85881C311yesmaximal, characteristic, complemented
7#3486⋊C3—31nomaximal
8C22×C3×SL(2,𝔽3)C2211yescharacteristic
9#1007⋊C3—22no—
10C2×D3×SL(2,𝔽3)—22no—
11C2×D3×SL(2,𝔽3)C2212yes—
12C2×D3×SL(2,𝔽3)C2211yescharacteristic, complemented
13C2×C3⋊(#41⋊C3)C2211yescharacteristic, complemented
14C3×#882—41nomaximal
15C3×C4×SL(2,𝔽3)C2212yescomplemented
16C2×#1007C2×C311yescharacteristic, complemented
17C22×D3×Q8C2×C311yescharacteristic, complemented
18C2×2−1+4⋊C3—31no—
19C23×SL(2,𝔽3)—31no—
20C3×#519C2×C311yescharacteristic, complemented
21#5576—32no—
22#5576C2×C312yescomplemented
23Q8×D12—32no—
24Q8×D12C2×C312yescomplemented
25#5526—31no—
26#5526—31no—
27#5524—31no—
28#5524—31no—
29D4×SL(2,𝔽3)—32no—
30#548⋊C3—32no—
31#519⋊C3—21no—
32#519⋊C3D311yescharacteristic, complemented
33C23×C3×D3—41no—
34C2×C3×#67—41no—
35C2×C3×D12—42no—
36C2×C3×SL(2,𝔽3)D412yescomplemented
37C2×C3×SL(2,𝔽3)C2311yescharacteristic
38D3×SL(2,𝔽3)—22no—
39D3×SL(2,𝔽3)—21no—
40C3⋊(#41⋊C3)—21no—
41C32×C22⋊C4—41no—
42C3×(C2×D3)⋊C4—42no—
43#3486—31noSylow
44C22×C3×Q8C22×C311yescharacteristic
45#1007—22no—
46#1007—62no—
47C2×#239—31no—
48C2×#239C22×C311yescharacteristic, complemented
49C2×D3×Q8—22no—
50C2×D3×Q8—32no—
51C2×D3×Q8—31no—
52C2×D3×Q8—62no—
53C2×D3×Q8C22×C312yes—
54C2×D3×Q8C22×C311yescharacteristic, complemented
55C2×#235—31no—
56C2×#235—31no—
57C22×C4×D3—31no—
58C22×Dic6—31no—
592−1+4⋊C3—62no—
60C2×#41⋊C3—31no—
61C22×SL(2,𝔽3)—21no—
62C22×SL(2,𝔽3)—32no—
63C22×SL(2,𝔽3)—31no—
64C22×SL(2,𝔽3)—62no—
65C22×SL(2,𝔽3)C2×D311yescharacteristic
66C3×#120—121no—
67C3×C4×Q8—32no—
68C3×C4×Q8C22×C312yescomplemented
69C3×#117—31no—
70C3×#117—31no—
71#897—34no—
72#897—32no—
73#897—32no—
74#897—32no—
75#897—32no—
76#895—32no—
77#895—32no—
78#886—32no—
79#886—31no—
80#886—31no—
81#882—31no—
82#882A411yescharacteristic
83#875—32no—
84#875—32no—
85C4×D12—34no—
86C4×D12—32no—
87C4×D12—32no—
88#869—32no—
89#869—32no—
90C4×SL(2,𝔽3)—22no—
91C4×SL(2,𝔽3)C2×D312yescomplemented
92C23×C32—41no—
93C22×C3×D3—42no—
94C22×C3×D3—41no—
95C22×C3×D3—84no—
96C22×C3×D3—82no—
97C2×C32×C4—42no—
98C3×#67—82no—
99C2×C3×Dic3—41no—
100C3×D12—84no—
101C3×SL(2,𝔽3)C2×D411yescharacteristic
102C2×2−1+4—31no—
103C23×Q8—31no—
104D4×Q8—32no—
105D4×Q8—92no—
106#548—32no—
107#548—92no—
108#536—91no—
109#536—91no—
110#525—91no—
111#525—91no—
112#519C3×D311yescharacteristic, complemented
113C24×C3—121no—
114C23×D3C2×A411yescharacteristic
115C2×C3×Q8—31no—
116C2×C3×Q8—32no—
117C2×C3×Q8C3×D412yescomplemented
118C2×C3×Q8C23×C311yescharacteristic
119C2×C3×D4—122no—
120C2×C3×D4—121no—
121C22×C3×C4—31no—
122C2×#67—31no—
123C2×#67C2×A411yescharacteristic
124C22×Dic3—31no—
125#239—21no—
126#239—62no—
127#239—62no—
128#239—61no—
129D3×Q8—22no—
130D3×Q8—21no—
131D3×Q8—64no—
132D3×Q8—62no—
133D3×Q8—64no—
134D3×Q8—62no—
135D3×Q8—62no—
136D3×Q8—61no—
137#235—62no—
138#235—62no—
139#235—62no—
140#235—62no—
141C2×D12—31no—
142C2×D12—32no—
143C2×D12—31no—
144C2×D12C2×A412yes—
145C2×C4×D3—32no—
146C2×C4×D3—32no—
147C2×C4×D3—32no—
148C2×C4×D3—31no—
149C2×C4×D3—31no—
150C2×C4×D3—62no—
151C2×C4×D3—62no—
152C2×Dic6—32no—
153C2×Dic6—31no—
154C2×Dic6—31no—
155C2×Dic6—32no—
156C2×Dic6—62no—
157C2×Dic6—62no—
158#41⋊C3—61no—
159C2×SL(2,𝔽3)—22no—
160C2×SL(2,𝔽3)—21no—
161C2×SL(2,𝔽3)—62no—
162C2×SL(2,𝔽3)—61no—
163C2×SL(2,𝔽3)D1212yescomplemented
164C2×SL(2,𝔽3)C22×D311yescharacteristic
165C3×C4⋊C4—32no—
166C3×C4⋊C4—32no—
167C3×C22⋊C4—31no—
168C3×C22⋊C4—41no—
169C3×C22⋊C4—81no—
170C3×C22⋊C4—122no—
171C3×C22⋊C4C2×A411yescharacteristic
172C3×C42—32no—
173C3×C42—32no—
174(C2×D3)⋊C4—34no—
175(C2×D3)⋊C4—32no—
176(C2×D3)⋊C4—32no—
177(C2×D3)⋊C4—32no—
178(C2×D3)⋊C4C2×A412yes—
179C4⋊Dic3—34no—
180C4⋊Dic3—32no—
181C4⋊Dic3—32no—
182C22×C32—42no—
183C22×C32—41no—
184C22×C32—82no—
185C2×C3×D3—44no—
186C2×C3×D3—82no—
187C2×C3×D3—82no—
188C2×C3×D3—88no—
189C2×C3×D3—82no—
190C2×C3×D3—81no—
191C32×C4—82no—
192C3×Dic3—81no—
1932−1+4—62no—
1942−1+4—182no—
195C2×#41—31no—
196C2×#41—91no—
197C2×#41—91no—
198C2×#41—91no—
199C22×Q8—32no—
200C22×Q8—31no—
201C22×Q8—62no—
202C22×Q8—91no—
203C22×Q8—91no—
204C22×Q8—92no—
205C22×Q8—182no—
206C22×Q8C2×C3×D311yescharacteristic
207C23×C4—91no—
208C4⋊Q8—92no—
209C4⋊Q8—92no—
210#124—92no—
211#124—92no—
212#123—91no—
213#123—92no—
214#123—91no—
215C22⋊Q8—92no—
216C22⋊Q8—94no—
217C22⋊Q8—92no—
218C22⋊Q8—92no—
219C22⋊Q8—92no—
220#121—92no—
221#121—92no—
222#120—31no—
223#120—91no—
224C4×Q8—32no—
225C4×Q8C2×C3×D312yescomplemented
226C4×D4—92no—
227C4×D4—92no—
228C4×D4—94no—
229#117—31no—
230#117—31no—
231C23×C3—41no—
232C23×C3—81no—
233C23×C3—122no—
234C23×C3—121no—
235C23×C3—242no—
236C23×C3—244no—
237C23×C3C22×A411yescharacteristic
238C22×D3—22no—
239C22×D3—24no—
240C22×D3—31no—
241C22×D3C22×A411yescharacteristic
242C22×D3C22×A412yes—
243C3×Q8—32no—
244C3×Q8—31no—
245C3×Q8—32no—
246C3×Q8C2×C3×D411yescharacteristic
247C3×D4—244no—
248C3×D4—242no—
249C2×C3×C4—32no—
250C2×C3×C4—32no—
251C2×C3×C4—31no—
252C2×C3×C4—31no—
253C2×C3×C4—32no—
254C2×C3×C4—42no—
255C2×C3×C4—82no—
256C2×C3×C4—121no—
257C2×C3×C4C22×A412yes—
258#67—22no—
259#67—62no—
260C2×Dic3—31no—
261C2×Dic3—32no—
262C2×Dic3—62no—
263C2×Dic3C22×A411yescharacteristic
264D12—24no—
265D12—62no—
266D12—61no—
267D12—61no—
268D12—64no—
269C4×D3—61no—
270C4×D3—61no—
271C4×D3—62no—
272C4×D3—64no—
273C4×D3—62no—
274C4×D3—61no—
275C4×D3—61no—
276C4×D3—64no—
277Dic6—62no—
278Dic6—64no—
279Dic6—62no—
280Dic6—62no—
281Dic6—61no—
282Dic6—61no—
283SL(2,𝔽3)—21no—
284SL(2,𝔽3)C2×D1211yescharacteristic
285C2×C32—41no—
286C2×C32—42no—
287C2×C32—82no—
288C3×D3—84no—
289C3×D3—161no—
290C24—31no—
291#41—61no—
292#41—182no—
293#41—182no—
294#41—182no—
295#41—182no—
296#41—182no—
297#41—182no—
298#41—181no—
299C2×Q8—31no—
300C2×Q8—32no—
301C2×Q8—62no—
302C2×Q8—61no—
303C2×Q8—91no—
304C2×Q8—92no—
305C2×Q8—92no—
306C2×Q8—91no—
307C2×Q8—182no—
308C2×Q8—182no—
309C2×Q8—182no—
310C2×Q8—184no—
311C2×Q8—182no—
312C2×Q8—182no—
313C2×Q8—181no—
314C2×Q8—184no—
315C2×Q8C3×D1212yescomplemented
316C2×Q8C22×C3×D311yescharacteristic
317C2×D4—31no—
318C2×D4—32no—
319C2×D4—92no—
320C2×D4—91no—
321C2×D4—91no—
322C2×D4—91no—
323C22×C4—31no—
324C22×C4—91no—
325C22×C4—92no—
326C22×C4—92no—
327C22×C4—91no—
328C22×C4—92no—
329C22×C4—91no—
330C22×C4—182no—
331C22×C4—182no—
332C4⋊C4—32no—
333C4⋊C4—32no—
334C4⋊C4—92no—
335C4⋊C4—94no—
336C4⋊C4—92no—
337C22⋊C4—31no—
338C22⋊C4—32no—
339C22⋊C4—92no—
340C22⋊C4—92no—
341C22⋊C4—94no—
342C22⋊C4—92no—
343C22⋊C4D3×A411yescharacteristic
344C42—32no—
345C42—32no—
346C22×C3—22no—
347C22×C3—42no—
348C22×C3—41no—
349C22×C3—82no—
350C22×C3—82no—
351C22×C3—81no—
352C22×C3—124no—
353C22×C3—162no—
354C22×C3—242no—
355C22×C3—242no—
356C22×C3—242no—
357C22×C3—241no—
358C22×C3—248no—
359C22×C3D4×A412yes—
360C22×C3C23×A411yescharacteristic
361C2×D3—28no—
362C2×D3—22no—
363C2×D3—22no—
364C2×D3—21no—
365C2×D3—22no—
366C2×D3—61no—
367C2×D3—62no—
368C2×D32−1+4⋊C314yescomplemented
369C3×C4—22no—
370C3×C4—32no—
371C3×C4—31no—
372C3×C4—31no—
373C3×C4—62no—
374C3×C4—82no—
375C3×C4—162no—
376C3×C4—241no—
377Dic3—21no—
378Dic3—62no—
379Dic3—61no—
380C32—41noSylow
381C23—32no—
382C23—31no—
383C23—64no—
384C23—62no—
385C23—91no—
386C23C2×D3×A411yescharacteristic
387Q8—32no—
388Q8—31no—
389Q8—32no—
390Q8—182no—
391Q8—182no—
392Q8—181no—
393Q8—182no—
394Q8—181no—
395Q8—184no—
396Q8C2×C3×D1211yescharacteristic
397D4—64no—
398D4—62no—
399D4—182no—
400D4—181no—
401D4—181no—
402D4—184no—
403D4—182no—
404C2×C4—32no—
405C2×C4—32no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—32no—
409C2×C4—31no—
410C2×C4—92no—
411C2×C4—91no—
412C2×C4—181no—
413C2×C4—181no—
414C2×C4—182no—
415C2×C4—182no—
416C2×C4—184no—
417C2×C4—181no—
418C2×C4—181no—
419C2×C4—184no—
420C2×C4—182no—
421C2×C4C2×D3×A412yes—
422C2×C3—22no—
423C2×C3—41no—
424C2×C3—42no—
425C2×C3—82no—
426C2×C3—81no—
427C2×C3—82no—
428C2×C3—162no—
429C2×C3—244no—
430C2×C3—481no—
431C2×C3C2×D4×A411yescharacteristic
432C2×C3C2×2−1+4⋊C312yes—
433D3—24no—
434D3—41no—
435D3—121no—
436C22—22no—
437C22—34no—
438C22—68no—
439C22—62no—
440C22—62no—
441C22—61no—
442C22—62no—
443C22—181no—
444C22—182no—
445C22A4×D1212yes—
446C22C22×D3×A411yescharacteristic
447C4—22no—
448C4—31no—
449C4—32no—
450C4—31no—
451C4—61no—
452C4—62no—
453C4—181no—
454C4—182no—
455C3—41no—
456C3—81no—
457C3#3486⋊C311yescharacteristic, complemented
458C2—22no—
459C2—64no—
460C2—121no—
461C2—361no—
462C2C2×A4×D1211yescharacteristic
463C2D3×2−1+4⋊C312yes—
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#519C2×C3
socleC22×C3C23×A4
hypercenterC22⋊C4D3×A4
hypocenterC3×Q8C2×C3×D4
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#1007⋊C3C2semidirect
2. C22×D3×SL(2,𝔽3)C2semidirect
3. C3×#519⋊C3C2semidirect
4. #5576⋊C3C2semidirect
5. SL(2,𝔽3)×D12C2semidirect
6. #85881C3semidirect
12. C2×D3×SL(2,𝔽3)C22semidirect
13. C2×C3⋊(#41⋊C3)C22semidirect
15. C3×C4×SL(2,𝔽3)C22semidirect
16. C2×#1007C2×C3semidirect
17. C22×D3×Q8C2×C3semidirect
20. C3×#519C2×C3semidirect
22. #5576C2×C3semidirect
24. Q8×D12C2×C3semidirect
32. #519⋊C3D3semidirect
36. C2×C3×SL(2,𝔽3)D4semidirect
48. C2×#239C22×C3semidirect
54. C2×D3×Q8C22×C3semidirect
68. C3×C4×Q8C22×C3semidirect
91. C4×SL(2,𝔽3)C2×D3semidirect
112. #519C3×D3semidirect
117. C2×C3×Q8C3×D4semidirect
163. C2×SL(2,𝔽3)D12semidirect
225. C4×Q8C2×C3×D3semidirect
315. C2×Q8C3×D12semidirect
368. C2×D32−1+4⋊C3semidirect
457. C3#3486⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×#1007⋊C32
C22×D3×SL(2,𝔽3)2
C3×#519⋊C32
#5576⋊C32
SL(2,𝔽3)×D122
#858813
#3486⋊C33
C3×#8824