Finite Group Databasebeta

← #85896⋊C3 #11923221

Subgroups — #85896⋊C3 #11923221

Order 1152.

555 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 555 conjugacy classes of subgroups — too many to draw. Only its 77 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×#520⋊C3C211yesmaximal, characteristic, complemented
2#5834⋊C3C211yesmaximal, characteristic, complemented
3#5834⋊C3C211yesmaximal, characteristic, complemented
4C4⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
5#5569⋊C3C211yesmaximal, characteristic, complemented
6#5574⋊C3C211yesmaximal, characteristic, complemented
7C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
8#85896C311yesmaximal, characteristic, complemented
9#3517⋊C3—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C2×C3⋊(#41⋊C3)C2211yescharacteristic
12C2×C3⋊(#41⋊C3)C2211yescharacteristic
13C3×#889—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3×C4×SL(2,𝔽3)C2211yescharacteristic
16Dic3×SL(2,𝔽3)C2211yescharacteristic
17Dic3×SL(2,𝔽3)C2211yescharacteristic
18C3×#520C2×C311yescharacteristic
19#5834C2×C311yescharacteristic
20#5834C2×C311yescharacteristic
21#5577C2×C311yescharacteristic
22#5574C2×C311yescharacteristic
23C4×#239C2×C311yescharacteristic
24#5569C2×C311yescharacteristic
25#5562—31no—
26#5558—31no—
27#5553—31no—
28#5548—31no—
29#5535—31no—
30#5528—31no—
31#5514—31no—
32#5507—31no—
33#557⋊C3—31no—
34C4⋊(#41⋊C3)—31no—
35C4⋊(#41⋊C3)—31no—
36C4⋊(#41⋊C3)—31no—
37#520⋊C3—21no—
38#520⋊C3—31no—
39#520⋊C3D311yescharacteristic, complemented
40C4×#41⋊C3—31no—
41C3×#41⋊C3—21no—
42C2×C3×SL(2,𝔽3)C2311yescharacteristic
43C3⋊(#41⋊C3)—21no—
44C3⋊(#41⋊C3)—21no—
45C32×C4⋊C4—41no—
46C3×C4⋊Dic3—41no—
47C3×Dic3⋊C4—41no—
48C3×Dic3⋊C4—41no—
49C3×Dic3⋊C4—41no—
50C3×Dic3⋊C4—41no—
51C3×C4×Dic3—41no—
52#3517—31noSylow
53C2×C3×#41C22×C311yescharacteristic
54C2×#239C22×C311yescharacteristic
55C2×#239C22×C311yescharacteristic
56C2×#235—32no—
57C2×#41⋊C3—21no—
58C2×#41⋊C3—31no—
59C2×#41⋊C3—31no—
60C2×#41⋊C3C2×D311yescharacteristic
61C3×#125—121no—
62C3×C4×Q8C22×C311yescharacteristic
63C3×C4×Q8C22×C311yescharacteristic
64C3×C4×D4—31no—
65C3×C4×D4—31no—
66C3×#117—31no—
67C2×C3×C4⋊C4—31no—
68Q8×Dic3C22×C311yescharacteristic
69Q8×Dic3C22×C311yescharacteristic
70#940—31no—
71#940—31no—
72#938—31no—
73#938—31no—
74#930—32no—
75#930—31no—
76#930—31no—
77#929—32no—
78C4×#67—32no—
79C4×#67—31no—
80C4×#67—31no—
81#898—31no—
82#896—31no—
83#895—31no—
84#895—31no—
85#894—32no—
86#894—31no—
87#893—31no—
88#893—31no—
89#892—31no—
90D3×C4⋊C4—31no—
91#890—31no—
92#889A411yescharacteristic
93#888—31no—
94#885—31no—
95#884—32no—
96#884—32no—
97#884—31no—
98#883—31no—
99#879—31no—
100#874—31no—
101C4×D12—31no—
102C4×D12—31no—
103C4×D12—31no—
104#872—31no—
105C42×D3—31no—
106#870—31no—
107C4×Dic6—31no—
108C4×SL(2,𝔽3)—21no—
109C4×SL(2,𝔽3)—21no—
110C4×SL(2,𝔽3)—31no—
111C4×SL(2,𝔽3)—31no—
112C4×SL(2,𝔽3)C2×D311yescharacteristic
113C4×SL(2,𝔽3)C2×D311yescharacteristic
114C2×C32×C4—41no—
115C2×C32×C4—41no—
116C2×C32×C4—41no—
117C2×C3×Dic3—41no—
118C2×C3×Dic3—41no—
119C2×C3×Dic3—41no—
120C2×C3×Dic3—41no—
121C3×SL(2,𝔽3)#4111yescharacteristic, complemented
122#557—31no—
123#553—91no—
124#553—91no—
125#550—31no—
126#550—31no—
127#550—31no—
128#550—91no—
129#547—91no—
130#540—91no—
131#540—91no—
132#532—91no—
133#529—91no—
134#520—31no—
135#520C3×D311yescharacteristic
136C4×#41—31no—
137C3×#41—21no—
138C3×#41—61no—
139C2×C3×Q8C23×C311yescharacteristic
140C2×C3×D4—31no—
141C22×C3×C4—32no—
142C22×C3×C4—31no—
143C2×#67—32no—
144C2×#67—32no—
145C2×#67—31no—
146C2×#67—31no—
147#239—21no—
148#239—21no—
149#239—61no—
150#239—61no—
151#235—124no—
152C2×D12—31no—
153C2×D12—31no—
154C2×D12—32no—
155C2×D12—31no—
156C2×C4×D3—31no—
157C2×C4×D3—31no—
158C2×C4×D3—31no—
159C2×C4×D3—32no—
160C2×C4×D3—31no—
161C2×Dic6—31no—
162#41⋊C3—22no—
163#41⋊C3—21no—
164#41⋊C3—61no—
165#41⋊C3—61no—
166C2×SL(2,𝔽3)—21no—
167C2×SL(2,𝔽3)C22×D311yescharacteristic
168C3×C4⋊C4—31no—
169C3×C4⋊C4—31no—
170C3×C4⋊C4—31no—
171C3×C4⋊C4—41no—
172C3×C4⋊C4—81no—
173C3×C4⋊C4—121no—
174C3×C4⋊C4—121no—
175C3×C4⋊C4—121no—
176C3×C4⋊C4—121no—
177C3×C4⋊C4—121no—
178C3×C4⋊C4C2×A411yescharacteristic
179C3×C22⋊C4—31no—
180C3×C22⋊C4—31no—
181C3×C42—31no—
182C3×C42—31no—
183C3×C42—121no—
184#217—31no—
185#217—31no—
186(C2×D3)⋊C4—32no—
187(C2×D3)⋊C4—31no—
188(C2×D3)⋊C4—31no—
189(C2×D3)⋊C4—31no—
190(C2×D3)⋊C4—31no—
191(C2×D3)⋊C4—31no—
192(C2×D3)⋊C4—31no—
193C4⋊Dic3—31no—
194C4⋊Dic3—31no—
195C4⋊Dic3—31no—
196C4⋊Dic3C2×A411yescharacteristic
197Dic3⋊C4—32no—
198Dic3⋊C4—31no—
199Dic3⋊C4—31no—
200Dic3⋊C4C2×A411yescharacteristic
201Dic3⋊C4C2×A411yescharacteristic
202Dic3⋊C4C2×A411yescharacteristic
203Dic3⋊C4C2×A411yescharacteristic
204C4×Dic3—31no—
205C4×Dic3—31no—
206C4×Dic3—31no—
207C4×Dic3C2×A411yescharacteristic
208C22×C32—41no—
209C32×C4—81no—
210C32×C4—81no—
211C32×C4—81no—
212C3×Dic3—42no—
213C3×Dic3—81no—
214C3×Dic3—81no—
215C3×Dic3—81no—
216C2×#41—31no—
217C2×#41—31no—
218C2×#41—92no—
219C2×#41C2×C3×D311yescharacteristic
220C4⋊Q8—91no—
221#127—91no—
222#127—91no—
223#127—91no—
224#126—91no—
225#126—91no—
226#125—31no—
227#125—91no—
228#125—91no—
229#124—91no—
230#123—91no—
231#123—92no—
232#123—92no—
233#123—91no—
234C22⋊Q8—91no—
235#121—91no—
236#121—92no—
237#121—92no—
238#121—92no—
239#121—91no—
240#121—91no—
241#121—91no—
242#121—91no—
243#121—91no—
244#121—91no—
245C4×Q8—31no—
246C4×Q8—31no—
247C4×Q8—91no—
248C4×Q8C2×C3×D311yescharacteristic
249C4×Q8C2×C3×D311yescharacteristic
250C4×D4—31no—
251C4×D4—31no—
252C4×D4—91no—
253C4×D4—91no—
254C4×D4—91no—
255C4×D4—91no—
256C4×D4—91no—
257C4×D4—91no—
258C4×D4—92no—
259C4×D4—91no—
260#117—31no—
261#117—91no—
262#117—91no—
263C2×C4⋊C4—31no—
264C2×C4⋊C4—91no—
265C2×C42—91no—
266C23×C3—31no—
267C22×D3—31no—
268C22×D3—31no—
269C3×Q8—31no—
270C3×Q8C3×#4111yescharacteristic
271C3×D4—61no—
272C3×D4—61no—
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—61no—
280C2×C3×C4—61no—
281C2×C3×C4—81no—
282C2×C3×C4—81no—
283C2×C3×C4—81no—
284C2×C3×C4—121no—
285C2×C3×C4—121no—
286C2×C3×C4—121no—
287C2×C3×C4—121no—
288C2×C3×C4—122no—
289C2×C3×C4C22×A411yescharacteristic
290C2×C3×C4C22×A411yescharacteristic
291C2×C3×C4C22×A411yescharacteristic
292#67—62no—
293#67—121no—
294#67—122no—
295#67—122no—
296C2×Dic3—31no—
297C2×Dic3—31no—
298C2×Dic3C22×A411yescharacteristic
299C2×Dic3C22×A411yescharacteristic
300C2×Dic3C22×A411yescharacteristic
301C2×Dic3C22×A411yescharacteristic
302D12—61no—
303D12—61no—
304D12—61no—
305D12—61no—
306D12—121no—
307D12—122no—
308C4×D3—61no—
309C4×D3—61no—
310C4×D3—61no—
311C4×D3—61no—
312C4×D3—62no—
313C4×D3—121no—
314C4×D3—122no—
315Dic6—62no—
316SL(2,𝔽3)—21no—
317SL(2,𝔽3)#23511yescharacteristic, complemented
318C2×C32—41no—
319C2×C32—41no—
320C2×C32—41no—
321#41—21no—
322#41—61no—
323#41—61no—
324#41—61no—
325#41—181no—
326#41—181no—
327#41—364no—
328C2×Q8—91no—
329C2×Q8C22×C3×D311yescharacteristic
330C2×D4—31no—
331C2×D4—91no—
332C2×D4—91no—
333C2×D4—91no—
334C2×D4—91no—
335C2×D4—92no—
336C2×D4—92no—
337C2×D4—92no—
338C2×D4—91no—
339C22×C4—32no—
340C22×C4—31no—
341C22×C4—91no—
342C22×C4—91no—
343C22×C4—91no—
344C22×C4—92no—
345C22×C4—91no—
346C4⋊C4—31no—
347C4⋊C4—31no—
348C4⋊C4—31no—
349C4⋊C4—31no—
350C4⋊C4—31no—
351C4⋊C4—31no—
352C4⋊C4—31no—
353C4⋊C4—31no—
354C4⋊C4—91no—
355C4⋊C4—91no—
356C4⋊C4—91no—
357C4⋊C4—91no—
358C4⋊C4—92no—
359C4⋊C4—91no—
360C4⋊C4D3×A411yescharacteristic
361C22⋊C4—31no—
362C22⋊C4—31no—
363C22⋊C4—91no—
364C22⋊C4—91no—
365C22⋊C4—91no—
366C22⋊C4—91no—
367C22⋊C4—91no—
368C22⋊C4—92no—
369C22⋊C4—91no—
370C22⋊C4—91no—
371C22⋊C4—91no—
372C42—31no—
373C42—31no—
374C42—31no—
375C42—91no—
376C42—91no—
377C42—91no—
378C22×C3—41no—
379C22×C3—61no—
380C22×C3—61no—
381C22×C3—61no—
382C22×C3—81no—
383C22×C3C23×A411yescharacteristic
384C2×D3—61no—
385C2×D3—61no—
386C2×D3—61no—
387C2×D3—61no—
388C2×D3—61no—
389C2×D3—61no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—31no—
394C3×C4—31no—
395C3×C4—61no—
396C3×C4—61no—
397C3×C4—81no—
398C3×C4—81no—
399C3×C4—81no—
400C3×C4—82no—
401C3×C4—122no—
402C3×C4—161no—
403C3×C4—161no—
404C3×C4—241no—
405C3×C4—241no—
406C3×C4—241no—
407Dic3—21no—
408Dic3—21no—
409Dic3—21no—
410Dic3—61no—
411Dic3—61no—
412Dic3#99412yes—
413C32—41noSylow
414C23—31no—
415C23—91no—
416C23—91no—
417Q8—31no—
418Q8—182no—
419Q8C3×#23511yescharacteristic
420D4—61no—
421D4—61no—
422D4—182no—
423D4—181no—
424D4—181no—
425D4—181no—
426D4—181no—
427D4—361no—
428D4—361no—
429D4—362no—
430D4—362no—
431D4—362no—
432C2×C4—31no—
433C2×C4—31no—
434C2×C4—31no—
435C2×C4—31no—
436C2×C4—31no—
437C2×C4—31no—
438C2×C4—31no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—91no—
442C2×C4—91no—
443C2×C4—122no—
444C2×C4—181no—
445C2×C4—181no—
446C2×C4—181no—
447C2×C4—181no—
448C2×C4—182no—
449C2×C4—361no—
450C2×C4—362no—
451C2×C4C2×D3×A411yescharacteristic
452C2×C4C2×D3×A411yescharacteristic
453C2×C4C2×D3×A411yescharacteristic
454C2×C3—41no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3—81no—
458C2×C3—81no—
459C2×C3—81no—
460C2×C3—121no—
461C2×C3A4×#4111yescharacteristic
462C2×C3C2×#99411yescharacteristic
463C2×C3#594511yescharacteristic
464D3—121no—
465D3—121no—
466C22—61no—
467C22—61no—
468C22—61no—
469C22—181no—
470C22—181no—
471C22—181no—
472C22—181no—
473C22—181no—
474C22—181no—
475C22C22×D3×A411yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—31no—
480C4—31no—
481C4—32no—
482C4—61no—
483C4—61no—
484C4—61no—
485C4—61no—
486C4—61no—
487C4—181no—
488C4—181no—
489C3—41no—
490C3—81no—
491C3#3517⋊C311yescharacteristic, complemented
492C2—121no—
493C2—361no—
494C2—361no—
495C2A4×#23511yescharacteristic
496C2D3×#99411yescharacteristic
497C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#520C2×C3
socleC22×C3C23×A4
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C3×#41
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×#520⋊C3C2semidirect
2. #5834⋊C3C2semidirect
3. #5834⋊C3C2semidirect
4. C4⋊(C3⋊(#41⋊C3))C2semidirect
5. #5569⋊C3C2semidirect
6. #5574⋊C3C2semidirect
7. C4×C3⋊(#41⋊C3)C2semidirect
8. #85896C3semidirect
39. #520⋊C3D3semidirect
121. C3×SL(2,𝔽3)#41semidirect
317. SL(2,𝔽3)#235semidirect
491. C3#3517⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#520⋊C32
#5834⋊C32
C4⋊(C3⋊(#41⋊C3))2
#5569⋊C32
#5574⋊C32
C4×C3⋊(#41⋊C3)2
#858963
#3517⋊C33
C3×#8894