Finite Group Databasebeta

← #11779546

Subgroups — #11779546

Order 1152.

522 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 522 conjugacy classes of subgroups — too many to draw. Only its 96 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×#5551C211yesmaximal, characteristic
2#10592707C211yesmaximal, characteristic, complemented
3#10590288C211yesmaximal, characteristic, complemented
4#10590268C211yesmaximal, characteristic, complemented
5C3×D4⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6#10589093C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic, complemented
8#71911—31nomaximal
9#68181—31nomaximal
10C2×C3×#235—21no—
11C32×C4×D4C2211yescharacteristic
12#64078C2211yescharacteristic
13C3×#938—21no—
14C3×#924—21no—
15C3×D3×C4⋊C4—21no—
16C3×#883—22no—
17C3×C4×D12C2211yescharacteristic
18C3×#869C2211yescharacteristic
19#63659—21no—
20#63635—21no—
21C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C3×C4⋊(C3⋊C8)C2211yescharacteristic
23C3×C4×C3⋊C8C2211yescharacteristic
24#63573—21no—
25#63567—22no—
26#63563—21no—
27C3×#547—31no—
28#5551D311yescharacteristic
29#5093—31no—
30#5008—31no—
31#5008—31no—
32#5008—31no—
33#5008—61no—
34#4842—31no—
35#4681—31no—
36C3×D4⋊C8—31no—
37#4547—31no—
38D4⋊(C3⋊C8)—31no—
39D4⋊(C3⋊C8)D311yescharacteristic, complemented
40#4482—31no—
41#4459—31no—
42C2×C32×D4D411yescharacteristic
43C22×C32×C4—21no—
44C2×#634—21no—
45C2×C3×#67—22no—
46C3×#235—42no—
47C3×#235—42no—
48C2×C3×D12D411yescharacteristic
49C2×C3×C4×D3—22no—
50C2×C3×C4×D3—21no—
51C2×C3×Dic6—21no—
52C32×C4⋊C4D411yescharacteristic
53C32×C22⋊C4—21no—
54C32×C42C2311yescharacteristic
55C4⋊(C3⋊Dic3)—21no—
56C4⋊(C3⋊Dic3)—21no—
57C2×C3⋊(C3⋊C8)—21no—
58C3×#217—22no—
59C3×(C2×D3)⋊C4—21no—
60C3×C4⋊Dic3—21no—
61C3×C4⋊Dic3D412yes—
62C3×C4⋊Dic3D411yescharacteristic
63C3×Dic3⋊C4—22no—
64C2×C3×C3⋊C8—21no—
65C2×C3×C3⋊C8—21no—
66#1635—91noSylow
67C2×C3×#41—61no—
68C2×#235—21no—
69C3×C4⋊Q8—31no—
70C3×#123—62no—
71C3×C22⋊Q8—61no—
72C3×#121—61no—
73C3×C4×D4—21no—
74C3×C4×D4—31no—
75C3×C4×D4C2×D311yescharacteristic
76C3×C4×D4C2×D311yescharacteristic
77C2×C3×C4⋊C4—61no—
78#938—21no—
79#924—21no—
80D3×C4⋊C4—21no—
81#883—22no—
82C4×D12C2×D311yescharacteristic
83#869—31no—
84#869—31no—
85#869—61no—
86#869C2×D311yescharacteristic
87C3×C4⋊C8—31no—
88C3×C4×C8—31no—
89#832—61no—
90#832—62no—
91#832—61no—
92#832—61no—
93#821—61no—
94#817—61no—
95#816—62no—
96#810—62no—
97#808—61no—
98#808—62no—
99#808—61no—
100#808—61no—
101C4⋊(C3⋊C8)—31no—
102C4⋊(C3⋊C8)—31no—
103C4⋊(C3⋊C8)—61no—
104C4⋊(C3⋊C8)C2×D311yescharacteristic
105C4×C3⋊C8C2×D311yescharacteristic
106C23×C32—21no—
107C22×C3×D3—21no—
108C32×D4—21no—
109C32×D4—21no—
110C2×C32×C4—21no—
111C2×C32×C4—41no—
112C2×C32×C4—41no—
113C2×C32×C4C2×D411yescharacteristic
114C2×C32×C4C2×D411yescharacteristic
115C2×C32×C4C2×D411yescharacteristic
116C2×C3⋊Dic3—41no—
117#634—42no—
118C3×#67—44no—
119C2×C3×Dic3—21no—
120C2×C3×Dic3—22no—
121C3×D12—21no—
122C3×D12SD1612yes—
123C3×C4×D3—44no—
124C3×C4×D3—41no—
125C3×C4×D3—41no—
126C3×Dic6—41no—
127C3×Dic6—41no—
128C3⋊(C3⋊C8)—41no—
129C3×C3⋊C8—41no—
130C3×C3⋊C8—41no—
131#547—31no—
132#487—91no—
133#476—91no—
134#476—91no—
135#332—91no—
136D4⋊C8—31no—
137D4⋊C8—91no—
138C3×#41—122no—
139C3×#41—122no—
140C2×C3×Q8—61no—
141C2×C3×D4—21no—
142C2×C3×D4—31no—
143C2×C3×D4—62no—
144C2×C3×D4D1211yescharacteristic
145C2×C3×D4#6711yescharacteristic
146C22×C3×C4—21no—
147C22×C3×C4—21no—
148C22×C3×C4—41no—
149C22×C3×C4—62no—
150C22×C3×C4—61no—
151C2×#67—22no—
152#235—42no—
153#235—42no—
154C2×D12#6711yescharacteristic
155C2×C4×D3—22no—
156C2×C4×D3—21no—
157C2×Dic6—21no—
158C2×Dic6—61no—
159C2×Dic6—61no—
160C2×Dic6—121no—
161C2×C3×C8—61no—
162C2×C3×C8—61no—
163C3×C4⋊C4—21no—
164C3×C4⋊C4—31no—
165C3×C4⋊C4—32no—
166C3×C4⋊C4—62no—
167C3×C4⋊C4—61no—
168C3×C4⋊C4D1211yescharacteristic
169C3×C4⋊C4#6711yescharacteristic
170C3×C22⋊C4—21no—
171C3×C22⋊C4—21no—
172C3×C22⋊C4—41no—
173C3×C22⋊C4—62no—
174C3×C22⋊C4—61no—
175C3×C42—21no—
176C3×C42C22×D311yescharacteristic
177C3×C42C22×D311yescharacteristic
178#217—22no—
179(C2×D3)⋊C4—21no—
180C4⋊Dic3—21no—
181C4⋊Dic3—62no—
182C4⋊Dic3—61no—
183C4⋊Dic3—61no—
184C4⋊Dic3—61no—
185C4⋊Dic3—61no—
186C4⋊Dic3—121no—
187C4⋊Dic3#6712yes—
188C4⋊Dic3#6711yescharacteristic
189Dic3⋊C4—22no—
190C2×C3⋊C8—21no—
191C2×C3⋊C8—21no—
192C2×C3⋊C8—62no—
193C2×C3⋊C8—61no—
194C2×C3⋊C8—61no—
195C22×C32—41no—
196C22×C32—41no—
197C22×C32—41no—
198C22×C32#12011yescharacteristic
199C2×C3×D3—22no—
200C2×C3×D3—41no—
201C2×C3×D3—41no—
202C32×C4—21no—
203C32×C4—41no—
204C32×C4C2×SD1611yescharacteristic
205C32×C4#13511yescharacteristic
206C32×C4AGL(1,ℤ/8)11yescharacteristic
207C32×C4#13711yescharacteristic
208C3⋊Dic3—81no—
209C3×Dic3—42no—
210C3×Dic3—41no—
211C2×#41—61no—
212C4⋊Q8—31no—
213C4⋊Q8—91no—
214#123—62no—
215C22⋊Q8—61no—
216#121—61no—
217C4×D4—31no—
218C4×D4D3211yescharacteristic
219C2×C4⋊C4—61no—
220#106—181no—
221#106—181no—
222C4⋊C8—31no—
223C4⋊C8—91no—
224#103—182no—
225#102—181no—
226#102—181no—
227C4×C8—31no—
228C23×C3—21no—
229C23×C3—21no—
230C23×C3—41no—
231C23×C3—61no—
232C22×D3—21no—
233C3×Q8—121no—
234C3×Q8—121no—
235C3×D4—22no—
236C3×D4—21no—
237C3×D4—21no—
238C3×D4—21no—
239C3×D4—21no—
240C3×D4—32no—
241C3×D4—41no—
242C3×D4—61no—
243C3×D4—124no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—61no—
255C2×C3×C4—62no—
256C2×C3×C4—81no—
257C2×C3×C4—81no—
258C2×C3×C4—124no—
259C2×C3×C4—121no—
260C2×C3×C4—121no—
261C2×C3×C4C2×D1211yescharacteristic
262C2×C3×C4D3×D411yescharacteristic
263C2×C3×C4D3×D411yescharacteristic
264C2×C3×C4C2×#6711yescharacteristic
265C2×C3×C4C2×#6711yescharacteristic
266C2×C3×C4C2×#6711yescharacteristic
267#67—44no—
268C2×Dic3—21no—
269C2×Dic3—22no—
270C2×Dic3—122no—
271C2×Dic3—121no—
272C2×Dic3—121no—
273D12—21no—
274D12#21412yes—
275C4×D3—44no—
276C4×D3—41no—
277C4×D3—41no—
278Dic6—41no—
279Dic6—41no—
280Dic6—122no—
281Dic6—122no—
282Dic6—242no—
283C3×C8—121no—
284C3×C8—121no—
285C3⋊C8—41no—
286C3⋊C8—41no—
287C3⋊C8—122no—
288C3⋊C8—121no—
289C3⋊C8—121no—
290C2×C32—81no—
291C2×C32#45011yescharacteristic
292C2×C32#45211yescharacteristic
293C2×C32#45411yescharacteristic
294C3×D3—42no—
295#41—122no—
296#41—122no—
297C2×Q8—61no—
298C2×Q8—181no—
299C2×D4—31no—
300C2×D4—62no—
301C2×D4#62611yescharacteristic
302C22×C4—21no—
303C22×C4—62no—
304C22×C4—61no—
305C2×C8—61no—
306C2×C8—61no—
307C2×C8—181no—
308C4⋊C4—32no—
309C4⋊C4—31no—
310C4⋊C4—62no—
311C4⋊C4—61no—
312C4⋊C4—181no—
313C4⋊C4—181no—
314C4⋊C4#62611yescharacteristic
315C22⋊C4—21no—
316C22⋊C4—62no—
317C22⋊C4—61no—
318C42C2×D3211yescharacteristic
319C22×C3—21no—
320C22×C3—42no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—62no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3—121no—
331C22×C3—121no—
332C22×C3#88211yescharacteristic
333C22×C3#95311yescharacteristic
334C2×D3—22no—
335C2×D3—41no—
336C2×D3—41no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—21no—
343C3×C4—41no—
344C3×C4—41no—
345C3×C4—41no—
346C3×C4—81no—
347C3×C4—121no—
348C3×C4—122no—
349C3×C4#90411yescharacteristic
350C3×C4#90911yescharacteristic
351C3×C4#91111yescharacteristic
352C3×C4D3×SD1611yescharacteristic
353C3×C4#93211yescharacteristic
354C3×C4C2×#21411yescharacteristic
355C3×C4#95011yescharacteristic
356C3×C4#95111yescharacteristic
357Dic3—42no—
358Dic3—41no—
359Dic3—242no—
360Dic3—241no—
361Dic3—241no—
362C32#163511yescharacteristic, complemented, Sylow
363C23—21no—
364C23—61no—
365Q8—121no—
366Q8—121no—
367Q8—362no—
368D4—21no—
369D4—21no—
370D4—32no—
371D4—61no—
372D4—124no—
373C2×C4—21no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—61no—
377C2×C4—62no—
378C2×C4—124no—
379C2×C4—121no—
380C2×C4—121no—
381C2×C4—361no—
382C2×C4C2×#62611yescharacteristic
383C2×C4D3×#6711yescharacteristic
384C2×C4D3×#6711yescharacteristic
385C8—121no—
386C8—121no—
387C8—361no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3—21no—
391C2×C3—82no—
392C2×C3—81no—
393C2×C3—81no—
394C2×C3—122no—
395C2×C3#473311yescharacteristic
396C2×C3#477611yescharacteristic
397C2×C3#482711yescharacteristic
398C2×C3#521811yescharacteristic
399C2×C3#523911yescharacteristic
400C2×C3#524511yescharacteristic
401D3—42no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—62no—
406C22—121no—
407C22—121no—
408C22#6397711yescharacteristic
409C4—21no—
410C4—41no—
411C4—121no—
412C4—122no—
413C4—721no—
414C4#6382911yescharacteristic
415C4#6383111yescharacteristic
416C4D3×#21411yescharacteristic
417C4#6393211yescharacteristic
418C3—21no—
419C3#6818111yescharacteristic, complemented
420C3#7191111yescharacteristic, complemented
421C2—81no—
422C2—122no—
423C2#1059117311yescharacteristic
424C2#1059145911yescharacteristic
425C2#1059168511yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4×D4C22
socleC22×C32#120
prime radicalC3×#5551C2
hypercenterC4×D4D32
hypocenterC32#1635
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. #10590288C2semidirect
4. #10590268C2semidirect
5. C3×D4⋊(C3⋊C8)C2semidirect
6. #10589093C2semidirect
7. #10589081C2semidirect
39. D4⋊(C3⋊C8)D3semidirect
362. C32#1635semidirect
419. C3#68181semidirect
420. C3#71911semidirect

Maximal subgroups

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

subgroupindex
C3×#55512
#105927072
#105902882
#105902682
C3×D4⋊(C3⋊C8)2
#105890932
#105890812
#719113
#681813