Finite Group Databasebeta

← #85900⋊C3 #11923228

Subgroups — #85900⋊C3 #11923228

Order 1152.

787 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 787 conjugacy classes of subgroups — too many to draw. Only its 89 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
1C2×D3×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×C3⋊#994C211yesmaximal, characteristic, complemented
3C3×#520⋊C3C211yesmaximal, characteristic, complemented
4#5576⋊C3C212yesmaximal, complemented
5C4⋊(C3⋊(#41⋊C3))C212yesmaximal, complemented
6#85900C311yesmaximal, characteristic, complemented
7#3488⋊C3—31nomaximal
8C2×C3×#41⋊C3C2211yescharacteristic
9D3×#41⋊C3—22no—
10C3⋊#994—22no—
11C2×D3×SL(2,𝔽3)C2211yescharacteristic, complemented
12C2×C3⋊(#41⋊C3)C2212yescomplemented
13C2×C3⋊(#41⋊C3)C2211yescharacteristic, complemented
14C3×#897—41nomaximal
15C3×C4×SL(2,𝔽3)C2212yescomplemented
16C2×#1009C2×C311yescharacteristic, complemented
17C2×D3×#41C2×C311yescharacteristic
18C2×#994—31no—
19C22×#41⋊C3—31no—
20C3×#520C2×C311yescharacteristic, complemented
21#5577C2×C312yescomplemented
22#5576C2×C312yes—
23#5554—32no—
24D4×D12—32no—
25#5525—31no—
26#5524—31no—
27#5510—31no—
28C2×#895—31no—
29C4⋊(#41⋊C3)—32no—
30#548⋊C3—32no—
31#520⋊C3—21no—
32#520⋊C3D311yescharacteristic, complemented
33C2×C3×C4×D3—41no—
34C2×C3×Dic6—41no—
35C3×#41⋊C3D412yescomplemented
36C2×C3×SL(2,𝔽3)C2311yescharacteristic
37D3×SL(2,𝔽3)—21no—
38C3⋊(#41⋊C3)—22no—
39C3⋊(#41⋊C3)—21no—
40C32×C4⋊C4—41no—
41C3×(C2×D3)⋊C4—42no—
42C3×C4⋊Dic3—42no—
43#3488—31noSylow
44C2×C3×#41C22×C311yescharacteristic
45#1009—22no—
46#1009—62no—
47D3×#41—22no—
48D3×#41—62no—
49C2×#239C22×C311yescharacteristic
50C2×#239C22×C312yes—
51C2×D3×Q8C22×C311yescharacteristic
52C2×#237—31no—
53C2×D3×D4—31no—
54C2×D3×D4—32no—
55C2×#235—31no—
56C2×#235—31no—
57C22×D12—32no—
58C22×D12—31no—
59C22×C4×D3—31no—
60#994—62no—
61C2×#41⋊C3—21no—
62C2×#41⋊C3—31no—
63C2×#41⋊C3—32no—
64C2×#41⋊C3—62no—
65C2×#41⋊C3C2×D311yescharacteristic
66C22×SL(2,𝔽3)—31no—
67C3×C22⋊Q8—121no—
68C3×C4×Q8C22×C312yes—
69C3×C4×D4—32no—
70C3×#117—31no—
71C2×C3×C4⋊C4—31no—
72#930—34no—
73C2×(C2×D3)⋊C4—32no—
74#897—31no—
75#897—32no—
76#897A411yescharacteristic
77#895—32no—
78#895—32no—
79#895—31no—
80#895—31no—
81#895—32no—
82#895—32no—
83#886—32no—
84#882—34no—
85#882—32no—
86#875—32no—
87#874—32no—
88C4×D12—32no—
89C4×D12—32no—
90C4×SL(2,𝔽3)—22no—
91C4×SL(2,𝔽3)C2×D312yescomplemented
92C22×C3×D3—41no—
93C2×C32×C4—41no—
94C2×C32×C4—42no—
95C2×C3×Dic3—41no—
96C2×C3×Dic3—42no—
97C3×C4×D3—82no—
98C3×Dic6—82no—
99C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
100C2×H(5,𝔽2)—31no—
101C22×#41—31no—
102#550—32no—
103#548—32no—
104#546—92no—
105D42—92no—
106#537—91no—
107#535—91no—
108#525—91no—
109C2×#121—91no—
110#520C3×D311yescharacteristic, complemented
111C23×D3—32no—
112C23×D3—31no—
113C3×#41—32no—
114C3×#41C3×D412yes—
115C2×C3×Q8—121no—
116C2×C3×Q8C23×C311yescharacteristic
117C2×C3×D4—31no—
118C22×C3×C4—31no—
119C22×C3×C4—32no—
120C22×C3×C4—121no—
121C2×#67—32no—
122C2×#67—32no—
123C22×Dic3—31no—
124#239—21no—
125#239—22no—
126#239—61no—
127#239—62no—
128D3×Q8—21no—
129D3×Q8—61no—
130#237—62no—
131#237—62no—
132D3×D4—62no—
133D3×D4—62no—
134D3×D4—64no—
135D3×D4—64no—
136#235—62no—
137#235—62no—
138#235—62no—
139#235—62no—
140C2×D12—31no—
141C2×D12—32no—
142C2×D12—32no—
143C2×D12—31no—
144C2×D12—34no—
145C2×D12—32no—
146C2×D12—62no—
147C2×D12—62no—
148C2×D12—124no—
149C2×C4×D3—31no—
150C2×C4×D3—31no—
151C2×C4×D3—31no—
152C2×C4×D3—32no—
153C2×C4×D3—62no—
154C2×C4×D3—62no—
155C2×C4×D3C2×A411yescharacteristic
156C2×Dic6—31no—
157C2×Dic6C2×A411yescharacteristic
158#41⋊C3—22no—
159#41⋊C3—61no—
160#41⋊C3—62no—
161#41⋊C3D1212yescomplemented
162C2×SL(2,𝔽3)—21no—
163C2×SL(2,𝔽3)—61no—
164C2×SL(2,𝔽3)C22×D311yescharacteristic
165C3×C4⋊C4—32no—
166C3×C4⋊C4—31no—
167C3×C4⋊C4—41no—
168C3×C4⋊C4—81no—
169C3×C4⋊C4—122no—
170C3×C4⋊C4C2×A411yescharacteristic
171C3×C22⋊C4—32no—
172C3×C22⋊C4—122no—
173C3×C42—32no—
174(C2×D3)⋊C4—34no—
175(C2×D3)⋊C4—32no—
176(C2×D3)⋊C4—32no—
177(C2×D3)⋊C4—32no—
178(C2×D3)⋊C4C2×A412yes—
179C4⋊Dic3—32no—
180C4⋊Dic3C2×A412yes—
181C22×C32—41no—
182C2×C3×D3—42no—
183C2×C3×D3—81no—
184C2×C3×D3—81no—
185C32×C4—42no—
186C32×C4—82no—
187C3×Dic3—82no—
188C3×Dic3—81no—
189H(5,𝔽2)—62no—
190H(5,𝔽2)—182no—
191C2×#41—32no—
192C2×#41—31no—
193C2×#41—62no—
194C2×#41—91no—
195C2×#41—91no—
196C2×#41—91no—
197C2×#41—182no—
198C2×#41C2×C3×D311yescharacteristic
199C22×Q8—31no—
200C22×D4—92no—
201C22×D4—91no—
202C22×D4—91no—
203C22×D4—92no—
204C23×C4—91no—
205#127—92no—
206#124—92no—
207#123—92no—
208C22⋊Q8—31no—
209C22⋊Q8—91no—
210C22⋊Q8—92no—
211#121—91no—
212#121—92no—
213#121—92no—
214#121—94no—
215#121—91no—
216#121—92no—
217#121—92no—
218#120—92no—
219#120—94no—
220C4×Q8C2×C3×D312yes—
221C4×D4—32no—
222C4×D4—92no—
223C4×D4—92no—
224#117—31no—
225C2×C4⋊C4—31no—
226C2×C22⋊C4—92no—
227C23×C3—31no—
228C23×C3—121no—
229C22×D3—32no—
230C22×D3—31no—
231C22×D3—62no—
232C22×D3—64no—
233C22×D3—62no—
234C22×D3—121no—
235C22×D3—122no—
236C22×D3—122no—
237C22×D3C22×A411yescharacteristic
238C3×Q8—31no—
239C3×Q8—242no—
240C3×Q8C2×C3×D411yescharacteristic
241C3×D4—32no—
242C3×D4—32no—
243C2×C3×C4—31no—
244C2×C3×C4—32no—
245C2×C3×C4—32no—
246C2×C3×C4—32no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—81no—
250C2×C3×C4—82no—
251C2×C3×C4—122no—
252C2×C3×C4—121no—
253C2×C3×C4—122no—
254C2×C3×C4—242no—
255C2×C3×C4C22×A412yes—
256C2×C3×C4C22×A411yescharacteristic
257#67—64no—
258#67—64no—
259C2×Dic3—31no—
260C2×Dic3—62no—
261C2×Dic3C22×A412yes—
262C2×Dic3C22×A411yescharacteristic
263D12—62no—
264D12—62no—
265D12—61no—
266D12—61no—
267D12—64no—
268D12—62no—
269D12—122no—
270D12—124no—
271C4×D3—22no—
272C4×D3—61no—
273C4×D3—61no—
274C4×D3—62no—
275C4×D3—62no—
276C4×D3—61no—
277C4×D3—61no—
278C4×D3—62no—
279Dic6—22no—
280Dic6—61no—
281Dic6—61no—
282SL(2,𝔽3)—21no—
283SL(2,𝔽3)C2×D1211yescharacteristic, complemented
284C2×C32—41no—
285C2×C32—41no—
286C2×C32—41no—
287C3×D3—82no—
288C24—91no—
289C24—92no—
290#41—32no—
291#41—61no—
292#41—62no—
293#41—182no—
294#41—182no—
295#41—182no—
296#41—182no—
297#41—182no—
298#41—182no—
299#41—181no—
300#41—182no—
301#41C3×D1212yes—
302C2×Q8—31no—
303C2×Q8—61no—
304C2×Q8—91no—
305C2×Q8—181no—
306C2×Q8C22×C3×D311yescharacteristic
307C2×D4—31no—
308C2×D4—92no—
309C2×D4—92no—
310C2×D4—92no—
311C2×D4—91no—
312C2×D4—94no—
313C2×D4—92no—
314C2×D4—91no—
315C2×D4—92no—
316C2×D4—182no—
317C2×D4—182no—
318C2×D4—182no—
319C2×D4—182no—
320C2×D4—184no—
321C2×D4—184no—
322C2×D4—364no—
323C22×C4—31no—
324C22×C4—32no—
325C22×C4—31no—
326C22×C4—91no—
327C22×C4—92no—
328C22×C4—91no—
329C22×C4—91no—
330C22×C4—91no—
331C22×C4—182no—
332C22×C4—182no—
333C4⋊C4—32no—
334C4⋊C4—31no—
335C4⋊C4—32no—
336C4⋊C4—92no—
337C4⋊C4D3×A411yescharacteristic
338C22⋊C4—32no—
339C22⋊C4—32no—
340C22⋊C4—92no—
341C22⋊C4—92no—
342C22⋊C4—94no—
343C22⋊C4—92no—
344C42—32no—
345C22×C3—32no—
346C22×C3—41no—
347C22×C3—61no—
348C22×C3—61no—
349C22×C3—81no—
350C22×C3—122no—
351C22×C3—241no—
352C22×C3—241no—
353C22×C3C23×A411yescharacteristic
354C2×D3—21no—
355C2×D3—21no—
356C2×D3—61no—
357C2×D3—62no—
358C2×D3—62no—
359C2×D3—61no—
360C2×D3—61no—
361C2×D3—62no—
362C2×D3—124no—
363C2×D3—244no—
364C2×D3#99412yescomplemented
365C3×C4—22no—
366C3×C4—31no—
367C3×C4—31no—
368C3×C4—42no—
369C3×C4—62no—
370C3×C4—82no—
371C3×C4—82no—
372C3×C4—162no—
373C3×C4—241no—
374C3×C4—242no—
375C3×C4D4×A412yes—
376Dic3—22no—
377Dic3—21no—
378Dic3—61no—
379C32—41noSylow
380C23—31no—
381C23—31no—
382C23—92no—
383C23—91no—
384C23—182no—
385C23—182no—
386C23—184no—
387C23—361no—
388C23—362no—
389C23—362no—
390Q8—31no—
391Q8—62no—
392Q8—181no—
393Q8—181no—
394Q8C2×C3×D1211yescharacteristic
395D4—32no—
396D4—32no—
397D4—182no—
398D4—182no—
399D4—181no—
400D4—181no—
401D4—184no—
402D4—184no—
403D4—184no—
404D4—182no—
405D4—362no—
406D4—364no—
407C2×C4—31no—
408C2×C4—32no—
409C2×C4—32no—
410C2×C4—32no—
411C2×C4—32no—
412C2×C4—31no—
413C2×C4—62no—
414C2×C4—91no—
415C2×C4—122no—
416C2×C4—181no—
417C2×C4—181no—
418C2×C4—182no—
419C2×C4—182no—
420C2×C4—181no—
421C2×C4—181no—
422C2×C4—182no—
423C2×C4—182no—
424C2×C4C2×D3×A412yes—
425C2×C4C2×D3×A411yescharacteristic
426C2×C3—41no—
427C2×C3—41no—
428C2×C3—41no—
429C2×C3—62no—
430C2×C3—81no—
431C2×C3—81no—
432C2×C3—81no—
433C2×C3—242no—
434C2×C3C2×D4×A411yescharacteristic
435C2×C3C2×#99411yescharacteristic
436C2×C3#594511yescharacteristic
437D3—22no—
438D3—121no—
439D3—122no—
440C22—32no—
441C22—32no—
442C22—61no—
443C22—61no—
444C22—61no—
445C22—61no—
446C22—181no—
447C22—182no—
448C22—182no—
449C22—181no—
450C22—181no—
451C22—182no—
452C22—364no—
453C22—724no—
454C22C22×D3×A411yescharacteristic
455C4—22no—
456C4—31no—
457C4—31no—
458C4—61no—
459C4—62no—
460C4—62no—
461C4—181no—
462C4A4×D1212yes—
463C3—41no—
464C3—81no—
465C3#3488⋊C311yescharacteristic, complemented
466C2—62no—
467C2—62no—
468C2—361no—
469C2—362no—
470C2C2×A4×D1211yescharacteristic
471C2D3×#99411yescharacteristic
472C2#5968⋊C311yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC3×#520C2×C3
socleC22×C3C23×A4
hypercenterC4⋊C4D3×A4
hypocenterC3×Q8C2×C3×D4
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. C2×D3×#41⋊C3C2semidirect
2. C2×C3⋊#994C2semidirect
3. C3×#520⋊C3C2semidirect
4. #5576⋊C3C2semidirect
5. C4⋊(C3⋊(#41⋊C3))C2semidirect
6. #85900C3semidirect
11. C2×D3×SL(2,𝔽3)C22semidirect
12. C2×C3⋊(#41⋊C3)C22semidirect
13. C2×C3⋊(#41⋊C3)C22semidirect
15. C3×C4×SL(2,𝔽3)C22semidirect
16. C2×#1009C2×C3semidirect
20. C3×#520C2×C3semidirect
21. #5577C2×C3semidirect
32. #520⋊C3D3semidirect
35. C3×#41⋊C3D4semidirect
91. C4×SL(2,𝔽3)C2×D3semidirect
99. C3×SL(2,𝔽3)C2×D4semidirect
110. #520C3×D3semidirect
161. #41⋊C3D12semidirect
283. SL(2,𝔽3)C2×D12semidirect
364. C2×D3#994semidirect
465. C3#3488⋊C3semidirect

Maximal subgroups

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

subgroupindex
C2×D3×#41⋊C32
C2×C3⋊#9942
C3×#520⋊C32
#5576⋊C32
C4⋊(C3⋊(#41⋊C3))2
#859003
#3488⋊C33
C3×#8974