Finite Group Databasebeta

← #85898⋊C3 #11923229

Subgroups — #85898⋊C3 #11923229

Order 1152.

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

normal subgroups only

This group has 531 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×#520⋊C3C211yesmaximal, characteristic, complemented
2#5831⋊C3C211yesmaximal, characteristic, complemented
3#5834⋊C3C211yesmaximal, characteristic, complemented
4#5576⋊C3C211yesmaximal, characteristic, complemented
5#5572⋊C3C211yesmaximal, characteristic, complemented
6#5570⋊C3C211yesmaximal, characteristic, complemented
7C4×C3⋊(#41⋊C3)C211yesmaximal, characteristic, complemented
8#85898C311yesmaximal, characteristic, complemented
9#3524⋊C3—31nomaximal
10C2×C3×#41⋊C3C2211yescharacteristic
11C2×D3×SL(2,𝔽3)C2211yescharacteristic
12C2×C3⋊(#41⋊C3)C2211yescharacteristic
13C3×#898—41nomaximal
14C3×C4×SL(2,𝔽3)C2211yescharacteristic
15C3×C4×SL(2,𝔽3)C2211yescharacteristic
16Dic3×SL(2,𝔽3)C2211yescharacteristic
17Dic3×SL(2,𝔽3)C2211yescharacteristic
18C3×#520C2×C311yescharacteristic, complemented
19#5834C2×C311yescharacteristic, complemented
20#5831C2×C311yescharacteristic
21#5576C2×C311yescharacteristic
22C4×#239C2×C311yescharacteristic, complemented
23#5572C2×C311yescharacteristic
24#5570C2×C311yescharacteristic, complemented
25#5563—31no—
26#5562—31no—
27#5559—31no—
28#5557—31no—
29#5534—31no—
30#5528—31no—
31#5514—31no—
32#5513—31no—
33#557⋊C3—31no—
34C4⋊(#41⋊C3)—31no—
35#548⋊C3—31no—
36#548⋊C3—31no—
37#520⋊C3—21no—
38#520⋊C3D311yescharacteristic, complemented
39#519⋊C3—31no—
40C4×#41⋊C3—31no—
41C3×#41⋊C3—21no—
42C2×C3×SL(2,𝔽3)C2311yescharacteristic
43D3×SL(2,𝔽3)—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×C3×#41C22×C311yescharacteristic
54C2×#239C22×C311yescharacteristic
55C2×D3×Q8C22×C311yescharacteristic
56C2×#41⋊C3—21no—
57C2×#41⋊C3—31no—
58C2×#41⋊C3C2×D311yescharacteristic
59C22×SL(2,𝔽3)—31no—
60C3×#126—121no—
61C3×C4×Q8C22×C311yescharacteristic
62C3×C4×Q8C22×C311yescharacteristic
63C3×C4×D4—31no—
64C3×C4×D4—31no—
65C3×#117—31no—
66C2×C3×C4⋊C4—31no—
67Q8×Dic3C22×C311yescharacteristic
68Q8×Dic3C22×C311yescharacteristic
69#940—31no—
70#938—31no—
71#938—31no—
72#936—31no—
73#930—32no—
74#930—31no—
75#929—32no—
76C4×#67—32no—
77C4×#67—31no—
78C4×#67—31no—
79#926—32no—
80#924—31no—
81#898—32no—
82#898A411yescharacteristic
83#897—31no—
84#896—31no—
85#895—31no—
86#894—31no—
87#893—31no—
88#892—31no—
89#892—31no—
90#890—31no—
91#890—31no—
92#889—31no—
93#887—31no—
94#885—32no—
95#885—31no—
96#884—31no—
97#883—32no—
98#883—31no—
99#879—31no—
100#875—31no—
101C4×D12—31no—
102C4×D12—31no—
103#872—31no—
104C42×D3—31no—
105#869—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×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#520C3×D311yescharacteristic, complemented
135#519—31no—
136C4×#41—31no—
137C3×#41—21no—
138C3×#41—61no—
139C2×C3×Q8C23×C311yescharacteristic
140C2×C3×D4—31no—
141C22×C3×C4—32no—
142C22×C3×C4—31no—
143C2×#67—32no—
144C2×#67—31no—
145C2×#67—31no—
146#239—21no—
147#239—61no—
148D3×Q8—21no—
149D3×Q8—61no—
150C2×D12—31no—
151C2×D12—31no—
152C2×C4×D3—31no—
153C2×C4×D3—31no—
154C2×C4×D3—31no—
155C2×C4×D3—31no—
156C2×Dic6—31no—
157C2×Dic6—31no—
158#41⋊C3—22no—
159#41⋊C3—21no—
160#41⋊C3—61no—
161C2×SL(2,𝔽3)—21no—
162C2×SL(2,𝔽3)—61no—
163C2×SL(2,𝔽3)C22×D311yescharacteristic
164C3×C4⋊C4—31no—
165C3×C4⋊C4—31no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—41no—
168C3×C4⋊C4—81no—
169C3×C4⋊C4—121no—
170C3×C4⋊C4—121no—
171C3×C4⋊C4C2×A411yescharacteristic
172C3×C22⋊C4—31no—
173C3×C22⋊C4—31no—
174C3×C22⋊C4—121no—
175C3×C22⋊C4—121no—
176C3×C22⋊C4—121no—
177C3×C42—31no—
178C3×C42—31no—
179C3×C42—121no—
180#217—32no—
181#217—31no—
182#217—31no—
183(C2×D3)⋊C4—32no—
184(C2×D3)⋊C4—32no—
185(C2×D3)⋊C4—31no—
186(C2×D3)⋊C4—31no—
187(C2×D3)⋊C4—31no—
188(C2×D3)⋊C4C2×A411yescharacteristic
189(C2×D3)⋊C4C2×A411yescharacteristic
190(C2×D3)⋊C4C2×A411yescharacteristic
191C4⋊Dic3—32no—
192C4⋊Dic3—31no—
193C4⋊Dic3—31no—
194C4⋊Dic3—31no—
195C4⋊Dic3C2×A411yescharacteristic
196Dic3⋊C4—31no—
197Dic3⋊C4—31no—
198Dic3⋊C4—31no—
199Dic3⋊C4—31no—
200Dic3⋊C4—31no—
201Dic3⋊C4C2×A411yescharacteristic
202C4×Dic3—32no—
203C4×Dic3—31no—
204C4×Dic3—31no—
205C4×Dic3—31no—
206C4×Dic3C2×A411yescharacteristic
207C22×C32—41no—
208C2×C3×D3—81no—
209C2×C3×D3—81no—
210C2×C3×D3—81no—
211C32×C4—81no—
212C32×C4—81no—
213C32×C4—81no—
214C3×Dic3—81no—
215C3×Dic3—81no—
216C3×Dic3—81no—
217C2×#41—31no—
218C2×#41C2×C3×D311yescharacteristic
219C22×Q8—31no—
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—91no—
229#124—91no—
230#124—92no—
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—91no—
240#121—91no—
241#121—92no—
242#121—91no—
243#121—91no—
244#121—91no—
245C4×Q8—31no—
246C4×Q8—31no—
247C4×Q8—91no—
248C4×Q8—91no—
249C4×Q8—91no—
250C4×Q8C2×C3×D311yescharacteristic
251C4×Q8C2×C3×D311yescharacteristic
252C4×D4—31no—
253C4×D4—31no—
254C4×D4—91no—
255C4×D4—91no—
256C4×D4—91no—
257C4×D4—91no—
258C4×D4—91no—
259C4×D4—92no—
260#117—31no—
261#117—91no—
262#117—92no—
263#117—91no—
264#117—91no—
265C2×C4⋊C4—31no—
266C2×C42—91no—
267C23×C3—31no—
268C23×C3—121no—
269C22×D3—31no—
270C22×D3C22×A411yescharacteristic
271C3×Q8—31no—
272C3×Q8C3×#4111yescharacteristic
273C3×D4—61no—
274C3×D4—61no—
275C2×C3×C4—31no—
276C2×C3×C4—31no—
277C2×C3×C4—31no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—61no—
282C2×C3×C4—61no—
283C2×C3×C4—81no—
284C2×C3×C4—81no—
285C2×C3×C4—81no—
286C2×C3×C4—121no—
287C2×C3×C4—121no—
288C2×C3×C4—121no—
289C2×C3×C4—122no—
290C2×C3×C4C22×A411yescharacteristic
291C2×C3×C4C22×A411yescharacteristic
292C2×C3×C4C22×A411yescharacteristic
293#67—121no—
294#67—121no—
295#67—122no—
296C2×Dic3—31no—
297C2×Dic3—31no—
298C2×Dic3—31no—
299C2×Dic3C22×A411yescharacteristic
300C2×Dic3C22×A411yescharacteristic
301C2×Dic3C22×A411yescharacteristic
302D12—61no—
303D12—61no—
304D12—121no—
305C4×D3—61no—
306C4×D3—61no—
307C4×D3—61no—
308C4×D3—61no—
309C4×D3—121no—
310C4×D3—121no—
311Dic6—61no—
312Dic6—61no—
313Dic6—121no—
314SL(2,𝔽3)—21no—
315SL(2,𝔽3)#23511yescharacteristic
316C2×C32—41no—
317C2×C32—41no—
318C2×C32—41no—
319C3×D3—161no—
320#41—21no—
321#41—61no—
322#41—61no—
323#41—181no—
324C2×Q8—61no—
325C2×Q8—91no—
326C2×Q8—91no—
327C2×Q8—181no—
328C2×Q8C22×C3×D311yescharacteristic
329C2×D4—31no—
330C2×D4—91no—
331C2×D4—91no—
332C2×D4—91no—
333C2×D4—91no—
334C2×D4—92no—
335C22×C4—32no—
336C22×C4—31no—
337C22×C4—91no—
338C22×C4—91no—
339C22×C4—91no—
340C22×C4—91no—
341C4⋊C4—31no—
342C4⋊C4—31no—
343C4⋊C4—31no—
344C4⋊C4—31no—
345C4⋊C4—31no—
346C4⋊C4—91no—
347C4⋊C4—91no—
348C4⋊C4—91no—
349C4⋊C4—91no—
350C4⋊C4—91no—
351C4⋊C4—91no—
352C4⋊C4—92no—
353C4⋊C4—91no—
354C4⋊C4—91no—
355C4⋊C4D3×A411yescharacteristic
356C22⋊C4—31no—
357C22⋊C4—31no—
358C22⋊C4—31no—
359C22⋊C4—31no—
360C22⋊C4—31no—
361C22⋊C4—91no—
362C22⋊C4—91no—
363C22⋊C4—91no—
364C22⋊C4—92no—
365C22⋊C4—92no—
366C22⋊C4—92no—
367C22⋊C4—91no—
368C22⋊C4—91no—
369C42—31no—
370C42—31no—
371C42—31no—
372C42—91no—
373C42—91no—
374C42—91no—
375C42—92no—
376C22×C3—41no—
377C22×C3—61no—
378C22×C3—61no—
379C22×C3—61no—
380C22×C3—81no—
381C22×C3—241no—
382C22×C3—241no—
383C22×C3—241no—
384C22×C3C23×A411yescharacteristic
385C2×D3—21no—
386C2×D3—21no—
387C2×D3—21no—
388C2×D3—61no—
389C2×D3—61no—
390C2×D3—61no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—31no—
395C3×C4—31no—
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—
412Dic3—61no—
413C32—41noSylow
414C23—31no—
415C23—31no—
416C23—91no—
417Q8—31no—
418Q8—181no—
419Q8—181no—
420Q8—361no—
421Q8C3×#23511yescharacteristic
422D4—61no—
423D4—61no—
424D4—181no—
425D4—181no—
426D4—361no—
427D4—361no—
428D4—361no—
429D4—362no—
430C2×C4—31no—
431C2×C4—31no—
432C2×C4—31no—
433C2×C4—31no—
434C2×C4—31no—
435C2×C4—31no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—91no—
439C2×C4—91no—
440C2×C4—91no—
441C2×C4—122no—
442C2×C4—181no—
443C2×C4—181no—
444C2×C4—181no—
445C2×C4—181no—
446C2×C4—361no—
447C2×C4—361no—
448C2×C4C2×D3×A411yescharacteristic
449C2×C4C2×D3×A411yescharacteristic
450C2×C4C2×D3×A411yescharacteristic
451C2×C3—41no—
452C2×C3—41no—
453C2×C3—41no—
454C2×C3—81no—
455C2×C3—81no—
456C2×C3—81no—
457C2×C3—121no—
458C2×C3—481no—
459C2×C3A4×#4111yescharacteristic
460C2×C3#594511yescharacteristic
461C2×C3#594511yescharacteristic
462D3—41no—
463D3—121no—
464C22—61no—
465C22—61no—
466C22—61no—
467C22—61no—
468C22—61no—
469C22—61no—
470C22—181no—
471C22—181no—
472C22—181no—
473C22C22×D3×A411yescharacteristic
474C4—21no—
475C4—21no—
476C4—21no—
477C4—31no—
478C4—31no—
479C4—61no—
480C4—61no—
481C4—61no—
482C4—61no—
483C4—61no—
484C4—181no—
485C4—181no—
486C4—181no—
487C3—41no—
488C3—81no—
489C3#3524⋊C311yescharacteristic, complemented
490C2—121no—
491C2—121no—
492C2—361no—
493C2A4×#23511yescharacteristic
494C2#1059662311yescharacteristic
495C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#520C2×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×#520⋊C3C2semidirect
2. #5831⋊C3C2semidirect
3. #5834⋊C3C2semidirect
4. #5576⋊C3C2semidirect
5. #5572⋊C3C2semidirect
6. #5570⋊C3C2semidirect
7. C4×C3⋊(#41⋊C3)C2semidirect
8. #85898C3semidirect
18. C3×#520C2×C3semidirect
19. #5834C2×C3semidirect
22. C4×#239C2×C3semidirect
24. #5570C2×C3semidirect
38. #520⋊C3D3semidirect
134. #520C3×D3semidirect
489. C3#3524⋊C3semidirect

Maximal subgroups

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

subgroupindex
C3×#520⋊C32
#5831⋊C32
#5834⋊C32
#5576⋊C32
#5572⋊C32
#5570⋊C32
C4×C3⋊(#41⋊C3)2
#858983
#3524⋊C33
C3×#8984