Finite Group Databasebeta

← #85842⋊C3 #11923201

Subgroups — #85842⋊C3 #11923201

Order 1152.

587 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 587 conjugacy classes of subgroups — too many to draw. Only its 87 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×C4×#41⋊C3C211yesmaximal, characteristic, complemented
2#5831⋊C3C211yesmaximal, characteristic, complemented
3#5834⋊C3C211yesmaximal, characteristic, complemented
4#5576⋊C3C211yesmaximal, characteristic, complemented
5C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
6#5569⋊C3C211yesmaximal, characteristic, complemented
7C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
8#85842C311yesmaximal, characteristic, complemented
9#3480⋊C3—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C2×D3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊(#41⋊C3)C2211yescharacteristic
13C3×#872—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3×C4×SL(2,𝔽3)C2211yescharacteristic
16Dic3×SL(2,𝔽3)C2211yescharacteristic
17Dic3×SL(2,𝔽3)C2211yescharacteristic
18C3×C4×#41C2×C311yescharacteristic, complemented
19#5834C2×C311yescharacteristic, complemented
20#5831C2×C311yescharacteristic
21#5576C2×C311yescharacteristic
22C4×#239C2×C311yescharacteristic, complemented
23C4×D3×Q8C2×C311yescharacteristic
24#5569C2×C311yescharacteristic, complemented
25#5559—31no—
26#5558—31no—
27#5557—31no—
28#5548—31no—
29#5533—31no—
30#5530—31no—
31#5479—31no—
32C4×#235—31no—
33#557⋊C3—31no—
34C4⋊(#41⋊C3)—31no—
35#548⋊C3—31no—
36#548⋊C3—31no—
37C4×#41⋊C3—21no—
38C4×#41⋊C3—31no—
39C4×#41⋊C3D311yescharacteristic, complemented
40C2×C4×SL(2,𝔽3)—31no—
41C2×C3×C4×D3—41no—
42C3×#41⋊C3—21no—
43C2×C3×SL(2,𝔽3)C2311yescharacteristic
44D3×SL(2,𝔽3)—21no—
45C3⋊(#41⋊C3)—21no—
46C32×C42—41no—
47C3×(C2×D3)⋊C4—41no—
48C3×(C2×D3)⋊C4—41no—
49C3×Dic3⋊C4—41no—
50C3×Dic3⋊C4—41no—
51C3×C4×Dic3—41no—
52#3480—31noSylow
53C2×C3×#41C22×C311yescharacteristic
54C2×#239C22×C311yescharacteristic
55C2×D3×Q8C22×C311yescharacteristic
56C2×#235—31no—
57C2×#41⋊C3—21no—
58C2×#41⋊C3—31no—
59C2×#41⋊C3C2×D311yescharacteristic
60C22×SL(2,𝔽3)—31no—
61C3×C4×Q8C22×C311yescharacteristic
62C3×C4×Q8C22×C311yescharacteristic
63C3×C4×D4—31no—
64C3×C4×D4—31no—
65C3×#117—31no—
66C3×#117—121no—
67C2×C3×C42—31no—
68Q8×Dic3C22×C311yescharacteristic
69Q8×Dic3C22×C311yescharacteristic
70#940—31no—
71#938—31no—
72#938—31no—
73#936—31no—
74#930—31no—
75#929—32no—
76C4×#67—32no—
77C4×#67—31no—
78C4×#67—31no—
79C4×#67—31no—
80C4×#67—31no—
81#926—31no—
82#924—31no—
83#897—31no—
84#895—31no—
85#893—31no—
86#892—31no—
87D3×C4⋊C4—31no—
88#890—31no—
89#888—31no—
90#887—31no—
91#885—32no—
92#884—32no—
93#883—32no—
94#879—32no—
95#876—32no—
96#875—31no—
97C4×D12—31no—
98C4×D12—31no—
99C4×D12—31no—
100#872—31no—
101#872A411yescharacteristic
102C42×D3—31no—
103C42×D3—31no—
104#870—31no—
105C4×Dic6—31no—
106C4×Dic6—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
114C22×C3×D3—41no—
115C2×C32×C4—41no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C3×Dic3—41no—
119C2×C3×Dic3—41no—
120C2×C3×Dic3—41no—
121C3×C4×D3—44no—
122C3×SL(2,𝔽3)#4111yescharacteristic, complemented
123#559—91no—
124#557—31no—
125#556—91no—
126#553—91no—
127#553—91no—
128#550—31no—
129#548—31no—
130#548—31no—
131#532—91no—
132#529—91no—
133#529—91no—
134C4×#41—31no—
135C4×#41—91no—
136C4×#41C3×D311yescharacteristic, complemented
137C2×C4×Q8—31no—
138C3×#41—21no—
139C3×#41—61no—
140C2×C3×Q8C23×C311yescharacteristic
141C2×C3×D4—31no—
142C22×C3×C4—31no—
143C22×C3×C4—31no—
144C22×C3×C4—31no—
145C22×C3×C4—121no—
146C2×#67—31no—
147C2×#67—32no—
148C2×#67—31no—
149#239—21no—
150#239—61no—
151D3×Q8—21no—
152D3×Q8—61no—
153#235—64no—
154C2×D12—31no—
155C2×D12—31no—
156C2×C4×D3—31no—
157C2×C4×D3—31no—
158C2×C4×D3—31no—
159C2×C4×D3—31no—
160C2×C4×D3—31no—
161C2×C4×D3C2×A411yescharacteristic
162C2×Dic6—31no—
163C2×Dic6—31no—
164#41⋊C3—22no—
165#41⋊C3—21no—
166#41⋊C3—61no—
167C2×SL(2,𝔽3)—21no—
168C2×SL(2,𝔽3)—61no—
169C2×SL(2,𝔽3)C22×D311yescharacteristic
170C3×C4⋊C4—31no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—121no—
173C3×C4⋊C4—121no—
174C3×C22⋊C4—32no—
175C3×C22⋊C4—121no—
176C3×C22⋊C4—121no—
177C3×C42—31no—
178C3×C42—31no—
179C3×C42—31no—
180C3×C42—41no—
181C3×C42—81no—
182C3×C42—121no—
183C3×C42C2×A411yescharacteristic
184#217—32no—
185#217—31no—
186#217—31no—
187(C2×D3)⋊C4—32no—
188(C2×D3)⋊C4—32no—
189(C2×D3)⋊C4—31no—
190(C2×D3)⋊C4—31no—
191(C2×D3)⋊C4C2×A411yescharacteristic
192(C2×D3)⋊C4C2×A411yescharacteristic
193C4⋊Dic3—31no—
194C4⋊Dic3—31no—
195C4⋊Dic3—31no—
196C4⋊Dic3—31no—
197Dic3⋊C4—32no—
198Dic3⋊C4—32no—
199Dic3⋊C4—31no—
200Dic3⋊C4—31no—
201Dic3⋊C4C2×A411yescharacteristic
202Dic3⋊C4C2×A411yescharacteristic
203C4×Dic3—31no—
204C4×Dic3—31no—
205C4×Dic3—31no—
206C4×Dic3—31no—
207C4×Dic3—31no—
208C4×Dic3C2×A411yescharacteristic
209C22×C32—41no—
210C2×C3×D3—42no—
211C2×C3×D3—81no—
212C2×C3×D3—81no—
213C32×C4—42no—
214C32×C4—81no—
215C32×C4—81no—
216C3×Dic3—42no—
217C3×Dic3—81no—
218C3×Dic3—81no—
219C2×#41—31no—
220C2×#41—91no—
221C2×#41C2×C3×D311yescharacteristic
222C22×Q8—31no—
223C4⋊Q8—91no—
224#127—91no—
225#126—92no—
226#126—92no—
227#125—91no—
228#125—91no—
229#124—91no—
230#124—92no—
231#124—91no—
232#123—92no—
233#123—92no—
234C22⋊Q8—91no—
235C22⋊Q8—91no—
236#121—92no—
237#121—91no—
238#121—91no—
239#121—91no—
240#121—91no—
241C4×Q8—31no—
242C4×Q8—31no—
243C4×Q8—91no—
244C4×Q8—91no—
245C4×Q8—91no—
246C4×Q8—91no—
247C4×Q8C2×C3×D311yescharacteristic
248C4×Q8C2×C3×D311yescharacteristic
249C4×D4—31no—
250C4×D4—31no—
251C4×D4—91no—
252C4×D4—91no—
253C4×D4—91no—
254C4×D4—91no—
255C4×D4—91no—
256C4×D4—91no—
257C4×D4—91no—
258C4×D4—91no—
259C4×D4—92no—
260#117—31no—
261#117—31no—
262#117—91no—
263#117—91no—
264#117—91no—
265C2×C4⋊C4—91no—
266C2×C42—31no—
267C2×C42—91no—
268C2×C42—91no—
269C23×C3—31no—
270C23×C3—121no—
271C22×D3—31no—
272C22×D3C22×A411yescharacteristic
273C3×Q8—31no—
274C3×Q8C3×#4111yescharacteristic
275C3×D4—61no—
276C3×D4—61no—
277C2×C3×C4—31no—
278C2×C3×C4—31no—
279C2×C3×C4—31no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—61no—
284C2×C3×C4—61no—
285C2×C3×C4—62no—
286C2×C3×C4—81no—
287C2×C3×C4—81no—
288C2×C3×C4—81no—
289C2×C3×C4—124no—
290C2×C3×C4—121no—
291C2×C3×C4—121no—
292C2×C3×C4—121no—
293C2×C3×C4—121no—
294C2×C3×C4C22×A411yescharacteristic
295C2×C3×C4C22×A411yescharacteristic
296C2×C3×C4C22×A411yescharacteristic
297#67—62no—
298#67—62no—
299#67—122no—
300C2×Dic3—31no—
301C2×Dic3—31no—
302C2×Dic3—31no—
303C2×Dic3C22×A411yescharacteristic
304C2×Dic3C22×A411yescharacteristic
305C2×Dic3C22×A411yescharacteristic
306D12—61no—
307D12—61no—
308D12—62no—
309C4×D3—61no—
310C4×D3—61no—
311C4×D3—61no—
312C4×D3—61no—
313C4×D3—62no—
314C4×D3—121no—
315C4×D3—121no—
316C4×D3#41⋊C314yescomplemented
317Dic6—62no—
318Dic6—61no—
319Dic6—61no—
320SL(2,𝔽3)—21no—
321SL(2,𝔽3)#23511yescharacteristic, complemented
322C2×C32—41no—
323C2×C32—41no—
324C2×C32—41no—
325C3×D3—82no—
326#41—21no—
327#41—61no—
328#41—61no—
329#41—184no—
330#41—181no—
331C2×Q8—61no—
332C2×Q8—91no—
333C2×Q8—91no—
334C2×Q8—181no—
335C2×Q8C22×C3×D311yescharacteristic
336C2×D4—31no—
337C2×D4—91no—
338C2×D4—91no—
339C2×D4—91no—
340C2×D4—91no—
341C2×D4—92no—
342C22×C4—31no—
343C22×C4—31no—
344C22×C4—31no—
345C22×C4—31no—
346C22×C4—91no—
347C22×C4—91no—
348C22×C4—91no—
349C22×C4—91no—
350C22×C4—91no—
351C4⋊C4—31no—
352C4⋊C4—31no—
353C4⋊C4—31no—
354C4⋊C4—31no—
355C4⋊C4—91no—
356C4⋊C4—91no—
357C4⋊C4—91no—
358C4⋊C4—91no—
359C4⋊C4—92no—
360C4⋊C4—92no—
361C4⋊C4—91no—
362C4⋊C4—91no—
363C22⋊C4—32no—
364C22⋊C4—31no—
365C22⋊C4—31no—
366C22⋊C4—91no—
367C22⋊C4—91no—
368C22⋊C4—92no—
369C22⋊C4—92no—
370C22⋊C4—92no—
371C22⋊C4—91no—
372C22⋊C4—91no—
373C42—31no—
374C42—31no—
375C42—31no—
376C42—31no—
377C42—91no—
378C42—91no—
379C42—91no—
380C42—91no—
381C42—91no—
382C42D3×A411yescharacteristic
383C22×C3—41no—
384C22×C3—61no—
385C22×C3—61no—
386C22×C3—61no—
387C22×C3—81no—
388C22×C3—122no—
389C22×C3—241no—
390C22×C3—241no—
391C22×C3C23×A411yescharacteristic
392C2×D3—21no—
393C2×D3—21no—
394C2×D3—61no—
395C2×D3—61no—
396C2×D3—61no—
397C2×D3C2×#41⋊C312yes—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—31no—
401C3×C4—31no—
402C3×C4—42no—
403C3×C4—61no—
404C3×C4—61no—
405C3×C4—81no—
406C3×C4—81no—
407C3×C4—82no—
408C3×C4—82no—
409C3×C4—82no—
410C3×C4—122no—
411C3×C4—241no—
412C3×C4—241no—
413C3×C4C2×#41⋊C312yes—
414Dic3—21no—
415Dic3—21no—
416Dic3—61no—
417Dic3—61no—
418Dic3—61no—
419Dic3C2×#41⋊C312yes—
420C32—41noSylow
421C23—31no—
422C23—31no—
423C23—91no—
424Q8—31no—
425Q8—182no—
426Q8—181no—
427Q8—181no—
428Q8C3×#23511yescharacteristic
429D4—61no—
430D4—61no—
431D4—182no—
432D4—181no—
433D4—181no—
434D4—182no—
435D4—182no—
436D4—362no—
437C2×C4—31no—
438C2×C4—31no—
439C2×C4—34no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—31no—
443C2×C4—31no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—62no—
447C2×C4—91no—
448C2×C4—91no—
449C2×C4—91no—
450C2×C4—121no—
451C2×C4—181no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4—181no—
455C2×C4—182no—
456C2×C4—361no—
457C2×C4—361no—
458C2×C4C2×D3×A411yescharacteristic
459C2×C4C2×D3×A411yescharacteristic
460C2×C4C2×D3×A411yescharacteristic
461C2×C3—41no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—81no—
465C2×C3—81no—
466C2×C3—81no—
467C2×C3—121no—
468C2×C3—242no—
469C2×C3C22×#41⋊C311yescharacteristic
470C2×C3A4×#4111yescharacteristic
471C2×C3#594511yescharacteristic
472D3—22no—
473D3—121no—
474C22—32no—
475C22—61no—
476C22—61no—
477C22—61no—
478C22—61no—
479C22—61no—
480C22—181no—
481C22—181no—
482C22—181no—
483C22C22×D3×A411yescharacteristic
484C4—21no—
485C4—21no—
486C4—31no—
487C4—31no—
488C4—32no—
489C4—61no—
490C4—61no—
491C4—61no—
492C4—61no—
493C4—181no—
494C4—181no—
495C4—181no—
496C4D3×#41⋊C312yes—
497C3—41no—
498C3—81no—
499C3#3480⋊C311yescharacteristic, complemented
500C2—62no—
501C2—121no—
502C2—361no—
503C2A4×#23511yescharacteristic
504C2C2×D3×#41⋊C311yescharacteristic
505C2#5964⋊C311yescharacteristic
subgroupG/H
centerC2×C4C2×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FrattiniC22C22×D3×A4
FittingC3×C4×#41C2×C3
socleC22×C3C23×A4
hypercenterC42D3×A4
hypocenterC3×Q8C3×#41
LCSG ▷ C2×C3×Q8 ▷ C3×Q8
UCSC1 ◁ C2×C4 ◁ C42

Complements

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

NH ≅ G/Ntype
1. C3×C4×#41⋊C3C2semidirect
2. #5831⋊C3C2semidirect
3. #5834⋊C3C2semidirect
4. #5576⋊C3C2semidirect
5. C4×D3×SL(2,𝔽3)C2semidirect
6. #5569⋊C3C2semidirect
7. C4×C3⋊(#41⋊C3)C2semidirect
8. #85842C3semidirect
18. C3×C4×#41C2×C3semidirect
19. #5834C2×C3semidirect
22. C4×#239C2×C3semidirect
24. #5569C2×C3semidirect
39. C4×#41⋊C3D3semidirect
122. C3×SL(2,𝔽3)#41semidirect
136. C4×#41C3×D3semidirect
316. C4×D3#41⋊C3semidirect
321. SL(2,𝔽3)#235semidirect
499. C3#3480⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#41⋊C32
#5831⋊C32
#5834⋊C32
#5576⋊C32
C4×D3×SL(2,𝔽3)2
#5569⋊C32
C4×C3⋊(#41⋊C3)2
#858423
#3480⋊C33
C3×#8724