Finite Group Databasebeta

← #87032⋊C3 #11923277

Subgroups — #87032⋊C3 #11923277

Order 1152.

527 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 527 conjugacy classes of subgroups — too many to draw. Only its 83 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×#557⋊C3C211yesmaximal, characteristic, complemented
2#5572⋊C3C211yesmaximal, characteristic
3#5572⋊C3C211yesmaximal, characteristic
4C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic
5#5569⋊C3C211yesmaximal, characteristic, complemented
6#5569⋊C3C211yesmaximal, characteristic, complemented
7SL(2,𝔽3)×Dic6C211yesmaximal, characteristic, complemented
8#87032C311yesmaximal, characteristic, complemented
9#3502⋊C3—31nomaximal
10C2×D3×SL(2,𝔽3)C2211yescharacteristic
11C3×#896—41nomaximal
12C3×C4×SL(2,𝔽3)C2211yescharacteristic
13C3×C4×SL(2,𝔽3)C2211yescharacteristic
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15Dic3×SL(2,𝔽3)C2211yescharacteristic
16Dic3×SL(2,𝔽3)C2211yescharacteristic
17Dic3×SL(2,𝔽3)C2211yescharacteristic
18C3×#557C2×C311yescharacteristic, complemented
19#5729—31no—
20#5722—31no—
21#5689—31no—
22#5688—31no—
23#5685—31no—
24#5685—31no—
25#5578—31no—
26#5572C2×C311yescharacteristic
27#5572C2×C311yescharacteristic
28C4×D3×Q8C2×C311yescharacteristic
29#5569C2×C311yescharacteristic, complemented
30#5569C2×C311yescharacteristic, complemented
31#5567—31no—
32Q8×Dic6C2×C311yescharacteristic, complemented
33Q8×SL(2,𝔽3)—31no—
34#557⋊C3—21no—
35#557⋊C3—31no—
36#557⋊C3—31no—
37#557⋊C3D311yescharacteristic, complemented
38#519⋊C3—31no—
39#519⋊C3—31no—
40C2×C4×SL(2,𝔽3)—31no—
41C2×C3×C4×D3—41no—
42C2×C3×Dic6—41no—
43C2×C3×SL(2,𝔽3)C2311yescharacteristic
44D3×SL(2,𝔽3)Q812yescomplemented
45C32×C4⋊C4—41no—
46C3×(C2×D3)⋊C4—41no—
47C3×(C2×D3)⋊C4—41no—
48C3×Dic3⋊C4—41no—
49C3×Dic3⋊C4—41no—
50#3502—31noSylow
51C2×D3×Q8C22×C311yescharacteristic
52C22×SL(2,𝔽3)—31no—
53C3×C4⋊Q8—31no—
54C3×#125—31no—
55C3×#125—31no—
56C3×C22⋊Q8—121no—
57C3×C4×Q8—31no—
58C3×C4×Q8C22×C311yescharacteristic
59C3×C4×Q8C22×C311yescharacteristic
60C3×C4×Q8C22×C311yescharacteristic
61#946—31no—
62Q8×Dic3C22×C311yescharacteristic
63Q8×Dic3C22×C311yescharacteristic
64Q8×Dic3C22×C311yescharacteristic
65#944—32no—
66#944—31no—
67#897—31no—
68#896—32no—
69#896—31no—
70#896A411yescharacteristic
71#892—31no—
72#892—31no—
73D3×C4⋊C4—31no—
74#890—31no—
75#890—31no—
76#889—32no—
77#889—32no—
78#889—31no—
79#889—31no—
80#888—31no—
81#888—31no—
82#888—32no—
83#888—31no—
84#888—31no—
85#888—31no—
86#887—32no—
87#887—32no—
88#887—31no—
89#887—31no—
90#887—31no—
91#872—31no—
92#872—31no—
93C42×D3—31no—
94#870—31no—
95#870—31no—
96#869—31no—
97C4×Dic6—31no—
98C4×Dic6—31no—
99C4×Dic6—31no—
100C4×Dic6—31no—
101C4×Dic6—31no—
102C4×Dic6—31no—
103C4×SL(2,𝔽3)—21no—
104C4×SL(2,𝔽3)—21no—
105C4×SL(2,𝔽3)—21no—
106C4×SL(2,𝔽3)—31no—
107C4×SL(2,𝔽3)—31no—
108C4×SL(2,𝔽3)—31no—
109C4×SL(2,𝔽3)C2×D311yescharacteristic
110C4×SL(2,𝔽3)C2×D311yescharacteristic
111C4×SL(2,𝔽3)C2×D311yescharacteristic
112C22×C3×D3—41no—
113C2×C32×C4—41no—
114C2×C32×C4—41no—
115C2×C32×C4—41no—
116C2×C3×Dic3—41no—
117C2×C3×Dic3—41no—
118C2×C3×Dic3—41no—
119C3×C4×D3—82no—
120C3×Dic6—82no—
121C3×SL(2,𝔽3)C2×Q811yescharacteristic
122Q82—31no—
123Q82—91no—
124#557—31no—
125#557—31no—
126#557—91no—
127#557—91no—
128#557—91no—
129#557C3×D311yescharacteristic, complemented
130#544—91no—
131#544—91no—
132#533—91no—
133#533—91no—
134#519—31no—
135#519—31no—
136C2×C4×Q8—31no—
137C2×C3×Q8—31no—
138C2×C3×Q8—121no—
139C2×C3×Q8C23×C311yescharacteristic
140C22×C3×C4—121no—
141D3×Q8—32no—
142D3×Q8C3×Q812yescomplemented
143C2×C4×D3—31no—
144C2×C4×D3—31no—
145C2×C4×D3C2×A411yescharacteristic
146C2×Dic6—31no—
147C2×Dic6—31no—
148C2×Dic6—32no—
149C2×Dic6—31no—
150C2×Dic6C2×A411yescharacteristic
151C2×SL(2,𝔽3)—21no—
152C2×SL(2,𝔽3)—32no—
153C2×SL(2,𝔽3)C22×D311yescharacteristic
154C3×C4⋊C4—31no—
155C3×C4⋊C4—31no—
156C3×C4⋊C4—31no—
157C3×C4⋊C4—32no—
158C3×C4⋊C4—31no—
159C3×C4⋊C4—31no—
160C3×C4⋊C4—41no—
161C3×C4⋊C4—81no—
162C3×C4⋊C4—121no—
163C3×C4⋊C4—121no—
164C3×C4⋊C4C2×A411yescharacteristic
165C3×C22⋊C4—121no—
166C3×C22⋊C4—121no—
167C3×C42—31no—
168C3×C42—31no—
169C3×C42—31no—
170(C2×D3)⋊C4—31no—
171(C2×D3)⋊C4—31no—
172(C2×D3)⋊C4C2×A411yescharacteristic
173(C2×D3)⋊C4C2×A411yescharacteristic
174C4⋊Dic3—31no—
175C4⋊Dic3—31no—
176C4⋊Dic3—31no—
177C4⋊Dic3—31no—
178C4⋊Dic3—31no—
179C4⋊Dic3—31no—
180Dic3⋊C4—32no—
181Dic3⋊C4—32no—
182Dic3⋊C4—32no—
183Dic3⋊C4—31no—
184Dic3⋊C4—31no—
185Dic3⋊C4—31no—
186Dic3⋊C4—31no—
187Dic3⋊C4—31no—
188Dic3⋊C4—31no—
189Dic3⋊C4—31no—
190Dic3⋊C4—31no—
191Dic3⋊C4C2×A411yescharacteristic
192Dic3⋊C4C2×A411yescharacteristic
193C4×Dic3—32no—
194C4×Dic3—32no—
195C4×Dic3—31no—
196C4×Dic3—31no—
197C4×Dic3—31no—
198C4×Dic3—31no—
199C4×Dic3—31no—
200C22×C32—41no—
201C2×C3×D3—42no—
202C2×C3×D3—81no—
203C2×C3×D3—81no—
204C32×C4—81no—
205C32×C4—81no—
206C32×C4—81no—
207C3×Dic3—42no—
208C3×Dic3—81no—
209C3×Dic3—81no—
210C22×Q8—31no—
211C4⋊Q8—31no—
212C4⋊Q8—91no—
213C4⋊Q8—91no—
214C4⋊Q8—91no—
215C4⋊Q8—91no—
216C4⋊Q8—91no—
217C4⋊Q8—92no—
218C4⋊Q8—91no—
219C4⋊Q8—91no—
220C4⋊Q8—92no—
221#125—31no—
222#125—31no—
223#125—91no—
224#125—91no—
225#125—91no—
226#125—91no—
227#125—91no—
228#125—92no—
229#125—92no—
230#125—91no—
231C22⋊Q8—31no—
232C22⋊Q8—91no—
233C22⋊Q8—91no—
234C22⋊Q8—91no—
235C22⋊Q8—92no—
236C4×Q8—31no—
237C4×Q8—31no—
238C4×Q8—31no—
239C4×Q8—31no—
240C4×Q8—91no—
241C4×Q8—91no—
242C4×Q8—91no—
243C4×Q8—91no—
244C4×Q8—91no—
245C4×Q8—91no—
246C4×Q8—92no—
247C4×Q8—92no—
248C4×Q8—91no—
249C4×Q8—91no—
250C4×Q8—91no—
251C4×Q8C2×C3×D311yescharacteristic
252C4×Q8C2×C3×D311yescharacteristic
253C4×Q8C2×C3×D311yescharacteristic
254#117—91no—
255#117—91no—
256#117—91no—
257#117—91no—
258C2×C4⋊C4—91no—
259C2×C42—91no—
260C23×C3—121no—
261C22×D3C22×A411yescharacteristic
262C3×Q8—31no—
263C3×Q8—121no—
264C3×Q8—242no—
265C3×Q8C2×C3×Q811yescharacteristic
266C2×C3×C4—31no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—31no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—81no—
274C2×C3×C4—81no—
275C2×C3×C4—81no—
276C2×C3×C4—121no—
277C2×C3×C4—121no—
278C2×C3×C4—121no—
279C2×C3×C4—242no—
280C2×C3×C4C22×A411yescharacteristic
281C2×C3×C4C22×A411yescharacteristic
282C2×C3×C4C22×A411yescharacteristic
283C2×Dic3—31no—
284C2×Dic3—31no—
285C2×Dic3—31no—
286C2×Dic3—31no—
287C2×Dic3C22×A411yescharacteristic
288C2×Dic3C22×A411yescharacteristic
289C2×Dic3C22×A411yescharacteristic
290C4×D3—22no—
291C4×D3—32no—
292C4×D3—32no—
293C4×D3—121no—
294Dic6—22no—
295Dic6—32no—
296Dic6—32no—
297Dic6—62no—
298Dic6—121no—
299Dic6—122no—
300SL(2,𝔽3)—21no—
301SL(2,𝔽3)D3×Q811yescharacteristic
302C2×C32—41no—
303C2×C32—41no—
304C2×C32—41no—
305C3×D3—82no—
306C2×Q8—31no—
307C2×Q8—31no—
308C2×Q8—32no—
309C2×Q8—91no—
310C2×Q8—91no—
311C2×Q8—92no—
312C2×Q8—91no—
313C2×Q8—92no—
314C2×Q8C22×C3×D311yescharacteristic
315C22×C4—31no—
316C22×C4—91no—
317C22×C4—91no—
318C4⋊C4—31no—
319C4⋊C4—31no—
320C4⋊C4—31no—
321C4⋊C4—32no—
322C4⋊C4—31no—
323C4⋊C4—31no—
324C4⋊C4—31no—
325C4⋊C4—31no—
326C4⋊C4—91no—
327C4⋊C4—91no—
328C4⋊C4—91no—
329C4⋊C4—91no—
330C4⋊C4—91no—
331C4⋊C4—91no—
332C4⋊C4—91no—
333C4⋊C4—91no—
334C4⋊C4—91no—
335C4⋊C4—92no—
336C4⋊C4—92no—
337C4⋊C4—92no—
338C4⋊C4—91no—
339C4⋊C4—91no—
340C4⋊C4—91no—
341C4⋊C4—91no—
342C4⋊C4—91no—
343C4⋊C4D3×A411yescharacteristic
344C22⋊C4—31no—
345C22⋊C4—31no—
346C22⋊C4—91no—
347C22⋊C4—91no—
348C42—31no—
349C42—31no—
350C42—31no—
351C42—91no—
352C42—91no—
353C42—91no—
354C42—91no—
355C42—92no—
356C42—92no—
357C42—91no—
358C22×C3—41no—
359C22×C3—81no—
360C22×C3—122no—
361C22×C3—241no—
362C22×C3—241no—
363C22×C3C23×A411yescharacteristic
364C2×D3—21no—
365C2×D3—21no—
366C2×D3Q8×A412yes—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—31no—
371C3×C4—31no—
372C3×C4—61no—
373C3×C4—61no—
374C3×C4—61no—
375C3×C4—81no—
376C3×C4—81no—
377C3×C4—81no—
378C3×C4—82no—
379C3×C4—122no—
380C3×C4—161no—
381C3×C4—161no—
382C3×C4—241no—
383C3×C4—241no—
384Dic3—21no—
385Dic3—21no—
386Dic3—32no—
387Dic3—61no—
388Dic3—61no—
389Dic3—61no—
390Dic32−1+4⋊C312yes—
391C32—41noSylow
392C23—31no—
393Q8—31no—
394Q8—62no—
395Q8—92no—
396Q8—92no—
397Q8—121no—
398Q8—182no—
399Q8—361no—
400Q8—362no—
401Q8C3×D3×Q811yescharacteristic
402C2×C4—31no—
403C2×C4—31no—
404C2×C4—31no—
405C2×C4—31no—
406C2×C4—31no—
407C2×C4—31no—
408C2×C4—31no—
409C2×C4—62no—
410C2×C4—92no—
411C2×C4—92no—
412C2×C4—91no—
413C2×C4—91no—
414C2×C4—91no—
415C2×C4—91no—
416C2×C4—361no—
417C2×C4C2×D3×A411yescharacteristic
418C2×C4C2×D3×A411yescharacteristic
419C2×C4C2×D3×A411yescharacteristic
420C2×C3—41no—
421C2×C3—41no—
422C2×C3—41no—
423C2×C3—81no—
424C2×C3—81no—
425C2×C3—81no—
426C2×C3—242no—
427C2×C3C2×Q8×A411yescharacteristic
428C2×C3C2×2−1+4⋊C311yescharacteristic
429C2×C3#594511yescharacteristic
430D3—22no—
431C22—32no—
432C22—61no—
433C22—61no—
434C22C22×D3×A411yescharacteristic
435C4—21no—
436C4—21no—
437C4—21no—
438C4—31no—
439C4—31no—
440C4—32no—
441C4—61no—
442C4—61no—
443C4—61no—
444C4—61no—
445C4—61no—
446C4—92no—
447C4—181no—
448C4—181no—
449C4—181no—
450C3—41no—
451C3—81no—
452C3#3502⋊C311yescharacteristic, complemented
453C2—62no—
454C2D3×Q8×A411yescharacteristic
455C2#5964⋊C311yescharacteristic
456C2D3×2−1+4⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#557C2×C3
socleC22×C3C23×A4
prime radicalC2×D3×SL(2,𝔽3)C22
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C2×C3×Q8
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
1. C3×#557⋊C3C2semidirect
5. #5569⋊C3C2semidirect
6. #5569⋊C3C2semidirect
7. SL(2,𝔽3)×Dic6C2semidirect
8. #87032C3semidirect
18. C3×#557C2×C3semidirect
29. #5569C2×C3semidirect
30. #5569C2×C3semidirect
32. Q8×Dic6C2×C3semidirect
37. #557⋊C3D3semidirect
44. D3×SL(2,𝔽3)Q8semidirect
129. #557C3×D3semidirect
142. D3×Q8C3×Q8semidirect
452. C3#3502⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
#5572⋊C32
C4×D3×SL(2,𝔽3)2
#5569⋊C32
SL(2,𝔽3)×Dic62
#870323
#3502⋊C33
C3×#8964