Finite Group Databasebeta

← #85884⋊C3 #11923216

Subgroups — #85884⋊C3 #11923216

Order 1152.

579 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 579 conjugacy classes of subgroups — too many to draw. Only its 79 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
2SL(2,𝔽3)×#67C211yesmaximal, characteristic, complemented
3SL(2,𝔽3)×#67C211yesmaximal, characteristic, complemented
4#5572⋊C3C211yesmaximal, characteristic, complemented
5SL(2,𝔽3)×D12C211yesmaximal, characteristic, complemented
6C4×D3×SL(2,𝔽3)C211yesmaximal, characteristic, complemented
7#5569⋊C3C211yesmaximal, characteristic, complemented
8#85884C311yesmaximal, characteristic, complemented
9#3523⋊C3—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11C2×D3×SL(2,𝔽3)C2211yescharacteristic
12C2×D3×SL(2,𝔽3)C2211yescharacteristic
13C3×#884—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, complemented
19Q8×#67C2×C311yescharacteristic, complemented
20Q8×#67C2×C311yescharacteristic, complemented
21#5583—31no—
22#5581—31no—
23#5578—31no—
24Q8×D12C2×C311yescharacteristic, complemented
25#5572C2×C311yescharacteristic, complemented
26C4×D3×Q8C2×C311yescharacteristic, complemented
27#5569C2×C311yescharacteristic, complemented
28#5567—31no—
29#5533—31no—
30#5529—31no—
31#5528—31no—
32#5518—31no—
33#557⋊C3—31no—
34D4×SL(2,𝔽3)—31no—
35D4×SL(2,𝔽3)—31no—
36D4×SL(2,𝔽3)—31no—
37#519⋊C3—21no—
38#519⋊C3—31no—
39#519⋊C3D311yescharacteristic, complemented
40C2×C4×SL(2,𝔽3)—31no—
41C2×C3×#67—41no—
42C2×C3×#67—41no—
43C2×C3×D12—41no—
44C2×C3×C4×D3—41no—
45C2×C3×SL(2,𝔽3)—21no—
46C2×C3×SL(2,𝔽3)C2311yescharacteristic
47D3×SL(2,𝔽3)—21no—
48D3×SL(2,𝔽3)—21no—
49C32×C22⋊C4—41no—
50C3×(C2×D3)⋊C4—41no—
51C3×Dic3⋊C4—41no—
52#3523—31noSylow
53C22×C3×Q8C22×C311yescharacteristic
54C2×D3×Q8C22×C311yescharacteristic
55C2×D3×Q8C22×C311yescharacteristic
56C22×SL(2,𝔽3)—21no—
57C22×SL(2,𝔽3)—31no—
58C22×SL(2,𝔽3)—31no—
59C22×SL(2,𝔽3)C2×D311yescharacteristic
60C3×#121—121no—
61C3×C4×Q8—31no—
62C3×C4×Q8—31no—
63C3×C4×Q8C22×C311yescharacteristic
64C3×C4×Q8C22×C311yescharacteristic
65C3×#117—31no—
66C3×#117—31no—
67#946—31no—
68#946—31no—
69Q8×Dic3C22×C311yescharacteristic
70Q8×Dic3C22×C311yescharacteristic
71#944—31no—
72#944—31no—
73C4×#67—31no—
74C4×#67—31no—
75#924—31no—
76#924—31no—
77#898—31no—
78#897—31no—
79#897—31no—
80#896—31no—
81#894—31no—
82#892—31no—
83D3×C4⋊C4—31no—
84#890—31no—
85#889—32no—
86#889—32no—
87#889—31no—
88#888—31no—
89#887—31no—
90#887—31no—
91#885—31no—
92#884A411yescharacteristic
93#883—31no—
94#879—32no—
95#879—31no—
96#876—32no—
97C4×D12—31no—
98#872—32no—
99#872—31no—
100C42×D3—31no—
101#870—32no—
102#870—31no—
103#869—31no—
104C4×Dic6—32no—
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
114C23×C32—41no—
115C22×C3×D3—41no—
116C22×C3×D3—41no—
117C2×C32×C4—41no—
118C2×C32×C4—41no—
119C3×#67—82no—
120C3×#67—82no—
121C2×C3×Dic3—41no—
122C2×C3×Dic3—41no—
123C3×D12—82no—
124C3×C4×D3—82no—
125C3×SL(2,𝔽3)#4111yescharacteristic
126#557—31no—
127#557—91no—
128#556—91no—
129#552—91no—
130#552—91no—
131D4×Q8—31no—
132D4×Q8—31no—
133D4×Q8—31no—
134#541—91no—
135#541—91no—
136#533—91no—
137#529—91no—
138#519—31no—
139#519C3×D311yescharacteristic, complemented
140C2×C4×Q8—31no—
141C2×C3×Q8—21no—
142C2×C3×Q8—31no—
143C2×C3×Q8—61no—
144C2×C3×Q8C23×C311yescharacteristic
145C2×C3×D4—121no—
146C2×C3×D4—121no—
147C2×C3×D4—121no—
148C22×C3×C4—31no—
149C22×C3×C4—121no—
150C2×#67C2×A411yescharacteristic
151C2×#67C2×A411yescharacteristic
152D3×Q8—21no—
153D3×Q8—21no—
154D3×Q8—61no—
155D3×Q8—61no—
156C2×D12C2×A411yescharacteristic
157C2×C4×D3—31no—
158C2×C4×D3—31no—
159C2×C4×D3—31no—
160C2×C4×D3C2×A411yescharacteristic
161C2×Dic6—31no—
162C2×Dic6—31no—
163C2×Dic6—31no—
164C2×SL(2,𝔽3)—22no—
165C2×SL(2,𝔽3)—21no—
166C2×SL(2,𝔽3)—21no—
167C2×SL(2,𝔽3)—61no—
168C2×SL(2,𝔽3)—61no—
169C2×SL(2,𝔽3)C22×D311yescharacteristic
170C3×C4⋊C4—31no—
171C3×C4⋊C4—31no—
172C3×C4⋊C4—31no—
173C3×C4⋊C4—31no—
174C3×C4⋊C4—121no—
175C3×C22⋊C4—31no—
176C3×C22⋊C4—41no—
177C3×C22⋊C4—81no—
178C3×C22⋊C4—121no—
179C3×C22⋊C4C2×A411yescharacteristic
180C3×C42—32no—
181C3×C42—31no—
182C3×C42—31no—
183#217—31no—
184#217—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)⋊C4C2×A411yescharacteristic
191C4⋊Dic3—32no—
192C4⋊Dic3—31no—
193C4⋊Dic3—31no—
194C4⋊Dic3—31no—
195C4⋊Dic3—31no—
196Dic3⋊C4—32no—
197Dic3⋊C4—32no—
198Dic3⋊C4—32no—
199Dic3⋊C4—32no—
200Dic3⋊C4—31no—
201Dic3⋊C4—31no—
202Dic3⋊C4—31no—
203Dic3⋊C4—31no—
204Dic3⋊C4—31no—
205Dic3⋊C4C2×A411yescharacteristic
206C4×Dic3—32no—
207C4×Dic3—31no—
208C4×Dic3—31no—
209C4×Dic3—31no—
210C4×Dic3—31no—
211C22×C32—41no—
212C22×C32—81no—
213C22×C32—81no—
214C22×C32—81no—
215C2×C3×D3—42no—
216C2×C3×D3—81no—
217C2×C3×D3—81no—
218C2×C3×D3—81no—
219C2×C3×D3—81no—
220C2×C3×D3—81no—
221C32×C4—81no—
222C32×C4—81no—
223C3×Dic3—42no—
224C3×Dic3—81no—
225C22×Q8—31no—
226C22×Q8—31no—
227C22×Q8C2×C3×D311yescharacteristic
228C4⋊Q8—91no—
229C4⋊Q8—91no—
230C4⋊Q8—91no—
231C4⋊Q8—91no—
232#126—92no—
233#126—91no—
234#126—92no—
235#126—91no—
236#125—91no—
237#125—92no—
238#125—91no—
239#125—91no—
240#125—92no—
241#125—92no—
242#124—91no—
243#123—91no—
244#123—91no—
245C22⋊Q8—91no—
246C22⋊Q8—91no—
247C22⋊Q8—91no—
248C22⋊Q8—91no—
249C22⋊Q8—91no—
250C22⋊Q8—91no—
251C22⋊Q8—91no—
252#121—31no—
253C4×Q8—31no—
254C4×Q8—31no—
255C4×Q8—31no—
256C4×Q8—31no—
257C4×Q8—91no—
258C4×Q8—92no—
259C4×Q8—91no—
260C4×Q8—91no—
261C4×Q8—91no—
262C4×Q8—91no—
263C4×Q8C2×C3×D311yescharacteristic
264C4×Q8C2×C3×D311yescharacteristic
265C4×D4—91no—
266C4×D4—91no—
267C4×D4—91no—
268#117—31no—
269#117—31no—
270#117—91no—
271#117—92no—
272#117—91no—
273C2×C4⋊C4—91no—
274C2×C42—91no—
275C23×C3—41no—
276C23×C3—81no—
277C23×C3—121no—
278C23×C3—121no—
279C23×C3C22×A411yescharacteristic
280C22×D3C22×A411yescharacteristic
281C22×D3C22×A411yescharacteristic
282C3×Q8—31no—
283C3×Q8—61no—
284C3×Q8—61no—
285C3×Q8C3×#4111yescharacteristic
286C3×D4—242no—
287C3×D4—242no—
288C3×D4—242no—
289C2×C3×C4—31no—
290C2×C3×C4—31no—
291C2×C3×C4—31no—
292C2×C3×C4—31no—
293C2×C3×C4—41no—
294C2×C3×C4—41no—
295C2×C3×C4—61no—
296C2×C3×C4—61no—
297C2×C3×C4—81no—
298C2×C3×C4—81no—
299C2×C3×C4—121no—
300C2×C3×C4—121no—
301C2×C3×C4—242no—
302C2×C3×C4C22×A411yescharacteristic
303C2×C3×C4C22×A411yescharacteristic
304#67—22no—
305#67—22no—
306C2×Dic3—31no—
307C2×Dic3—31no—
308C2×Dic3—31no—
309C2×Dic3—31no—
310C2×Dic3C22×A411yescharacteristic
311C2×Dic3C22×A411yescharacteristic
312D12—22no—
313C4×D3—22no—
314C4×D3—61no—
315C4×D3—61no—
316C4×D3—61no—
317C4×D3—61no—
318C4×D3—62no—
319Dic6—62no—
320Dic6—61no—
321Dic6—61no—
322Dic6—61no—
323Dic6—61no—
324SL(2,𝔽3)—21no—
325SL(2,𝔽3)#23511yescharacteristic
326C2×C32—41no—
327C2×C32—41no—
328C2×C32—41no—
329C2×C32—161no—
330C3×D3—82no—
331C3×D3—161no—
332C2×Q8—21no—
333C2×Q8—31no—
334C2×Q8—61no—
335C2×Q8—61no—
336C2×Q8—61no—
337C2×Q8—91no—
338C2×Q8—91no—
339C2×Q8—91no—
340C2×Q8—181no—
341C2×Q8—181no—
342C2×Q8C22×C3×D311yescharacteristic
343C2×D4—31no—
344C2×D4—31no—
345C2×D4—31no—
346C22×C4—31no—
347C22×C4—31no—
348C22×C4—91no—
349C22×C4—91no—
350C22×C4—91no—
351C4⋊C4—31no—
352C4⋊C4—31no—
353C4⋊C4—31no—
354C4⋊C4—31no—
355C4⋊C4—31no—
356C4⋊C4—91no—
357C4⋊C4—91no—
358C4⋊C4—91no—
359C4⋊C4—91no—
360C4⋊C4—91no—
361C4⋊C4—91no—
362C4⋊C4—92no—
363C4⋊C4—92no—
364C4⋊C4—92no—
365C4⋊C4—92no—
366C4⋊C4—92no—
367C4⋊C4—91no—
368C4⋊C4—91no—
369C4⋊C4—91no—
370C22⋊C4—31no—
371C22⋊C4—31no—
372C22⋊C4—91no—
373C22⋊C4—91no—
374C22⋊C4—91no—
375C22⋊C4—91no—
376C22⋊C4—91no—
377C22⋊C4—91no—
378C22⋊C4—91no—
379C22⋊C4D3×A411yescharacteristic
380C42—32no—
381C42—31no—
382C42—31no—
383C42—91no—
384C42—91no—
385C42—91no—
386C42—92no—
387C42—91no—
388C22×C3—21no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—41no—
392C22×C3—81no—
393C22×C3—81no—
394C22×C3—81no—
395C22×C3—82no—
396C22×C3—81no—
397C22×C3—122no—
398C22×C3—161no—
399C22×C3—161no—
400C22×C3—241no—
401C22×C3—241no—
402C22×C3—241no—
403C22×C3—241no—
404C22×C3—241no—
405C22×C3C23×A411yescharacteristic
406C2×D3—21no—
407C2×D3—21no—
408C2×D3—21no—
409C2×D3—21no—
410C2×D3—21no—
411C2×D32−1+4⋊C312yes—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—31no—
415C3×C4—31no—
416C3×C4—61no—
417C3×C4—61no—
418C3×C4—61no—
419C3×C4—81no—
420C3×C4—81no—
421C3×C4—122no—
422C3×C4—161no—
423C3×C4—161no—
424C3×C4—241no—
425Dic3—21no—
426Dic3—61no—
427Dic3—61no—
428Dic3—61no—
429Dic3—61no—
430Dic32−1+4⋊C312yes—
431C32—41noSylow
432C23—31no—
433C23—31no—
434C23C2×D3×A411yescharacteristic
435Q8—31no—
436Q8—61no—
437Q8—61no—
438Q8—182no—
439Q8—181no—
440Q8—181no—
441Q8—181no—
442Q8—181no—
443Q8C3×#23511yescharacteristic
444D4—62no—
445D4—62no—
446D4—62no—
447C2×C4—31no—
448C2×C4—31no—
449C2×C4—31no—
450C2×C4—31no—
451C2×C4—31no—
452C2×C4—31no—
453C2×C4—62no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—91no—
457C2×C4—91no—
458C2×C4—91no—
459C2×C4—91no—
460C2×C4—181no—
461C2×C4—181no—
462C2×C4—181no—
463C2×C4—181no—
464C2×C4—182no—
465C2×C4C2×D3×A411yescharacteristic
466C2×C4C2×D3×A411yescharacteristic
467C2×C3—41no—
468C2×C3—41no—
469C2×C3—41no—
470C2×C3—41no—
471C2×C3—81no—
472C2×C3—81no—
473C2×C3—81no—
474C2×C3—161no—
475C2×C3—162no—
476C2×C3—242no—
477C2×C3—481no—
478C2×C3A4×#4111yescharacteristic
479C2×C3C2×2−1+4⋊C311yescharacteristic
480C2×C3C2×2−1+4⋊C311yescharacteristic
481D3—22no—
482D3—41no—
483C22—21no—
484C22—21no—
485C22—21no—
486C22—32no—
487C22—61no—
488C22—61no—
489C22—61no—
490C22—61no—
491C22—61no—
492C22C22×D3×A411yescharacteristic
493C4—21no—
494C4—21no—
495C4—31no—
496C4—31no—
497C4—32no—
498C4—61no—
499C4—61no—
500C4—61no—
501C4—61no—
502C4—181no—
503C4—181no—
504C4—181no—
505C4—181no—
506C3—41no—
507C3—81no—
508C3#3523⋊C311yescharacteristic, complemented
509C2—41no—
510C2—62no—
511C2—121no—
512C2A4×#23511yescharacteristic
513C2D3×2−1+4⋊C311yescharacteristic
514C2D3×2−1+4⋊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. SL(2,𝔽3)×#67C2semidirect
3. SL(2,𝔽3)×#67C2semidirect
4. #5572⋊C3C2semidirect
5. SL(2,𝔽3)×D12C2semidirect
6. C4×D3×SL(2,𝔽3)C2semidirect
7. #5569⋊C3C2semidirect
8. #85884C3semidirect
18. C3×#519C2×C3semidirect
19. Q8×#67C2×C3semidirect
20. Q8×#67C2×C3semidirect
24. Q8×D12C2×C3semidirect
25. #5572C2×C3semidirect
26. C4×D3×Q8C2×C3semidirect
27. #5569C2×C3semidirect
39. #519⋊C3D3semidirect
139. #519C3×D3semidirect
508. C3#3523⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#519⋊C32
SL(2,𝔽3)×#672
#5572⋊C32
SL(2,𝔽3)×D122
C4×D3×SL(2,𝔽3)2
#5569⋊C32
#858843
#3523⋊C33
C3×#8844