Finite Group Databasebeta

← #85879⋊C3 #11923211

Subgroups — #85879⋊C3 #11923211

Order 1152.

535 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 535 conjugacy classes of subgroups — too many to draw. Only its 75 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×#519⋊C3C211yesmaximal, characteristic, complemented
2#5817⋊C3C211yesmaximal, characteristic, complemented
3#5817⋊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#85879C311yesmaximal, characteristic, complemented
9#3524⋊C3—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C2×C3⋊(#41⋊C3)C2211yescharacteristic
12C2×C3⋊(#41⋊C3)C2211yescharacteristic
13C3×#879—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3×C4×SL(2,𝔽3)C2211yescharacteristic
16Dic3×SL(2,𝔽3)C2211yescharacteristic
17Dic3×SL(2,𝔽3)C2211yescharacteristic
18C3×#519C2×C311yescharacteristic
19#5817C2×C311yescharacteristic
20#5817C2×C311yescharacteristic
21#5585—31no—
22#5582—31no—
23#5581—31no—
24#5580—31no—
25#5577C2×C311yescharacteristic, complemented
26#5574C2×C311yescharacteristic, complemented
27C4×#239C2×C311yescharacteristic, complemented
28#5569C2×C311yescharacteristic, complemented
29#5535—31no—
30#5532—31no—
31#5530—31no—
32#5528—31no—
33#557⋊C3—31no—
34C4⋊(#41⋊C3)—31no—
35#548⋊C3—31no—
36#548⋊C3—31no—
37#520⋊C3—31no—
38#519⋊C3—21no—
39#519⋊C3D311yescharacteristic, complemented
40C4×#41⋊C3—31no—
41C2×C3×SL(2,𝔽3)—21no—
42C2×C3×SL(2,𝔽3)C2311yescharacteristic
43C3⋊(#41⋊C3)—21no—
44C3⋊(#41⋊C3)—21no—
45C32×C22⋊C4—41no—
46C3×#217—41no—
47C3×#217—41no—
48C3×C4⋊Dic3—41no—
49C3×Dic3⋊C4—41no—
50C3×Dic3⋊C4—41no—
51C3×C4×Dic3—41no—
52#3524—31noSylow
53C22×C3×Q8C22×C311yescharacteristic
54C2×#239C22×C311yescharacteristic
55C2×#239C22×C311yescharacteristic
56C2×#41⋊C3—31no—
57C2×#41⋊C3—31no—
58C22×SL(2,𝔽3)—21no—
59C22×SL(2,𝔽3)C2×D311yescharacteristic
60C3×#126—121no—
61C3×C4×Q8—31no—
62C3×C4×Q8—31no—
63C3×C4×Q8C22×C311yescharacteristic
64C3×C4×Q8C22×C311yescharacteristic
65C3×#117—31no—
66C3×#117—31no—
67#947—31no—
68#947—31no—
69#946—31no—
70#946—31no—
71Q8×Dic3C22×C311yescharacteristic
72Q8×Dic3C22×C311yescharacteristic
73#930—31no—
74#930—31no—
75C4×#67—31no—
76C4×#67—31no—
77#898—31no—
78#896—31no—
79#895—31no—
80#895—31no—
81#894—32no—
82#894—32no—
83#894—31no—
84#893—31no—
85#893—31no—
86#892—31no—
87D3×C4⋊C4—31no—
88#890—31no—
89#889—31no—
90#888—31no—
91#885—31no—
92#884—32no—
93#884—31no—
94#883—31no—
95#879A411yescharacteristic
96#876—32no—
97#875—32no—
98#874—31no—
99C4×D12—32no—
100C4×D12—31no—
101C4×D12—31no—
102C4×D12—31no—
103#872—32no—
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
114C23×C32—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
122#559—91no—
123#557—31no—
124#555—91no—
125#553—91no—
126#552—91no—
127#550—31no—
128#548—31no—
129#548—31no—
130#542—91no—
131#540—91no—
132#529—91no—
133#529—91no—
134#520—31no—
135#519C3×D311yescharacteristic
136C4×#41—31no—
137C2×C3×Q8—21no—
138C2×C3×Q8—31no—
139C2×C3×Q8—61no—
140C2×C3×Q8C23×C311yescharacteristic
141C22×C3×C4—31no—
142C2×#67—31no—
143C2×#67—31no—
144#239—21no—
145#239—21no—
146#239—61no—
147#239—61no—
148C2×D12—31no—
149C2×D12—31no—
150C2×D12—32no—
151C2×D12—31no—
152C2×C4×D3—31no—
153C2×C4×D3—31no—
154C2×C4×D3—32no—
155C2×C4×D3—31no—
156C2×C4×D3—31no—
157C2×Dic6—31no—
158#41⋊C3—61no—
159#41⋊C3—61no—
160C2×SL(2,𝔽3)—22no—
161C2×SL(2,𝔽3)—21no—
162C2×SL(2,𝔽3)—21no—
163C2×SL(2,𝔽3)C22×D311yescharacteristic
164C3×C4⋊C4—31no—
165C3×C4⋊C4—31no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—31no—
168C3×C4⋊C4—121no—
169C3×C4⋊C4—121no—
170C3×C4⋊C4—121no—
171C3×C22⋊C4—31no—
172C3×C22⋊C4—41no—
173C3×C22⋊C4—81no—
174C3×C22⋊C4—121no—
175C3×C22⋊C4—121no—
176C3×C22⋊C4C2×A411yescharacteristic
177C3×C42—32no—
178C3×C42—31no—
179C3×C42—31no—
180C3×C42—121no—
181#217C2×A411yescharacteristic
182#217C2×A411yescharacteristic
183(C2×D3)⋊C4—32no—
184(C2×D3)⋊C4—32no—
185(C2×D3)⋊C4—32no—
186(C2×D3)⋊C4—31no—
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—
192C4⋊Dic3—31no—
193C4⋊Dic3—31no—
194C4⋊Dic3—31no—
195C4⋊Dic3C2×A411yescharacteristic
196Dic3⋊C4—32no—
197Dic3⋊C4—31no—
198Dic3⋊C4—31no—
199Dic3⋊C4—31no—
200Dic3⋊C4—31no—
201Dic3⋊C4C2×A411yescharacteristic
202Dic3⋊C4C2×A411yescharacteristic
203C4×Dic3—31no—
204C4×Dic3—31no—
205C4×Dic3—31no—
206C4×Dic3C2×A411yescharacteristic
207C22×C32—41no—
208C22×C32—81no—
209C22×C32—81no—
210C22×C32—81no—
211C32×C4—81no—
212C32×C4—81no—
213C3×Dic3—81no—
214C3×Dic3—81no—
215C3×Dic3—81no—
216C3×Dic3—81no—
217C2×#41—31no—
218C2×#41—31no—
219C22×Q8C2×C3×D311yescharacteristic
220C4⋊Q8—91no—
221#127—91no—
222#126—31no—
223#126—91no—
224#126—92no—
225#125—91no—
226#125—91no—
227#125—91no—
228#124—92no—
229#124—91no—
230#124—91no—
231#124—91no—
232#123—91no—
233#123—91no—
234#123—92no—
235#123—92no—
236C22⋊Q8—91no—
237C22⋊Q8—91no—
238C22⋊Q8—91no—
239#121—92no—
240#121—91no—
241#121—91no—
242#121—91no—
243#121—91no—
244#121—91no—
245C4×Q8—31no—
246C4×Q8—31no—
247C4×Q8—31no—
248C4×Q8—31no—
249C4×Q8—91no—
250C4×Q8C2×C3×D311yescharacteristic
251C4×Q8C2×C3×D311yescharacteristic
252C4×D4—91no—
253C4×D4—92no—
254C4×D4—91no—
255C4×D4—91no—
256C4×D4—91no—
257C4×D4—91no—
258C4×D4—91no—
259C4×D4—91no—
260#117—31no—
261#117—31no—
262#117—91no—
263#117—92no—
264#117—91no—
265C2×C4⋊C4—91no—
266C2×C42—91no—
267C23×C3—41no—
268C23×C3—81no—
269C23×C3C22×A411yescharacteristic
270C22×D3—31no—
271C22×D3—31no—
272C3×Q8—31no—
273C3×Q8—61no—
274C3×Q8—61no—
275C3×Q8C3×#4111yescharacteristic
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—31no—
279C2×C3×C4—31no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—61no—
283C2×C3×C4—61no—
284C2×C3×C4—81no—
285C2×C3×C4—81no—
286C2×C3×C4—121no—
287C2×C3×C4—121no—
288C2×C3×C4—121no—
289C2×C3×C4—121no—
290C2×C3×C4C22×A411yescharacteristic
291C2×C3×C4C22×A411yescharacteristic
292#67—121no—
293#67—121no—
294C2×Dic3—31no—
295C2×Dic3—31no—
296C2×Dic3C22×A411yescharacteristic
297C2×Dic3C22×A411yescharacteristic
298C2×Dic3C22×A411yescharacteristic
299C2×Dic3C22×A411yescharacteristic
300D12—61no—
301D12—61no—
302D12—61no—
303D12—61no—
304D12—121no—
305D12—122no—
306C4×D3—61no—
307C4×D3—61no—
308C4×D3—61no—
309C4×D3—61no—
310C4×D3—121no—
311C4×D3—121no—
312C4×D3—122no—
313Dic6—121no—
314SL(2,𝔽3)—21no—
315SL(2,𝔽3)#23511yescharacteristic
316C2×C32—41no—
317C2×C32—41no—
318C2×C32—41no—
319C2×C32—161no—
320#41—61no—
321#41—61no—
322#41—181no—
323#41—181no—
324C2×Q8—21no—
325C2×Q8—31no—
326C2×Q8—61no—
327C2×Q8—91no—
328C2×Q8C22×C3×D311yescharacteristic
329C2×D4—91no—
330C2×D4—91no—
331C2×D4—92no—
332C2×D4—91no—
333C2×D4—91no—
334C2×D4—91no—
335C22×C4—31no—
336C22×C4—91no—
337C22×C4—91no—
338C22×C4—91no—
339C22×C4—92no—
340C22×C4—91no—
341C4⋊C4—31no—
342C4⋊C4—31no—
343C4⋊C4—31no—
344C4⋊C4—31no—
345C4⋊C4—31no—
346C4⋊C4—31no—
347C4⋊C4—31no—
348C4⋊C4—91no—
349C4⋊C4—91no—
350C4⋊C4—91no—
351C4⋊C4—91no—
352C4⋊C4—91no—
353C4⋊C4—91no—
354C4⋊C4—92no—
355C4⋊C4—91no—
356C22⋊C4—31no—
357C22⋊C4—31no—
358C22⋊C4—31no—
359C22⋊C4—91no—
360C22⋊C4—91no—
361C22⋊C4—91no—
362C22⋊C4—91no—
363C22⋊C4—91no—
364C22⋊C4—92no—
365C22⋊C4—92no—
366C22⋊C4—92no—
367C22⋊C4—91no—
368C22⋊C4D3×A411yescharacteristic
369C42—32no—
370C42—31no—
371C42—31no—
372C42—31no—
373C42—91no—
374C42—91no—
375C42—91no—
376C22×C3—21no—
377C22×C3—21no—
378C22×C3—21no—
379C22×C3—41no—
380C22×C3—81no—
381C22×C3—81no—
382C22×C3—81no—
383C22×C3—82no—
384C22×C3—81no—
385C22×C3—161no—
386C22×C3—161no—
387C22×C3C23×A411yescharacteristic
388C2×D3—61no—
389C2×D3—61no—
390C2×D3—61no—
391C2×D3—61no—
392C2×D3—61no—
393C2×D3—61no—
394C3×C4—21no—
395C3×C4—21no—
396C3×C4—31no—
397C3×C4—31no—
398C3×C4—61no—
399C3×C4—61no—
400C3×C4—61no—
401C3×C4—81no—
402C3×C4—81no—
403C3×C4—161no—
404C3×C4—161no—
405C3×C4—241no—
406C3×C4—241no—
407C3×C4—241no—
408C3×C4—241no—
409Dic3—21no—
410Dic3—21no—
411Dic3—21no—
412Dic3—21no—
413Dic3—61no—
414Dic3—61no—
415C32—41noSylow
416C23—91no—
417C23—91no—
418C23C2×D3×A411yescharacteristic
419Q8—31no—
420Q8—61no—
421Q8—61no—
422Q8—361no—
423Q8C3×#23511yescharacteristic
424D4—181no—
425D4—181no—
426D4—181no—
427D4—181no—
428D4—361no—
429D4—362no—
430D4—361no—
431D4—361no—
432C2×C4—31no—
433C2×C4—31no—
434C2×C4—31no—
435C2×C4—31no—
436C2×C4—31no—
437C2×C4—31no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—91no—
443C2×C4—91no—
444C2×C4—181no—
445C2×C4—181no—
446C2×C4—181no—
447C2×C4—181no—
448C2×C4—361no—
449C2×C4—361no—
450C2×C4—362no—
451C2×C4C2×D3×A411yescharacteristic
452C2×C4C2×D3×A411yescharacteristic
453C2×C3—41no—
454C2×C3—41no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3—81no—
458C2×C3—81no—
459C2×C3—81no—
460C2×C3—161no—
461C2×C3—162no—
462C2×C3A4×#4111yescharacteristic
463C2×C3#594511yescharacteristic
464C2×C3#594511yescharacteristic
465D3—121no—
466D3—121no—
467C22—21no—
468C22—21no—
469C22—21no—
470C22—181no—
471C22—181no—
472C22—181no—
473C22—181no—
474C22—181no—
475C22—181no—
476C22C22×D3×A411yescharacteristic
477C4—21no—
478C4—21no—
479C4—31no—
480C4—31no—
481C4—61no—
482C4—61no—
483C4—61no—
484C4—61no—
485C4—61no—
486C4—61no—
487C4—61no—
488C4—181no—
489C4—181no—
490C3—41no—
491C3—81no—
492C3#3524⋊C311yescharacteristic, complemented
493C2—41no—
494C2—361no—
495C2—361no—
496C2A4×#23511yescharacteristic
497C2#5968⋊C311yescharacteristic
498C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#519C2×C3
socleC22×C3C23×A4
hypercenterC22⋊C4D3×A4
hypocenterC3×Q8C3×#41
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. C3×#519⋊C3C2semidirect
2. #5817⋊C3C2semidirect
3. #5817⋊C3C2semidirect
4. C4⋊(C3⋊(#41⋊C3))C2semidirect
5. #5569⋊C3C2semidirect
6. #5574⋊C3C2semidirect
7. C4×C3⋊(#41⋊C3)C2semidirect
8. #85879C3semidirect
25. #5577C2×C3semidirect
26. #5574C2×C3semidirect
27. C4×#239C2×C3semidirect
28. #5569C2×C3semidirect
39. #519⋊C3D3semidirect
492. C3#3524⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#519⋊C32
#5817⋊C32
C4⋊(C3⋊(#41⋊C3))2
#5569⋊C32
#5574⋊C32
C4×C3⋊(#41⋊C3)2
#858793
#3524⋊C33
C3×#8794