Finite Group Databasebeta

← #87026⋊C3 #11923287

Subgroups — #87026⋊C3 #11923287

Order 1152.

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

normal subgroups only

This group has 529 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×#557⋊C3C211yesmaximal, characteristic, complemented
2#5576⋊C3C211yesmaximal, characteristic, complemented
3#5576⋊C3C211yesmaximal, characteristic, complemented
4#5572⋊C3C211yesmaximal, characteristic, complemented
5C4⋊(C3⋊(#41⋊C3))C211yesmaximal, characteristic, complemented
6#5574⋊C3C211yesmaximal, characteristic, complemented
7C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
8#87026C311yesmaximal, characteristic, complemented
9#3524⋊C3—31nomaximal
10C2×D3×SL(2,𝔽3)C2211yescharacteristic
11C2×C3⋊(#41⋊C3)C2211yescharacteristic
12C2×C3⋊(#41⋊C3)C2211yescharacteristic
13C3×#898—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3×C4×SL(2,𝔽3)C2211yescharacteristic
16C3×C4×SL(2,𝔽3)C2211yescharacteristic
17Dic3×SL(2,𝔽3)C2211yescharacteristic
18C3×#557C2×C311yescharacteristic, complemented
19#5734—31no—
20#5733—31no—
21#5697—31no—
22#5695—31no—
23#5694—31no—
24#5691—31no—
25#5585—31no—
26#5580—31no—
27#5577C2×C311yescharacteristic, complemented
28#5576C2×C311yescharacteristic
29#5576C2×C311yescharacteristic
30#5574C2×C311yescharacteristic, complemented
31C4×#239C2×C311yescharacteristic, complemented
32#5572C2×C311yescharacteristic
33#557⋊C3—21no—
34#557⋊C3D311yescharacteristic, complemented
35C4⋊(#41⋊C3)—31no—
36#548⋊C3—31no—
37#548⋊C3—31no—
38#520⋊C3—31no—
39#519⋊C3—31no—
40C4×#41⋊C3—31no—
41C2×C3×SL(2,𝔽3)C2311yescharacteristic
42D3×SL(2,𝔽3)—21no—
43C3⋊(#41⋊C3)—21no—
44C3⋊(#41⋊C3)—21no—
45C32×C4⋊C4—41no—
46C3×(C2×D3)⋊C4—41no—
47C3×(C2×D3)⋊C4—41no—
48C3×(C2×D3)⋊C4—41no—
49C3×C4⋊Dic3—41no—
50C3×Dic3⋊C4—41no—
51C3×C4×Dic3—41no—
52#3524—31noSylow
53C2×#239C22×C311yescharacteristic
54C2×#239C22×C311yescharacteristic
55C2×D3×Q8C22×C311yescharacteristic
56C2×#41⋊C3—31no—
57C2×#41⋊C3—31no—
58C22×SL(2,𝔽3)—31no—
59C3×C4⋊Q8—31no—
60C3×#126—121no—
61C3×#125—31no—
62C3×#125—31no—
63C3×C4×Q8—31no—
64C3×C4×Q8C22×C311yescharacteristic
65C3×C4×Q8C22×C311yescharacteristic
66C3×C4×Q8C22×C311yescharacteristic
67#947—32no—
68#947—31no—
69#946—31no—
70Q8×Dic3C22×C311yescharacteristic
71#898—32no—
72#898—31no—
73#898A411yescharacteristic
74#897—31no—
75#897—31no—
76#895—31no—
77#895—31no—
78#895—32no—
79#895—31no—
80#895—31no—
81#895—31no—
82#894—32no—
83#894—32no—
84#894—31no—
85#894—31no—
86#893—32no—
87#893—31no—
88#893—31no—
89#892—32no—
90#892—31no—
91#892—31no—
92D3×C4⋊C4—31no—
93#890—31no—
94#887—31no—
95#875—31no—
96#875—31no—
97#874—31no—
98C4×D12—31no—
99C4×D12—31no—
100C4×D12—31no—
101C4×D12—31no—
102C4×D12—31no—
103#872—31no—
104#872—31no—
105C42×D3—31no—
106C4×Dic6—31no—
107C4×SL(2,𝔽3)—21no—
108C4×SL(2,𝔽3)—21no—
109C4×SL(2,𝔽3)—21no—
110C4×SL(2,𝔽3)—31no—
111C4×SL(2,𝔽3)C2×D311yescharacteristic
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×SL(2,𝔽3)#4111yescharacteristic
122#559—91no—
123#557C3×D311yescharacteristic, complemented
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#519—31no—
136C4×#41—31no—
137C2×C3×Q8—31no—
138C2×C3×Q8C23×C311yescharacteristic
139#239—21no—
140#239—21no—
141#239—61no—
142#239—61no—
143D3×Q8—21no—
144D3×Q8—61no—
145C2×D12—31no—
146C2×D12—31no—
147C2×D12—32no—
148C2×D12—31no—
149C2×D12—31no—
150C2×D12—31no—
151C2×C4×D3—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)—21no—
161C2×SL(2,𝔽3)—61no—
162C2×SL(2,𝔽3)C22×D311yescharacteristic
163C3×C4⋊C4—31no—
164C3×C4⋊C4—31no—
165C3×C4⋊C4—31no—
166C3×C4⋊C4—32no—
167C3×C4⋊C4—31no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—41no—
170C3×C4⋊C4—81no—
171C3×C4⋊C4—121no—
172C3×C4⋊C4—121no—
173C3×C4⋊C4C2×A411yescharacteristic
174C3×C22⋊C4—121no—
175C3×C22⋊C4—121no—
176C3×C22⋊C4—121no—
177C3×C42—31no—
178C3×C42—31no—
179C3×C42—31no—
180C3×C42—121no—
181(C2×D3)⋊C4—32no—
182(C2×D3)⋊C4—32no—
183(C2×D3)⋊C4—32no—
184(C2×D3)⋊C4—31no—
185(C2×D3)⋊C4—31no—
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)⋊C4C2×A411yescharacteristic
192(C2×D3)⋊C4C2×A411yescharacteristic
193(C2×D3)⋊C4C2×A411yescharacteristic
194C4⋊Dic3—31no—
195C4⋊Dic3—31no—
196C4⋊Dic3—31no—
197C4⋊Dic3C2×A411yescharacteristic
198Dic3⋊C4—31no—
199Dic3⋊C4—31no—
200Dic3⋊C4—31no—
201Dic3⋊C4C2×A411yescharacteristic
202C4×Dic3—32no—
203C4×Dic3—31no—
204C4×Dic3—31no—
205C4×Dic3C2×A411yescharacteristic
206C22×C32—41no—
207C2×C3×D3—81no—
208C2×C3×D3—81no—
209C2×C3×D3—81no—
210C32×C4—81no—
211C32×C4—81no—
212C32×C4—81no—
213C3×Dic3—81no—
214C3×Dic3—81no—
215C3×Dic3—81no—
216C2×#41—31no—
217C2×#41—31no—
218C22×Q8—31no—
219C4⋊Q8—31no—
220#127—91no—
221#126—31no—
222#126—91no—
223#126—92no—
224#125—31no—
225#125—31no—
226#125—91no—
227#124—91no—
228#124—91no—
229#124—91no—
230#124—92no—
231#123—91no—
232#123—92no—
233#123—92no—
234#123—91no—
235C22⋊Q8—91no—
236C22⋊Q8—91no—
237C22⋊Q8—91no—
238#121—91no—
239#121—91no—
240#121—91no—
241#121—92no—
242#121—91no—
243#121—91no—
244C4×Q8—31no—
245C4×Q8—31no—
246C4×Q8—91no—
247C4×Q8—91no—
248C4×Q8C2×C3×D311yescharacteristic
249C4×Q8C2×C3×D311yescharacteristic
250C4×Q8C2×C3×D311yescharacteristic
251C4×D4—91no—
252C4×D4—91no—
253C4×D4—91no—
254C4×D4—91no—
255C4×D4—91no—
256C4×D4—92no—
257C4×D4—91no—
258C4×D4—91no—
259#117—91no—
260#117—91no—
261#117—92no—
262#117—91no—
263#117—91no—
264C2×C4⋊C4—91no—
265C2×C42—91no—
266C23×C3—121no—
267C22×D3—31no—
268C22×D3—31no—
269C22×D3C22×A411yescharacteristic
270C3×Q8—31no—
271C3×Q8—121no—
272C3×Q8C3×#4111yescharacteristic
273C2×C3×C4—31no—
274C2×C3×C4—31no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—81no—
281C2×C3×C4—81no—
282C2×C3×C4—81no—
283C2×C3×C4—121no—
284C2×C3×C4—121no—
285C2×C3×C4—121no—
286C2×C3×C4C22×A411yescharacteristic
287C2×C3×C4C22×A411yescharacteristic
288C2×C3×C4C22×A411yescharacteristic
289C2×Dic3—31no—
290C2×Dic3—31no—
291C2×Dic3C22×A411yescharacteristic
292C2×Dic3C22×A411yescharacteristic
293C2×Dic3C22×A411yescharacteristic
294D12—61no—
295D12—61no—
296D12—61no—
297D12—61no—
298D12—121no—
299D12—121no—
300D12—121no—
301D12—122no—
302C4×D3—61no—
303C4×D3—61no—
304C4×D3—61no—
305C4×D3—61no—
306C4×D3—61no—
307C4×D3—61no—
308C4×D3—121no—
309C4×D3—121no—
310C4×D3—122no—
311Dic6—61no—
312Dic6—61no—
313SL(2,𝔽3)—21no—
314SL(2,𝔽3)#23911yescharacteristic
315C2×C32—41no—
316C2×C32—41no—
317C2×C32—41no—
318C3×D3—161no—
319#41—61no—
320#41—61no—
321#41—181no—
322#41—181no—
323C2×Q8—31no—
324C2×Q8—61no—
325C2×Q8—91no—
326C2×Q8—181no—
327C2×Q8C22×C3×D311yescharacteristic
328C2×D4—91no—
329C2×D4—91no—
330C2×D4—91no—
331C2×D4—91no—
332C2×D4—92no—
333C2×D4—91no—
334C22×C4—91no—
335C22×C4—91no—
336C22×C4—91no—
337C22×C4—91no—
338C22×C4—92no—
339C22×C4—91no—
340C4⋊C4—31no—
341C4⋊C4—31no—
342C4⋊C4—31no—
343C4⋊C4—32no—
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⋊C4D3×A411yescharacteristic
355C22⋊C4—31no—
356C22⋊C4—31no—
357C22⋊C4—31no—
358C22⋊C4—91no—
359C22⋊C4—91no—
360C22⋊C4—91no—
361C22⋊C4—91no—
362C22⋊C4—92no—
363C22⋊C4—92no—
364C22⋊C4—92no—
365C22⋊C4—91no—
366C22⋊C4—91no—
367C22⋊C4—91no—
368C42—31no—
369C42—31no—
370C42—31no—
371C42—31no—
372C42—91no—
373C42—91no—
374C42—92no—
375C22×C3—41no—
376C22×C3—81no—
377C22×C3—241no—
378C22×C3—241no—
379C22×C3—241no—
380C22×C3C23×A411yescharacteristic
381C2×D3—21no—
382C2×D3—21no—
383C2×D3—21no—
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—61no—
398C3×C4—81no—
399C3×C4—81no—
400C3×C4—81no—
401C3×C4—82no—
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—
412C32—41noSylow
413C23—31no—
414C23—91no—
415C23—91no—
416Q8—31no—
417Q8—121no—
418Q8—181no—
419Q8—181no—
420Q8C3×#23911yescharacteristic
421D4—181no—
422D4—181no—
423D4—181no—
424D4—181no—
425D4—361no—
426D4—361no—
427D4—361no—
428D4—362no—
429C2×C4—31no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—31no—
434C2×C4—31no—
435C2×C4—31no—
436C2×C4—91no—
437C2×C4—91no—
438C2×C4—181no—
439C2×C4—181no—
440C2×C4—181no—
441C2×C4—181no—
442C2×C4—181no—
443C2×C4—181no—
444C2×C4—361no—
445C2×C4—361no—
446C2×C4—362no—
447C2×C4C2×D3×A411yescharacteristic
448C2×C4C2×D3×A411yescharacteristic
449C2×C4C2×D3×A411yescharacteristic
450C2×C3—41no—
451C2×C3—41no—
452C2×C3—41no—
453C2×C3—81no—
454C2×C3—81no—
455C2×C3—81no—
456C2×C3—481no—
457C2×C3A4×#4111yescharacteristic
458C2×C3#594511yescharacteristic
459C2×C3#594511yescharacteristic
460D3—41no—
461D3—121no—
462D3—121no—
463C22—61no—
464C22—61no—
465C22—61no—
466C22—181no—
467C22—181no—
468C22—181no—
469C22—181no—
470C22—181no—
471C22—181no—
472C22C22×D3×A411yescharacteristic
473C4—21no—
474C4—21no—
475C4—21no—
476C4—31no—
477C4—31no—
478C4—61no—
479C4—61no—
480C4—61no—
481C4—61no—
482C4—61no—
483C4—61no—
484C4—181no—
485C4—181no—
486C3—41no—
487C3—81no—
488C3#3524⋊C311yescharacteristic, complemented
489C2—121no—
490C2—361no—
491C2—361no—
492C2A4×#23911yescharacteristic
493C2#5964⋊C311yescharacteristic
494C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#557C2×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×#557⋊C3C2semidirect
2. #5576⋊C3C2semidirect
3. #5576⋊C3C2semidirect
4. #5572⋊C3C2semidirect
5. C4⋊(C3⋊(#41⋊C3))C2semidirect
6. #5574⋊C3C2semidirect
7. C4×C3⋊(#41⋊C3)C2semidirect
8. #87026C3semidirect
18. C3×#557C2×C3semidirect
27. #5577C2×C3semidirect
30. #5574C2×C3semidirect
31. C4×#239C2×C3semidirect
34. #557⋊C3D3semidirect
123. #557C3×D3semidirect
488. C3#3524⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#557⋊C32
#5576⋊C32
#5572⋊C32
C4⋊(C3⋊(#41⋊C3))2
#5574⋊C32
C4×C3⋊(#41⋊C3)2
#870263
#3524⋊C33
C3×#8984