Finite Group Databasebeta

← #11779310

Subgroups — #11779310

Order 1152.

682 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 682 conjugacy classes of subgroups — too many to draw. Only its 104 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×D4×D12C211yesmaximal, characteristic
2#10592707C211yesmaximal, characteristic, complemented
3#10590369C211yesmaximal, characteristic, complemented
4#10590268C211yesmaximal, characteristic, complemented
5C3×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#10589090C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic, complemented
8#71907—31nomaximal
9#68195—31nomaximal
10C2×C3×D3×D4—21no—
11C22×C3×D12—21no—
12C32×C4×D4C2211yescharacteristic
13#64078C2211yescharacteristic, complemented
14C3×#930—21no—
15C3×#895—21no—
16C3×#882—22no—
17C3×#874C2211yescharacteristic
18C3×C4×D12C2211yescharacteristic
19C2×#3716—22no—
20#63659—21no—
21#63635—21no—
22C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic, complemented
23C3×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
24C3×C4×C3⋊C8C2211yescharacteristic, complemented
25#63573—21no—
26#63563—21no—
27C3×D42—31no—
28D4×D12D311yescharacteristic
29#5076—31no—
30#5008—31no—
31#5008—31no—
32#5008—31no—
33#5008—61no—
34#4842—31no—
35#4693—31no—
36C3×D4⋊C8—31no—
37#4544—31no—
38D4⋊(C3⋊C8)—31no—
39D4⋊(C3⋊C8)D311yescharacteristic, complemented
40#4485—31no—
41#4459—31no—
42C23×C3×D3—22no—
43C2×C32×D4D411yescharacteristic
44C22×C32×C4—21no—
45C2×#634—21no—
46C2×C3×#67—22no—
47C3×D3×D4—42no—
48C3×D3×D4—42no—
49C2×C3×D12—22no—
50C2×C3×D12—21no—
51C2×C3×D12—21no—
52C2×C3×D12—42no—
53C2×C3×D12—42no—
54C2×C3×D12D411yescharacteristic
55C2×C3×D12D412yes—
56C2×C3×C4×D3—21no—
57C32×C4⋊C4D411yescharacteristic
58C32×C22⋊C4—21no—
59C32×C42C2311yescharacteristic
60C4⋊(C3⋊Dic3)—21no—
61C4⋊(C3⋊Dic3)—21no—
62C2×C3⋊(C3⋊C8)—21no—
63C3×(C2×D3)⋊C4—22no—
64C3×(C2×D3)⋊C4—21no—
65C3×C4⋊Dic3D411yescharacteristic
66C2×C3×C3⋊C8—21no—
67C2×C3×C3⋊C8—21no—
68#3716—44no—
69#1629—91noSylow
70C22×C3×D4—61no—
71C22×C3×D4—61no—
72C2×D3×D4—21no—
73C22×D12—21no—
74C3×#127—31no—
75C3×#121—61no—
76C3×#121—61no—
77C3×#120—62no—
78C3×C4×D4—21no—
79C3×C4×D4—31no—
80C3×C4×D4C2×D311yescharacteristic
81C3×C4×D4C2×D311yescharacteristic
82C2×#214—62no—
83#930—21no—
84C2×#204—62no—
85#895—21no—
86#882—22no—
87#874C2×D311yescharacteristic
88C4×D12C2×D311yescharacteristic
89#869—31no—
90#869—31no—
91#869—61no—
92C3×C4⋊C8—31no—
93C3×C4×C8—31no—
94#832—61no—
95#832—62no—
96#832—61no—
97#832—61no—
98#821—61no—
99#817—61no—
100#808—61no—
101#808—62no—
102#808—61no—
103#808—61no—
104C4⋊(C3⋊C8)—31no—
105C4⋊(C3⋊C8)—31no—
106C4⋊(C3⋊C8)—61no—
107C4⋊(C3⋊C8)C2×D311yescharacteristic, complemented
108C4×C3⋊C8C2×D311yescharacteristic, complemented
109C23×C32—21no—
110C22×C3×D3—21no—
111C22×C3×D3—22no—
112C22×C3×D3—44no—
113C22×C3×D3—44no—
114C22×C3×D3—82no—
115C32×D4—21no—
116C32×D4SD1612yescomplemented
117C2×C32×C4—21no—
118C2×C32×C4—41no—
119C2×C32×C4—41no—
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×D411yescharacteristic
123C2×C3⋊Dic3—41no—
124#634—42no—
125C3×#67—44no—
126C2×C3×Dic3—21no—
127C3×D12—21no—
128C3×D12—42no—
129C3×D12—44no—
130C3×D12—41no—
131C3×D12—41no—
132C3×D12—41no—
133C3×D12—41no—
134C3×D12SD1612yescomplemented
135C3×C4×D3—41no—
136C3×C4×D3—41no—
137C3⋊(C3⋊C8)—41no—
138C3×C3⋊C8—22no—
139C3×C3⋊C8—41no—
140D42—31no—
141#492—91no—
142#476—91no—
143#476—91no—
144#331—91no—
145D4⋊C8—31no—
146D4⋊C8—91no—
147C24×C3—62no—
148C23×D3—22no—
149C2×C3×D4—21no—
150C2×C3×D4—31no—
151C2×C3×D4—32no—
152C2×C3×D4—62no—
153C2×C3×D4—62no—
154C2×C3×D4—61no—
155C2×C3×D4—61no—
156C2×C3×D4—122no—
157C2×C3×D4—122no—
158C2×C3×D4—122no—
159C2×C3×D4—122no—
160C2×C3×D4D1211yescharacteristic
161C2×C3×D4#6711yescharacteristic
162C22×C3×C4—21no—
163C22×C3×C4—21no—
164C22×C3×C4—41no—
165C22×C3×C4—61no—
166C2×#67—22no—
167D3×D4—42no—
168D3×D4—42no—
169C2×D12—22no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12—42no—
173C2×D12—42no—
174C2×D12#6711yescharacteristic
175C2×D12#6712yes—
176C2×C4×D3—21no—
177C2×Dic6—61no—
178C2×Dic6—61no—
179C2×Dic6—121no—
180C2×C3×C8—61no—
181C2×C3×C8—61no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—31no—
184C3×C4⋊C4D1211yescharacteristic
185C3×C4⋊C4#6711yescharacteristic
186C3×C22⋊C4—21no—
187C3×C22⋊C4—21no—
188C3×C22⋊C4—41no—
189C3×C22⋊C4—62no—
190C3×C22⋊C4—61no—
191C3×C42—21no—
192C3×C42C22×D311yescharacteristic
193C3×C42C22×D311yescharacteristic
194#214—124no—
195(C2×D3)⋊C4—22no—
196(C2×D3)⋊C4—21no—
197C4⋊Dic3—62no—
198C4⋊Dic3—61no—
199C4⋊Dic3—61no—
200C4⋊Dic3—61no—
201C4⋊Dic3—61no—
202C4⋊Dic3—121no—
203C4⋊Dic3#6711yescharacteristic
204C2×C3⋊C8—21no—
205C2×C3⋊C8—21no—
206C2×C3⋊C8—62no—
207C2×C3⋊C8—61no—
208C2×C3⋊C8—61no—
209#204—124no—
210C22×C32—22no—
211C22×C32—41no—
212C22×C32—41no—
213C22×C32#12011yescharacteristic
214C2×C3×D3—22no—
215C2×C3×D3—41no—
216C2×C3×D3—41no—
217C2×C3×D3—42no—
218C2×C3×D3—42no—
219C2×C3×D3—42no—
220C2×C3×D3—88no—
221C32×C4—21no—
222C32×C4—41no—
223C32×C4C2×SD1611yescharacteristic
224C32×C4C2×SD1611yescharacteristic
225C32×C4AGL(1,ℤ/8)11yescharacteristic
226C32×C4AGL(1,ℤ/8)11yescharacteristic
227C3⋊Dic3—81no—
228C3×Dic3—41no—
229C22×D4—61no—
230C22×D4—61no—
231C2×SD16—182no—
232C4⋊Q8—91no—
233#127—31no—
234#121—61no—
235#121—61no—
236#120—62no—
237C4×D4—31no—
238C4×D4D3211yescharacteristic
239#106—181no—
240#106—181no—
241C4⋊C8—31no—
242C4⋊C8—91no—
243#102—181no—
244#102—181no—
245C4×C8—31no—
246C23×C3—21no—
247C23×C3—21no—
248C23×C3—41no—
249C23×C3—61no—
250C23×C3—62no—
251C23×C3—124no—
252C23×C3—124no—
253C23×C3—242no—
254C22×D3—21no—
255C22×D3—22no—
256C22×D3—44no—
257C22×D3—44no—
258C22×D3—82no—
259C3×D4—21no—
260C3×D4—21no—
261C3×D4—22no—
262C3×D4—32no—
263C3×D4—41no—
264C3×D4—61no—
265C3×D4—122no—
266C3×D4—124no—
267C3×D4—124no—
268C3×D4—121no—
269C3×D4—121no—
270C3×D4—121no—
271C3×D4—121no—
272C3×D4#20412yescomplemented
273C3×D4#21412yescomplemented
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4—41no—
282C2×C3×C4—41no—
283C2×C3×C4—41no—
284C2×C3×C4—61no—
285C2×C3×C4—81no—
286C2×C3×C4—81no—
287C2×C3×C4—121no—
288C2×C3×C4—121no—
289C2×C3×C4C2×D1211yescharacteristic
290C2×C3×C4D3×D411yescharacteristic
291C2×C3×C4D3×D411yescharacteristic
292C2×C3×C4C2×#6711yescharacteristic
293C2×C3×C4C2×#6711yescharacteristic
294C2×C3×C4C2×#6711yescharacteristic
295#67—44no—
296C2×Dic3—21no—
297C2×Dic3—122no—
298C2×Dic3—121no—
299C2×Dic3—121no—
300D12—21no—
301D12—42no—
302D12—44no—
303D12—41no—
304D12—41no—
305D12—41no—
306D12—41no—
307D12#21412yescomplemented
308C4×D3—41no—
309C4×D3—41no—
310Dic6—122no—
311Dic6—122no—
312Dic6—242no—
313C3×C8—62no—
314C3×C8—121no—
315C3⋊C8—22no—
316C3⋊C8—41no—
317C3⋊C8—122no—
318C3⋊C8—121no—
319C3⋊C8—121no—
320C2×C32—42no—
321C2×C32#45011yescharacteristic
322C2×C32#45011yescharacteristic
323C2×C32#45311yescharacteristic
324C3×D3—42no—
325C3×D3—82no—
326C24—62no—
327C2×Q8—181no—
328C2×D4—31no—
329C2×D4—32no—
330C2×D4—62no—
331C2×D4—62no—
332C2×D4—61no—
333C2×D4—61no—
334C2×D4—122no—
335C2×D4—122no—
336C2×D4—122no—
337C2×D4—122no—
338C2×D4#62611yescharacteristic
339C22×C4—21no—
340C22×C4—61no—
341SD16—364no—
342C2×C8—61no—
343C2×C8—61no—
344C2×C8—181no—
345C4⋊C4—31no—
346C4⋊C4—181no—
347C4⋊C4—181no—
348C4⋊C4#62611yescharacteristic
349C22⋊C4—21no—
350C22⋊C4—62no—
351C22⋊C4—61no—
352C42C2×D3211yescharacteristic
353C22×C3—22no—
354C22×C3—22no—
355C22×C3—21no—
356C22×C3—42no—
357C22×C3—41no—
358C22×C3—41no—
359C22×C3—41no—
360C22×C3—41no—
361C22×C3—62no—
362C22×C3—81no—
363C22×C3—81no—
364C22×C3—121no—
365C22×C3—121no—
366C22×C3—122no—
367C22×C3—122no—
368C22×C3—122no—
369C22×C3—248no—
370C22×C3#88211yescharacteristic
371C22×C3#95311yescharacteristic
372C2×D3—22no—
373C2×D3—41no—
374C2×D3—41no—
375C2×D3—42no—
376C2×D3—42no—
377C2×D3—42no—
378C2×D3—88no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—21no—
384C3×C4—21no—
385C3×C4—41no—
386C3×C4—41no—
387C3×C4—41no—
388C3×C4—81no—
389C3×C4—121no—
390C3×C4C2×#20411yescharacteristic
391C3×C4#90811yescharacteristic
392C3×C4#91111yescharacteristic
393C3×C4D3×SD1611yescharacteristic
394C3×C4#93211yescharacteristic
395C3×C4#93211yescharacteristic
396C3×C4C2×#21411yescharacteristic
397C3×C4C2×#21411yescharacteristic
398Dic3—41no—
399Dic3—242no—
400Dic3—241no—
401Dic3—241no—
402C32#162911yescharacteristic, complemented, Sylow
403C23—21no—
404C23—61no—
405C23—62no—
406C23—124no—
407C23—124no—
408C23—242no—
409Q8—362no—
410D4—21no—
411D4—32no—
412D4—61no—
413D4—122no—
414D4—124no—
415D4—124no—
416D4—121no—
417D4—121no—
418D4—121no—
419D4—121no—
420D4#371612yescomplemented
421C2×C4—21no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—61no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—361no—
428C2×C4C2×#62611yescharacteristic
429C2×C4D3×#6711yescharacteristic
430C2×C4D3×#6711yescharacteristic
431C8—62no—
432C8—121no—
433C8—361no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—42no—
438C2×C3—42no—
439C2×C3—82no—
440C2×C3—122no—
441C2×C3—242no—
442C2×C3#473111yescharacteristic
443C2×C3#477611yescharacteristic
444C2×C3#482211yescharacteristic
445C2×C3#521811yescharacteristic
446C2×C3#521811yescharacteristic
447C2×C3#524111yescharacteristic
448D3—42no—
449D3—82no—
450C22—22no—
451C22—41no—
452C22—41no—
453C22—62no—
454C22—121no—
455C22—121no—
456C22—122no—
457C22—122no—
458C22—122no—
459C22—248no—
460C22#6397711yescharacteristic
461C4—21no—
462C4—41no—
463C4—121no—
464C4—721no—
465C4C2×#371611yescharacteristic
466C4#6383011yescharacteristic
467C4D3×#21411yescharacteristic
468C4#6393211yescharacteristic
469C3—21no—
470C3#6819511yescharacteristic, complemented
471C3#7190711yescharacteristic, complemented
472C2—42no—
473C2—122no—
474C2—242no—
475C2#1059117211yescharacteristic
476C2#1059145911yescharacteristic
477C2#1059166611yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×D4C22
socleC22×C32#120
prime radicalC3×D4×D12C2
hypercenterC4×D4D32
hypocenterC32#1629
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. #10592707C2semidirect
3. #10590369C2semidirect
4. #10590268C2semidirect
5. C3×D4⋊(C3⋊C8)C2semidirect
6. #10589090C2semidirect
7. #10589081C2semidirect
13. #64078C22semidirect
22. C4⋊(C3⋊(C3⋊C8))C22semidirect
23. C3×C4⋊(C3⋊C8)C22semidirect
24. C3×C4×C3⋊C8C22semidirect
39. D4⋊(C3⋊C8)D3semidirect
107. C4⋊(C3⋊C8)C2×D3semidirect
108. C4×C3⋊C8C2×D3semidirect
116. C32×D4SD16semidirect
134. C3×D12SD16semidirect
272. C3×D4#204semidirect
273. C3×D4#214semidirect
307. D12#214semidirect
402. C32#1629semidirect
420. D4#3716semidirect
470. C3#68195semidirect
471. C3#71907semidirect

Maximal subgroups

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

subgroupindex
C3×D4×D122
#105927072
#105903692
#105902682
C3×D4⋊(C3⋊C8)2
#105890902
#105890812
#719073
#681953