Finite Group Databasebeta

← #87022⋊C3 #11923284

Subgroups — #87022⋊C3 #11923284

Order 1152.

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

normal subgroups only

This group has 553 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×#557⋊C3C211yesmaximal, characteristic, complemented
2#5576⋊C3C211yesmaximal, characteristic, complemented
3#5576⋊C3C211yesmaximal, characteristic, complemented
4#5572⋊C3C211yesmaximal, characteristic, complemented
5SL(2,𝔽3)×D12C211yesmaximal, characteristic, complemented
6C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
7#5574⋊C3C211yesmaximal, characteristic, complemented
8#87022C311yesmaximal, characteristic, complemented
9#3516⋊C3—31nomaximal
10C2×D3×SL(2,𝔽3)C2211yescharacteristic
11C2×D3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊(#41⋊C3)C2211yescharacteristic
13C3×#894—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#5733—31no—
20#5730—31no—
21#5698—31no—
22#5695—31no—
23#5694—31no—
24#5691—31no—
25#5584—31no—
26#5580—31no—
27#5576C2×C311yescharacteristic, complemented
28#5576C2×C311yescharacteristic, complemented
29Q8×D12C2×C311yescharacteristic
30#5574C2×C311yescharacteristic, complemented
31#5572C2×C311yescharacteristic, complemented
32C4×D3×Q8C2×C311yescharacteristic, complemented
33#557⋊C3—21no—
34#557⋊C3D311yescharacteristic, complemented
35D4×SL(2,𝔽3)—31no—
36#548⋊C3—31no—
37#548⋊C3—31no—
38#520⋊C3—31no—
39#519⋊C3—31no—
40C2×C4×SL(2,𝔽3)—31no—
41C2×C3×D12—41no—
42C2×C3×C4×D3—41no—
43C2×C3×SL(2,𝔽3)C2311yescharacteristic
44D3×SL(2,𝔽3)—21no—
45D3×SL(2,𝔽3)—21no—
46C3⋊(#41⋊C3)—21no—
47C32×C4⋊C4—41no—
48C3×(C2×D3)⋊C4—41no—
49C3×(C2×D3)⋊C4—41no—
50C3×(C2×D3)⋊C4—41no—
51C3×Dic3⋊C4—41no—
52#3516—31noSylow
53C2×#239C22×C311yescharacteristic
54C2×D3×Q8C22×C311yescharacteristic
55C2×D3×Q8C22×C311yescharacteristic
56C2×#41⋊C3—31no—
57C22×SL(2,𝔽3)—31no—
58C22×SL(2,𝔽3)—31no—
59C3×C4⋊Q8—31no—
60C3×#125—31no—
61C3×#125—31no—
62C3×#123—121no—
63C3×C4×Q8—31no—
64C3×C4×Q8C22×C311yescharacteristic
65C3×C4×Q8C22×C311yescharacteristic
66C3×C4×Q8C22×C311yescharacteristic
67#947—31no—
68#946—32no—
69#946—31no—
70Q8×Dic3C22×C311yescharacteristic
71#898—32no—
72#898—32no—
73#898—31no—
74#898—31no—
75#897—31no—
76#897—31no—
77#897—31no—
78#897—31no—
79#895—31no—
80#895—31no—
81#895—31no—
82#894—32no—
83#894—31no—
84#894A411yescharacteristic
85#893—32no—
86#893—31no—
87#892—32no—
88#892—31no—
89D3×C4⋊C4—31no—
90D3×C4⋊C4—31no—
91#890—32no—
92#890—31no—
93#887—31no—
94#887—31no—
95#875—31no—
96#875—31no—
97C4×D12—31no—
98C4×D12—31no—
99C4×D12—31no—
100C4×D12—31no—
101#872—31no—
102#872—31no—
103C42×D3—31no—
104#869—31no—
105C4×Dic6—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—
115C22×C3×D3—41no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4—41no—
119C2×C3×Dic3—41no—
120C2×C3×Dic3—41no—
121C3×D12—82no—
122C3×C4×D3—82no—
123C3×SL(2,𝔽3)#4111yescharacteristic
124#557C3×D311yescharacteristic, complemented
125#556—91no—
126#554—91no—
127#553—91no—
128#552—91no—
129D4×Q8—31no—
130#548—31no—
131#548—31no—
132#542—91no—
133#541—91no—
134#529—91no—
135#529—91no—
136#520—31no—
137#519—31no—
138C2×C4×Q8—31no—
139C2×C3×Q8—31no—
140C2×C3×Q8C23×C311yescharacteristic
141C2×C3×D4—121no—
142C22×C3×C4—121no—
143#239—21no—
144#239—61no—
145D3×Q8—21no—
146D3×Q8—21no—
147D3×Q8—61no—
148D3×Q8—61no—
149C2×D12—31no—
150C2×D12—31no—
151C2×D12—31no—
152C2×D12C2×A411yescharacteristic
153C2×C4×D3—31no—
154C2×C4×D3—31no—
155C2×C4×D3—31no—
156C2×C4×D3—32no—
157C2×C4×D3—31no—
158C2×C4×D3C2×A411yescharacteristic
159C2×Dic6—31no—
160C2×Dic6—31no—
161#41⋊C3—61no—
162C2×SL(2,𝔽3)—21no—
163C2×SL(2,𝔽3)—61no—
164C2×SL(2,𝔽3)—61no—
165C2×SL(2,𝔽3)C22×D311yescharacteristic
166C3×C4⋊C4—31no—
167C3×C4⋊C4—31no—
168C3×C4⋊C4—31no—
169C3×C4⋊C4—32no—
170C3×C4⋊C4—31no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—41no—
173C3×C4⋊C4—81no—
174C3×C4⋊C4—121no—
175C3×C4⋊C4C2×A411yescharacteristic
176C3×C22⋊C4—121no—
177C3×C22⋊C4—121no—
178C3×C22⋊C4—121no—
179C3×C42—31no—
180C3×C42—31no—
181C3×C42—31no—
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—32no—
195C4⋊Dic3—31no—
196C4⋊Dic3—31no—
197C4⋊Dic3—31no—
198C4⋊Dic3—31no—
199Dic3⋊C4—32no—
200Dic3⋊C4—31no—
201Dic3⋊C4—31no—
202Dic3⋊C4—31no—
203Dic3⋊C4C2×A411yescharacteristic
204C4×Dic3—32no—
205C4×Dic3—31no—
206C4×Dic3—31no—
207C4×Dic3—31no—
208C22×C32—41no—
209C2×C3×D3—42no—
210C2×C3×D3—81no—
211C2×C3×D3—81no—
212C2×C3×D3—81no—
213C2×C3×D3—81no—
214C2×C3×D3—81no—
215C32×C4—81no—
216C32×C4—81no—
217C32×C4—81no—
218C3×Dic3—81no—
219C3×Dic3—81no—
220C2×#41—31no—
221C22×Q8—31no—
222C22×Q8—31no—
223C4⋊Q8—31no—
224C4⋊Q8—91no—
225#126—91no—
226#126—92no—
227#126—92no—
228#126—91no—
229#125—31no—
230#125—31no—
231#125—91no—
232#125—92no—
233#124—91no—
234#124—91no—
235#124—91no—
236#123—31no—
237#123—91no—
238#123—92no—
239C22⋊Q8—91no—
240C22⋊Q8—91no—
241C22⋊Q8—91no—
242C22⋊Q8—92no—
243C22⋊Q8—91no—
244C22⋊Q8—91no—
245#121—91no—
246#121—91no—
247#121—91no—
248C4×Q8—31no—
249C4×Q8—31no—
250C4×Q8—91no—
251C4×Q8—91no—
252C4×Q8—91no—
253C4×Q8—91no—
254C4×Q8C2×C3×D311yescharacteristic
255C4×Q8C2×C3×D311yescharacteristic
256C4×Q8C2×C3×D311yescharacteristic
257C4×D4—91no—
258C4×D4—91no—
259C4×D4—91no—
260C4×D4—91no—
261C4×D4—92no—
262C4×D4—91no—
263#117—91no—
264#117—91no—
265#117—92no—
266#117—91no—
267C2×C4⋊C4—91no—
268C2×C4⋊C4—91no—
269C2×C42—91no—
270C23×C3—121no—
271C23×C3—121no—
272C22×D3—31no—
273C22×D3C22×A411yescharacteristic
274C22×D3C22×A411yescharacteristic
275C3×Q8—31no—
276C3×Q8—121no—
277C3×Q8C3×#4111yescharacteristic
278C3×D4—242no—
279C2×C3×C4—31no—
280C2×C3×C4—31no—
281C2×C3×C4—31no—
282C2×C3×C4—31no—
283C2×C3×C4—41no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—81no—
287C2×C3×C4—81no—
288C2×C3×C4—81no—
289C2×C3×C4—121no—
290C2×C3×C4—121no—
291C2×C3×C4—242no—
292C2×C3×C4C22×A411yescharacteristic
293C2×C3×C4C22×A411yescharacteristic
294C2×C3×C4C22×A411yescharacteristic
295C2×Dic3—31no—
296C2×Dic3—31no—
297C2×Dic3—31no—
298C2×Dic3C22×A411yescharacteristic
299C2×Dic3C22×A411yescharacteristic
300D12—22no—
301D12—61no—
302D12—61no—
303D12—62no—
304D12—121no—
305C4×D3—22no—
306C4×D3—61no—
307C4×D3—61no—
308C4×D3—61no—
309C4×D3—61no—
310C4×D3—61no—
311C4×D3—61no—
312C4×D3—121no—
313C4×D3—122no—
314Dic6—61no—
315Dic6—61no—
316Dic6—61no—
317Dic6—61no—
318SL(2,𝔽3)—21no—
319SL(2,𝔽3)#23911yescharacteristic
320C2×C32—41no—
321C2×C32—41no—
322C2×C32—41no—
323C3×D3—82no—
324C3×D3—161no—
325#41—61no—
326#41—181no—
327C2×Q8—31no—
328C2×Q8—61no—
329C2×Q8—61no—
330C2×Q8—91no—
331C2×Q8—91no—
332C2×Q8—181no—
333C2×Q8—181no—
334C2×Q8C22×C3×D311yescharacteristic
335C2×D4—31no—
336C2×D4—91no—
337C2×D4—91no—
338C2×D4—91no—
339C22×C4—31no—
340C22×C4—91no—
341C22×C4—91no—
342C22×C4—91no—
343C22×C4—91no—
344C22×C4—92no—
345C4⋊C4—31no—
346C4⋊C4—31no—
347C4⋊C4—31no—
348C4⋊C4—32no—
349C4⋊C4—31no—
350C4⋊C4—31no—
351C4⋊C4—31no—
352C4⋊C4—91no—
353C4⋊C4—91no—
354C4⋊C4—91no—
355C4⋊C4—91no—
356C4⋊C4—91no—
357C4⋊C4—92no—
358C4⋊C4—92no—
359C4⋊C4—91no—
360C4⋊C4—91no—
361C4⋊C4D3×A411yescharacteristic
362C22⋊C4—31no—
363C22⋊C4—31no—
364C22⋊C4—31no—
365C22⋊C4—91no—
366C22⋊C4—91no—
367C22⋊C4—91no—
368C22⋊C4—91no—
369C22⋊C4—92no—
370C22⋊C4—92no—
371C22⋊C4—91no—
372C22⋊C4—91no—
373C22⋊C4—91no—
374C42—31no—
375C42—31no—
376C42—31no—
377C42—91no—
378C42—91no—
379C42—92no—
380C42—91no—
381C22×C3—41no—
382C22×C3—81no—
383C22×C3—122no—
384C22×C3—241no—
385C22×C3—241no—
386C22×C3—241no—
387C22×C3—241no—
388C22×C3—241no—
389C22×C3C23×A411yescharacteristic
390C2×D3—21no—
391C2×D3—21no—
392C2×D3—21no—
393C2×D3—21no—
394C2×D3—21no—
395C2×D3—61no—
396C2×D3—61no—
397C2×D3—61no—
398C2×D32−1+4⋊C312yes—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—31no—
403C3×C4—31no—
404C3×C4—61no—
405C3×C4—61no—
406C3×C4—61no—
407C3×C4—81no—
408C3×C4—81no—
409C3×C4—81no—
410C3×C4—82no—
411C3×C4—161no—
412C3×C4—161no—
413C3×C4—241no—
414C3×C4—241no—
415Dic3—21no—
416Dic3—21no—
417Dic3—61no—
418Dic3—61no—
419Dic3—61no—
420C32—41noSylow
421C23—31no—
422C23—31no—
423C23—91no—
424Q8—31no—
425Q8—121no—
426Q8—181no—
427Q8—181no—
428Q8—181no—
429Q8—181no—
430Q8C3×#23911yescharacteristic
431D4—62no—
432D4—181no—
433D4—181no—
434D4—182no—
435D4—361no—
436C2×C4—31no—
437C2×C4—31no—
438C2×C4—31no—
439C2×C4—31no—
440C2×C4—31no—
441C2×C4—31no—
442C2×C4—62no—
443C2×C4—91no—
444C2×C4—91no—
445C2×C4—91no—
446C2×C4—181no—
447C2×C4—181no—
448C2×C4—181no—
449C2×C4—181no—
450C2×C4—181no—
451C2×C4—181no—
452C2×C4—361no—
453C2×C4—362no—
454C2×C4C2×D3×A411yescharacteristic
455C2×C4C2×D3×A411yescharacteristic
456C2×C4C2×D3×A411yescharacteristic
457C2×C3—41no—
458C2×C3—41no—
459C2×C3—41no—
460C2×C3—81no—
461C2×C3—81no—
462C2×C3—81no—
463C2×C3—242no—
464C2×C3—481no—
465C2×C3A4×#4111yescharacteristic
466C2×C3C2×2−1+4⋊C311yescharacteristic
467C2×C3#594511yescharacteristic
468D3—22no—
469D3—41no—
470D3—121no—
471C22—32no—
472C22—61no—
473C22—61no—
474C22—61no—
475C22—61no—
476C22—61no—
477C22—181no—
478C22—181no—
479C22—181no—
480C22C22×D3×A411yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—31no—
485C4—31no—
486C4—61no—
487C4—61no—
488C4—61no—
489C4—61no—
490C4—61no—
491C4—181no—
492C4—181no—
493C4—181no—
494C3—41no—
495C3—81no—
496C3#3516⋊C311yescharacteristic, complemented
497C2—62no—
498C2—121no—
499C2—361no—
500C2A4×#23911yescharacteristic
501C2#5964⋊C311yescharacteristic
502C2D3×2−1+4⋊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. SL(2,𝔽3)×D12C2semidirect
6. C4×D3×SL(2,𝔽3)C2semidirect
7. #5574⋊C3C2semidirect
8. #87022C3semidirect
18. C3×#557C2×C3semidirect
27. #5576C2×C3semidirect
28. #5576C2×C3semidirect
30. #5574C2×C3semidirect
31. #5572C2×C3semidirect
32. C4×D3×Q8C2×C3semidirect
34. #557⋊C3D3semidirect
124. #557C3×D3semidirect
496. C3#3516⋊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
SL(2,𝔽3)×D122
C4×D3×SL(2,𝔽3)2
#5574⋊C32
#870223
#3516⋊C33
C3×#8944