Finite Group Databasebeta

← #87275⋊C3 #11923378

Subgroups — #87275⋊C3 #11923378

Order 1152.

809 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 809 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C22×C3×#41⋊C3C211yesmaximal, characteristic, complemented
2C2×C3⋊#994C211yesmaximal, characteristic, complemented
3#5827⋊C3C211yesmaximal, characteristic, complemented
4#5834⋊C3C212yesmaximal, complemented
5#5817⋊C3C212yesmaximal, complemented
6#87275C311yesmaximal, characteristic, complemented
7#3488⋊C3—31nomaximal
8C2×C3×#41⋊C3—22no—
9C2×C3×#41⋊C3C2211yescharacteristic
10C2×C3×#41⋊C3C2211yescharacteristic, complemented
11C22×C3×SL(2,𝔽3)C2211yescharacteristic, complemented
12C3⋊#994—22no—
13C2×C3⋊(#41⋊C3)C2212yescomplemented
14C3×#924—41nomaximal
15Dic3×SL(2,𝔽3)C2212yescomplemented
16C22×C3×#41C2×C311yescharacteristic
17C2×#1009C2×C311yescharacteristic, complemented
18C2×#994—31no—
19C22×#41⋊C3—21no—
20C22×#41⋊C3D311yescharacteristic, complemented
21#5834C2×C312yescomplemented
22#5833—31no—
23#5830—31no—
24#5829—32no—
25#5827C2×C311yescharacteristic, complemented
26#5817C2×C312yes—
27D4×#67—32no—
28#5791—31no—
29C2×#930—31no—
30C4⋊(#41⋊C3)—32no—
31#548⋊C3—32no—
32#520⋊C3—31no—
33C22×C32×C4—41no—
34C2×C3×Dic6—41no—
35C3×#41⋊C3—21no—
36C3×#41⋊C3D412yescomplemented
37C2×C3×SL(2,𝔽3)—21no—
38C2×C3×SL(2,𝔽3)C2311yescharacteristic
39C3⋊(#41⋊C3)—22no—
40C3×#217—42no—
41C3×C4⋊Dic3—41no—
42C3×Dic3⋊C4—42no—
43#3488—31noSylow
44C2×C3×#41—22no—
45C2×C3×#41—31no—
46C2×C3×#41—31no—
47C2×C3×#41—62no—
48C2×C3×#41C22×C311yescharacteristic
49C2×C3×#41C22×C311yescharacteristic
50C22×C3×Q8C22×C311yescharacteristic
51C22×C3×D4—31no—
52C23×C3×C4—31no—
53C22×#67—32no—
54#1009—22no—
55#1009—62no—
56C2×#239C22×C312yes—
57C2×D3×D4—32no—
58C2×#235—31no—
59C22×D12—31no—
60#994—62no—
61C2×#41⋊C3—24no—
62C2×#41⋊C3—22no—
63C2×#41⋊C3—21no—
64C2×#41⋊C3—21no—
65C2×#41⋊C3—32no—
66C2×#41⋊C3C2×D311yescharacteristic
67C2×#41⋊C3C2×D311yescharacteristic, complemented
68C22×SL(2,𝔽3)—21no—
69C22×SL(2,𝔽3)C2×D311yescharacteristic, complemented
70C3×C22⋊Q8—121no—
71#953—32no—
72#947—32no—
73#946—32no—
74Q8×Dic3C22×C312yes—
75#940—32no—
76#939—34no—
77#938—32no—
78#937—34no—
79D4×Dic3—32no—
80#930—32no—
81#930—31no—
82#930—32no—
83#930—32no—
84#930—31no—
85#929—32no—
86C4×#67—32no—
87C4×#67—32no—
88C2×(C2×D3)⋊C4—32no—
89#926—31no—
90C2×C4⋊Dic3—31no—
91#924—31no—
92#924A411yescharacteristic
93C4×SL(2,𝔽3)—32no—
94C23×C32—41no—
95C2×C32×C4—41no—
96C2×C32×C4—41no—
97C2×C32×C4—82no—
98C2×C3×Dic3—42no—
99C2×C3×Dic3—42no—
100C3×Dic6—82no—
101C3×SL(2,𝔽3)C2×D411yescharacteristic, complemented
102C2×H(5,𝔽2)—31no—
103C22×#41C3×D311yescharacteristic
104#550—32no—
105#548—32no—
106#546—92no—
107D42—92no—
108#537—91no—
109#535—91no—
110#525—91no—
111C2×#121—91no—
112#520—31no—
113C24×C3—31no—
114C23×D3—32no—
115C3×#41—21no—
116C3×#41—32no—
117C3×#41—62no—
118C3×#41—62no—
119C3×#41—62no—
120C3×#41—62no—
121C3×#41—61no—
122C3×#41C3×D412yes—
123C2×C3×Q8—21no—
124C2×C3×Q8—31no—
125C2×C3×Q8—61no—
126C2×C3×Q8—121no—
127C2×C3×Q8C23×C311yescharacteristic
128C2×C3×D4—31no—
129C2×C3×D4—31no—
130C2×C3×D4—32no—
131C2×C3×D4—62no—
132C2×C3×D4—62no—
133C22×C3×C4—31no—
134C22×C3×C4—31no—
135C22×C3×C4—31no—
136C22×C3×C4—31no—
137C22×C3×C4—31no—
138C22×C3×C4—41no—
139C22×C3×C4—62no—
140C22×C3×C4—62no—
141C22×C3×C4—81no—
142C22×C3×C4C2×A411yescharacteristic
143C2×#67—32no—
144C2×#67—34no—
145C2×#67—32no—
146C2×#67—32no—
147C2×#67—124no—
148C22×Dic3—32no—
149#239—22no—
150#239—62no—
151D3×D4—64no—
152D3×D4—64no—
153#235—62no—
154#235—62no—
155C2×D12—32no—
156C2×D12—31no—
157C2×D12—62no—
158C2×D12—62no—
159C2×C4×D3—32no—
160C2×Dic6C2×A411yescharacteristic
161#41⋊C3—22no—
162#41⋊C3—21no—
163#41⋊C3—22no—
164#41⋊C3—62no—
165#41⋊C3#6712yescomplemented
166C2×SL(2,𝔽3)—22no—
167C2×SL(2,𝔽3)—21no—
168C2×SL(2,𝔽3)—21no—
169C2×SL(2,𝔽3)C22×D311yescharacteristic
170C3×C4⋊C4—121no—
171C3×C4⋊C4—122no—
172C3×C22⋊C4—122no—
173#217—32no—
174#217—32no—
175#217C2×A412yes—
176(C2×D3)⋊C4—34no—
177(C2×D3)⋊C4—32no—
178(C2×D3)⋊C4—32no—
179C4⋊Dic3—31no—
180C4⋊Dic3—32no—
181C4⋊Dic3C2×A411yescharacteristic
182Dic3⋊C4—32no—
183Dic3⋊C4C2×A412yes—
184C4×Dic3—32no—
185C22×C32—42no—
186C22×C32—41no—
187C22×C32—81no—
188C22×C32—81no—
189C32×C4—42no—
190C32×C4—81no—
191C3×Dic3—82no—
192C3×Dic3—82no—
193H(5,𝔽2)—62no—
194H(5,𝔽2)—182no—
195C2×#41—22no—
196C2×#41—31no—
197C2×#41—31no—
198C2×#41—32no—
199C2×#41—62no—
200C2×#41—91no—
201C2×#41C2×C3×D311yescharacteristic
202C2×#41C2×C3×D311yescharacteristic
203C22×Q8C2×C3×D311yescharacteristic
204C22×D4—31no—
205C22×D4—91no—
206C22×D4—92no—
207C22×D4—92no—
208C23×C4—31no—
209#127—92no—
210#124—92no—
211#123—92no—
212C22⋊Q8—31no—
213C22⋊Q8—91no—
214C22⋊Q8—92no—
215#121—91no—
216#121—92no—
217#121—92no—
218#121—94no—
219#121—91no—
220#121—92no—
221#121—92no—
222#120—92no—
223#120—94no—
224C4×Q8—32no—
225C4×D4—92no—
226C4×D4—92no—
227C4×D4—92no—
228#117—91no—
229C2×C4⋊C4—91no—
230C2×C22⋊C4—92no—
231C23×C3—31no—
232C23×C3—31no—
233C23×C3—41no—
234C23×C3—62no—
235C23×C3—62no—
236C23×C3—81no—
237C23×C3—121no—
238C23×C3C22×A411yescharacteristic
239C22×D3—32no—
240C22×D3—64no—
241C22×D3—122no—
242C22×D3—122no—
243C3×Q8—31no—
244C3×Q8—61no—
245C3×Q8—61no—
246C3×Q8—242no—
247C3×Q8C2×C3×D411yescharacteristic
248C3×D4—32no—
249C3×D4—32no—
250C3×D4—61no—
251C3×D4—61no—
252C3×D4—64no—
253C2×C3×C4—22no—
254C2×C3×C4—31no—
255C2×C3×C4—32no—
256C2×C3×C4—32no—
257C2×C3×C4—31no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—61no—
261C2×C3×C4—61no—
262C2×C3×C4—62no—
263C2×C3×C4—61no—
264C2×C3×C4—61no—
265C2×C3×C4—62no—
266C2×C3×C4—82no—
267C2×C3×C4—84no—
268C2×C3×C4—81no—
269C2×C3×C4—81no—
270C2×C3×C4—122no—
271C2×C3×C4—122no—
272C2×C3×C4C22×A411yescharacteristic
273C2×C3×C4C22×A411yescharacteristic
274#67—64no—
275#67—64no—
276#67—122no—
277#67—124no—
278C2×Dic3—32no—
279C2×Dic3—122no—
280C2×Dic3C22×A412yes—
281C2×Dic3C22×A412yes—
282D12—62no—
283D12—62no—
284D12—62no—
285C4×D3—62no—
286C4×D3—62no—
287Dic6—22no—
288SL(2,𝔽3)—21no—
289SL(2,𝔽3)C2×#6711yescharacteristic, complemented
290C2×C32—41no—
291C2×C32—41no—
292C2×C32—41no—
293C2×C32—82no—
294C24—31no—
295C24—92no—
296#41—21no—
297#41—32no—
298#41—62no—
299#41—62no—
300#41—62no—
301#41—62no—
302#41—61no—
303#41—62no—
304#41—182no—
305#41—182no—
306#41—182no—
307#41C3×#6712yes—
308C2×Q8—21no—
309C2×Q8—31no—
310C2×Q8—31no—
311C2×Q8—61no—
312C2×Q8C22×C3×D311yescharacteristic
313C2×D4—31no—
314C2×D4—31no—
315C2×D4—32no—
316C2×D4—62no—
317C2×D4—62no—
318C2×D4—92no—
319C2×D4—92no—
320C2×D4—92no—
321C2×D4—92no—
322C2×D4—94no—
323C2×D4—91no—
324C2×D4—182no—
325C2×D4—182no—
326C2×D4—184no—
327C2×D4—184no—
328C2×D4—364no—
329C22×C4—31no—
330C22×C4—31no—
331C22×C4—31no—
332C22×C4—31no—
333C22×C4—31no—
334C22×C4—62no—
335C22×C4—62no—
336C22×C4—92no—
337C22×C4—92no—
338C22×C4D3×A411yescharacteristic
339C4⋊C4—31no—
340C4⋊C4—32no—
341C4⋊C4—92no—
342C4⋊C4—92no—
343C4⋊C4—91no—
344C22⋊C4—32no—
345C22⋊C4—92no—
346C22⋊C4—92no—
347C22⋊C4—92no—
348C22⋊C4—94no—
349C22⋊C4—92no—
350C42—92no—
351C22×C3—21no—
352C22×C3—21no—
353C22×C3—32no—
354C22×C3—42no—
355C22×C3—41no—
356C22×C3—61no—
357C22×C3—61no—
358C22×C3—61no—
359C22×C3—61no—
360C22×C3—61no—
361C22×C3—81no—
362C22×C3—81no—
363C22×C3—82no—
364C22×C3—82no—
365C22×C3—82no—
366C22×C3—81no—
367C22×C3—124no—
368C22×C3#99412yescomplemented
369C22×C3C23×A411yescharacteristic
370C2×D3—62no—
371C2×D3—62no—
372C2×D3—62no—
373C2×D3—244no—
374C3×C4—21no—
375C3×C4—31no—
376C3×C4—31no—
377C3×C4—42no—
378C3×C4—61no—
379C3×C4—81no—
380C3×C4—82no—
381C3×C4—82no—
382C3×C4—242no—
383C3×C4—242no—
384C3×C4D4×A412yes—
385Dic3—22no—
386Dic3—22no—
387Dic3—62no—
388C32—41noSylow
389C23—31no—
390C23—31no—
391C23—62no—
392C23—62no—
393C23—92no—
394C23—121no—
395C23—184no—
396C23—362no—
397C23—362no—
398C23C2×D3×A411yescharacteristic
399Q8—31no—
400Q8—61no—
401Q8—61no—
402Q8—62no—
403Q8C2×C3×#6711yescharacteristic
404D4—32no—
405D4—32no—
406D4—61no—
407D4—61no—
408D4—64no—
409D4—182no—
410D4—182no—
411D4—184no—
412D4—184no—
413D4—182no—
414D4—362no—
415D4—364no—
416C2×C4—22no—
417C2×C4—31no—
418C2×C4—32no—
419C2×C4—32no—
420C2×C4—32no—
421C2×C4—32no—
422C2×C4—31no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—62no—
426C2×C4—61no—
427C2×C4—61no—
428C2×C4—62no—
429C2×C4—92no—
430C2×C4—182no—
431C2×C4—182no—
432C2×C4—362no—
433C2×C4C2×D3×A411yescharacteristic
434C2×C4C2×D3×A411yescharacteristic
435C2×C3—22no—
436C2×C3—41no—
437C2×C3—41no—
438C2×C3—41no—
439C2×C3—62no—
440C2×C3—82no—
441C2×C3—84no—
442C2×C3—81no—
443C2×C3—81no—
444C2×C3—81no—
445C2×C3—121no—
446C2×C3C2×D4×A411yescharacteristic
447C2×C3C2×#99411yescharacteristic
448C2×C3#594511yescharacteristic
449D3—122no—
450C22—21no—
451C22—21no—
452C22—32no—
453C22—61no—
454C22—61no—
455C22—61no—
456C22—61no—
457C22—61no—
458C22—124no—
459C22—182no—
460C22—182no—
461C22—182no—
462C22—724no—
463C22C3⋊#99412yescomplemented
464C22C22×D3×A411yescharacteristic
465C4—21no—
466C4—31no—
467C4—31no—
468C4—62no—
469C4—61no—
470C4—62no—
471C4—182no—
472C4A4×#6712yes—
473C3—41no—
474C3—81no—
475C3#3488⋊C311yescharacteristic, complemented
476C2—22no—
477C2—62no—
478C2—121no—
479C2—362no—
480C2C2×C3⋊#99411yescharacteristic
481C2#5964⋊C311yescharacteristic
482C2C2×A4×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×A4
commutator, cosocleC2×C3×Q8C23×C3
FittingC22×C3×#41C2×C3
socleC22×C3C23×A4
hypercenterC22×C4D3×A4
hypocenterC3×Q8C2×C3×D4
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. C22×C3×#41⋊C3C2semidirect
2. C2×C3⋊#994C2semidirect
3. #5827⋊C3C2semidirect
4. #5834⋊C3C2semidirect
5. #5817⋊C3C2semidirect
6. #87275C3semidirect
10. C2×C3×#41⋊C3C22semidirect
11. C22×C3×SL(2,𝔽3)C22semidirect
13. C2×C3⋊(#41⋊C3)C22semidirect
15. Dic3×SL(2,𝔽3)C22semidirect
17. C2×#1009C2×C3semidirect
20. C22×#41⋊C3D3semidirect
21. #5834C2×C3semidirect
25. #5827C2×C3semidirect
36. C3×#41⋊C3D4semidirect
67. C2×#41⋊C3C2×D3semidirect
69. C22×SL(2,𝔽3)C2×D3semidirect
101. C3×SL(2,𝔽3)C2×D4semidirect
165. #41⋊C3#67semidirect
289. SL(2,𝔽3)C2×#67semidirect
368. C22×C3#994semidirect
463. C22C3⋊#994semidirect
475. C3#3488⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C3×#41⋊C32
C2×C3⋊#9942
#5827⋊C32
#5834⋊C32
#5817⋊C32
#872753
#3488⋊C33
C3×#9244