Finite Group Databasebeta

← #11922222

Subgroups — #11922222

Order 1152.

663 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 663 conjugacy classes of subgroups — too many to draw. Only its 83 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×#1007⋊C3C211yesmaximal, characteristic
2#10593654C211yesmaximal, characteristic, complemented
3C2×C3×#859C211yesmaximal, characteristic, complemented
4D3×#859C211yesmaximal, characteristic, complemented
5#10593209C211yesmaximal, characteristic, complemented
6#10593200C211yesmaximal, characteristic, complemented
7#10593191C211yesmaximal, characteristic, complemented
8#85534—31nomaximal
9#78790—31nomaximal
10C22×C3×SL(2,𝔽3)C2211yescharacteristic
11#1007⋊C3C414yescomplemented
12C2×D3×SL(2,𝔽3)C2211yescharacteristic
13C2×C3⋊(#41⋊C3)C2211yescharacteristic
14Dic3×#67—41nomaximal
15C3⋊#859C2211yescharacteristic
16C3⋊#859C2211yescharacteristic
17C3×#859C2211yescharacteristic
18C3×#859C2211yescharacteristic
19C2×#1007D311yescharacteristic
20C2×2−1+4⋊C3—31no—
21C2×#901—31no—
22#5504—31no—
23#5426—31no—
24#5424—31no—
25#5420—31no—
26#5420—31no—
27#5420—61no—
28C2×#859—31no—
29C2×#859D311yescharacteristic, complemented
30C2×C3×#103—31no—
31#5225—31no—
32#5113—31no—
33#4966—31no—
34#4804—31no—
35#4804—31no—
36#4803—31no—
37#4803—31no—
38#4802—31no—
39#4802—31no—
40D3×#103—31no—
41D3×#103—31no—
42C2×C3×#67—41no—
43C22×C3×Dic3—41no—
44C2×C3×SL(2,𝔽3)C2×C412yescomplemented
45C2×C3×SL(2,𝔽3)C2311yescharacteristic
46C2×D3×Dic3—41no—
47D3×SL(2,𝔽3)C2×C412yescomplemented
48C3⋊(#41⋊C3)C2×C412yescomplemented
49#3757—41no—
50Dic3⋊Dic3—41no—
51(C2×D3)⋊Dic3—41no—
52Dic32—41no—
53#2898—91noSylow
54C22×C3×Q8C2×D311yescharacteristic
55#1007—34no—
56#1007Dic314yescomplemented
57C2×#239—31no—
58C2×#239C2×D311yescharacteristic
59C2×D3×Q8—31no—
60C2×D3×Q8C2×D311yescharacteristic
61C2×#235—31no—
62C2×#235—31no—
632−1+4⋊C3—34no—
64C2×#41⋊C3—31no—
65C22×SL(2,𝔽3)—21no—
66C22×SL(2,𝔽3)—31no—
67C22×SL(2,𝔽3)C2×D311yescharacteristic
68C22×C3×C8—31no—
69C2×C3×C4⋊C4—31no—
70D4×Dic3—121no—
71C4×#67—61no—
72C4×#67—61no—
73#926—31no—
74C2×#208—31no—
75#901—64no—
76C2×ΓL(1,𝔽25)—31no—
77C2×ΓL(1,𝔽25)—31no—
78C2×D3×C8—31no—
79C2×D3×C8—31no—
80#893—61no—
81#892—31no—
82#892—31no—
83D3×C4⋊C4—31no—
84D3×C4⋊C4—31no—
85#887—61no—
86#859—31no—
87#859—31no—
88#859—61no—
89#859—61no—
90#859C2×D311yescharacteristic
91#859C2×D311yescharacteristic
92C3×#103—31no—
93C3×#103—31no—
94C3×#103—31no—
95C3×#103—31no—
96#835—31no—
97#835—31no—
98#835—31no—
99#835—31no—
100#821—31no—
101#821—31no—
102#816—31no—
103#816—31no—
104#810—31no—
105#810—31no—
106#809—31no—
107#809—31no—
108C23×C32—41no—
109C22×C3×D3—41no—
110C2×C3⋊Dic3—41no—
111C2×C3⋊Dic3—41no—
112C3×#67—44no—
113C2×C3×Dic3—41no—
114C2×C3×Dic3—41no—
115C2×C3×Dic3—41no—
116C2×C3×Dic3—82no—
117C3×SL(2,𝔽3)C22×C411yescharacteristic
118D3×Dic3—82no—
119C2×2−1+4—31no—
120C2×#131—91no—
121#520—91no—
122#419—91no—
123#419—91no—
124#419—91no—
125#417—91no—
126#417—91no—
127#417—91no—
128#416—91no—
129#416—91no—
130#416—91no—
131C2×#103—31no—
132C2×#103—91no—
133C2×#103—91no—
134C2×C3×Q8—31no—
135C2×C3×Q8—32no—
136C2×C3×Q8C2×Dic312yescomplemented
137C2×C3×Q8C22×D311yescharacteristic
138C2×C3×D4—121no—
139C22×C3×C4—31no—
140C22×C3×C4—61no—
141C2×#67—31no—
142C2×#67S411yescharacteristic
143C22×Dic3—41no—
144C22×Dic3—121no—
145#239—32no—
146#239—34no—
147#239—62no—
148#239C2×Dic312yescomplemented
149D3×Q8—34no—
150D3×Q8—32no—
151D3×Q8—62no—
152D3×Q8C2×Dic312yescomplemented
153#235—34no—
154#235—34no—
155#235—34no—
156#235—62no—
157C2×D12—31no—
158C2×D12—31no—
159C2×C4×D3—31no—
160C2×C4×D3—31no—
161C2×C4×D3—61no—
162C2×C4×D3—61no—
163C2×C4×D3—61no—
164C2×Dic6—31no—
165C2×Dic6—31no—
166#41⋊C3—32no—
167C2×SL(2,𝔽3)—22no—
168C2×SL(2,𝔽3)—21no—
169C2×SL(2,𝔽3)—32no—
170C2×SL(2,𝔽3)C4×D312yescomplemented
171C2×SL(2,𝔽3)C22×D311yescharacteristic
172C2×C3×C8—31no—
173C2×C3×C8—31no—
174C2×C3×C8—62no—
175C3×C4⋊C4—31no—
176C3×C4⋊C4—31no—
177C3×C4⋊C4—61no—
178#217—61no—
179#217—121no—
180#217—121no—
181#217—241no—
182(C2×D3)⋊C4—61no—
183(C2×D3)⋊C4—61no—
184C4⋊Dic3—31no—
185C4⋊Dic3—31no—
186C4⋊Dic3—121no—
187Dic3⋊C4—61no—
188Dic3⋊C4—61no—
189C4×Dic3—61no—
190C4×Dic3—61no—
191C4×Dic3—61no—
192C4×Dic3—121no—
193#208—62no—
194C2×C3⋊C8—31no—
195C2×C3⋊C8—31no—
196ΓL(1,𝔽25)—62no—
197ΓL(1,𝔽25)—62no—
198D3×C8—62no—
199D3×C8—62no—
200C22×C32—42no—
201C22×C32—41no—
202C22×C32—81no—
203C22×C32—81no—
204C2×C3×D3—42no—
205C2×C3×D3—81no—
206C2×C3×D3—81no—
207C3⋊Dic3—81no—
208C3⋊Dic3—81no—
209C3×Dic3—42no—
210C3×Dic3—42no—
211C3×Dic3—81no—
2122−1+4—34no—
2132−1+4—94no—
214C2×#41—31no—
215C2×#41—91no—
216C2×#41—91no—
217C2×#41—91no—
218C22×Q8—31no—
219C22×Q8—91no—
220C22×Q8D3211yescharacteristic
221#131—184no—
222C2×M16—91no—
223C2×M16—91no—
224C2×M16—91no—
225C22×C8—31no—
226C22×C8—91no—
227C22×C8—91no—
228C4×Q8—181no—
229C4×D4—181no—
230C4×D4—181no—
231C4×D4—181no—
232#117—91no—
233#117—91no—
234#117—91no—
235C2×C4⋊C4—31no—
236C2×C4⋊C4—91no—
237C2×C4⋊C4—91no—
238#103—31no—
239#103—31no—
240#103—31no—
241#103—31no—
242#103—91no—
243#103—91no—
244#103—91no—
245#103—91no—
246#103—91no—
247#103—91no—
248#103—91no—
249#103—91no—
250#102—91no—
251#102—91no—
252#102—91no—
253#102—91no—
254C23×C3—41no—
255C23×C3—81no—
256C23×C3—121no—
257C23×C3C2×S411yescharacteristic
258C22×D3—31no—
259C22×D3C2×S411yescharacteristic
260C3×Q8—32no—
261C3×Q8—31no—
262C3×Q8—61no—
263C3×Q8C22×Dic311yescharacteristic
264C3×D4—124no—
265C2×C3×C4—32no—
266C2×C3×C4—32no—
267C2×C3×C4—31no—
268C2×C3×C4—31no—
269C2×C3×C4—61no—
270C2×C3×C4—61no—
271C2×C3×C4—121no—
272C2×C3×C4—241no—
273#67—34no—
274#67SL(2,ℤ/4)14yes—
275C2×Dic3—31no—
276C2×Dic3—41no—
277C2×Dic3—41no—
278C2×Dic3—61no—
279C2×Dic3—61no—
280C2×Dic3—82no—
281C2×Dic3—121no—
282C2×Dic3—121no—
283C2×Dic3—241no—
284C2×Dic3—241no—
285C2×Dic3—242no—
286C2×Dic3C2×S411yescharacteristic
287D12—32no—
288D12—34no—
289D12—61no—
290C4×D3—32no—
291C4×D3—32no—
292C4×D3—32no—
293C4×D3—32no—
294C4×D3—64no—
295C4×D3—241no—
296C4×D3—241no—
297Dic6—34no—
298Dic6—32no—
299Dic6—61no—
300SL(2,𝔽3)—21no—
301SL(2,𝔽3)C2×C4×D311yescharacteristic
302C3×C8—61no—
303C3×C8—61no—
304C3⋊C8—61no—
305C3⋊C8—61no—
306C2×C32—41no—
307C2×C32—41no—
308C2×C32—41no—
309C2×C32—82no—
310C3×D3—82no—
311#41—32no—
312#41—94no—
313#41—94no—
314#41—94no—
315#41—92no—
316#41—94no—
317#41—182no—
318#41—182no—
319C2×Q8—31no—
320C2×Q8—32no—
321C2×Q8—32no—
322C2×Q8—91no—
323C2×Q8—91no—
324C2×Q8—92no—
325C2×Q8—94no—
326C2×Q8—182no—
327C2×Q8D3×Dic312yescomplemented
328C2×Q8C2×D3211yescharacteristic
329C2×D4—31no—
330C2×D4—91no—
331C2×D4—91no—
332C2×D4—91no—
333C22×C4—31no—
334C22×C4—61no—
335C22×C4—91no—
336C22×C4—91no—
337C22×C4—181no—
338C22×C4—181no—
339C22×C4—181no—
340M16—182no—
341M16—182no—
342M16—182no—
343C2×C8—31no—
344C2×C8—31no—
345C2×C8—62no—
346C2×C8—91no—
347C2×C8—91no—
348C2×C8—182no—
349C2×C8—182no—
350C4⋊C4—31no—
351C4⋊C4—31no—
352C4⋊C4—61no—
353C4⋊C4—91no—
354C4⋊C4—91no—
355C4⋊C4—181no—
356C4⋊C4—181no—
357C22⋊C4—181no—
358C22⋊C4—181no—
359C22⋊C4—181no—
360C42—181no—
361C42—181no—
362C42—181no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—41no—
366C22×C3—42no—
367C22×C3—81no—
368C22×C3—81no—
369C22×C3—81no—
370C22×C3—82no—
371C22×C3—122no—
372C22×C3—161no—
373C22×C3—161no—
374C22×C3—241no—
375C22×C3—241no—
376C22×C3C2×SL(2,ℤ/4)12yes—
377C22×C3C22×S411yescharacteristic
378C2×D3—21no—
379C2×D3—21no—
380C2×D3—32no—
381C2×D3—61no—
382C2×D3—61no—
383C2×D3C2×SL(2,ℤ/4)12yes—
384C3×C4—31no—
385C3×C4—31no—
386C3×C4—32no—
387C3×C4—122no—
388C3×C4—121no—
389C3×C4—121no—
390Dic3—32no—
391Dic3—42no—
392Dic3—81no—
393Dic3—121no—
394Dic3—121no—
395Dic3—241no—
396Dic3—241no—
397Dic3—481no—
398Dic3—481no—
399Dic3C2×SL(2,ℤ/4)12yes—
400C32—41noSylow
401C23—31no—
402C23—91no—
403C23D3×S411yescharacteristic
404Q8—32no—
405Q8—31no—
406Q8—61no—
407Q8—94no—
408Q8—92no—
409Q8—181no—
410Q8C2×D3×Dic311yescharacteristic
411D4—34no—
412D4—94no—
413D4—92no—
414D4—94no—
415D4—181no—
416C2×C4—31no—
417C2×C4—31no—
418C2×C4—31no—
419C2×C4—32no—
420C2×C4—32no—
421C2×C4—61no—
422C2×C4—61no—
423C2×C4—92no—
424C2×C4—92no—
425C2×C4—92no—
426C2×C4—92no—
427C2×C4—91no—
428C2×C4—184no—
429C2×C4—181no—
430C2×C4—181no—
431C2×C4—241no—
432C2×C4—721no—
433C2×C4—721no—
434C8—61no—
435C8—61no—
436C8—181no—
437C8—181no—
438C2×C3—22no—
439C2×C3—41no—
440C2×C3—41no—
441C2×C3—41no—
442C2×C3—82no—
443C2×C3—81no—
444C2×C3—81no—
445C2×C3—81no—
446C2×C3—162no—
447C2×C3—242no—
448C2×C3#592611yescharacteristic
449C2×C3#592711yescharacteristic
450C2×C3C22×SL(2,ℤ/4)11yescharacteristic
451D3—22no—
452D3—62no—
453C22—21no—
454C22—21no—
455C22—32no—
456C22—61no—
457C22—61no—
458C22—92no—
459C22—181no—
460C22—181no—
461C22D3×SL(2,ℤ/4)12yes—
462C22C2×D3×S411yescharacteristic
463C4—31no—
464C4—31no—
465C4—32no—
466C4—32no—
467C4—92no—
468C4—121no—
469C4—121no—
470C4—361no—
471C4—361no—
472C3—41no—
473C3—81no—
474C3#8553411yescharacteristic, complemented
475C2—22no—
476C2—62no—
477C2—182no—
478C2#1059643911yescharacteristic
479C2#1059644511yescharacteristic
480C2C2×D3×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, FrattiniC22C2×D3×S4
commutator, hypocenterC3×SL(2,𝔽3)C22×C4
FittingC22×C3×Q8C2×D3
socleC22×C3C22×S4
prime radicalC2×#1007⋊C3C2
cosocleC2×C3×SL(2,𝔽3)C23
hypercenterC23D3×S4
LCSG ▷ C3×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C23

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #10593654C2semidirect
3. C2×C3×#859C2semidirect
4. D3×#859C2semidirect
5. #10593209C2semidirect
6. #10593200C2semidirect
7. #10593191C2semidirect
11. #1007⋊C3C4semidirect
29. C2×#859D3semidirect
44. C2×C3×SL(2,𝔽3)C2×C4semidirect
47. D3×SL(2,𝔽3)C2×C4semidirect
48. C3⋊(#41⋊C3)C2×C4semidirect
56. #1007Dic3semidirect
136. C2×C3×Q8C2×Dic3semidirect
148. #239C2×Dic3semidirect
152. D3×Q8C2×Dic3semidirect
170. C2×SL(2,𝔽3)C4×D3semidirect
327. C2×Q8D3×Dic3semidirect
474. C3#85534semidirect

Maximal subgroups

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

subgroupindex
C2×#1007⋊C32
#105936542
C2×C3×#8592
D3×#8592
#105932092
#105932002
#105931912
#855343
#787903
Dic3×#674