Finite Group Databasebeta

← #11780649

Subgroups — #11780649

Order 1152.

446 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 76 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×#5700C211yesmaximal, characteristic
2C3×#5064C211yesmaximal, characteristic, complemented
3#10590376C211yesmaximal, characteristic, complemented
4#10590280C211yesmaximal, characteristic, complemented
5#10589409C211yesmaximal, characteristic, complemented
6#10589086C211yesmaximal, characteristic, complemented
7#10589086C211yesmaximal, characteristic, complemented
8#71953—31nomaximal
9#71392—31nomaximal
10C32×#125C2211yescharacteristic
11C3×#896—21no—
12C3×#896—21no—
13C3×#895—21no—
14C3×#895—21no—
15C3×#894—21no—
16C3×#894—21no—
17C3×#875C2211yescharacteristic
18C3×#875C2211yescharacteristic
19C3×#869C2211yescharacteristic
20C2×#3718—21no—
21C2×#3715—21no—
22#63630C2211yescharacteristic
23#63630C2211yescharacteristic
24C3×#808—21no—
25C3×#807—21no—
26C3×#803C2211yescharacteristic
27#63565—21no—
28#63562—21no—
29C3×#562—31no—
30#5700D311yescharacteristic
31C3×#501—31no—
32#5089—31no—
33#5086—31no—
34#5064D311yescharacteristic, complemented
35#5060—31no—
36#5055—31no—
37#4543—31no—
38#4543—31no—
39#4543—61no—
40#4540—31no—
41#4540—31no—
42#4464—31no—
43#4464—31no—
44C2×C3×D12—21no—
45C2×C3×D12—21no—
46C2×C3×D12D411yescharacteristic
47C2×C3×D12D411yescharacteristic
48C2×C3×C4×D3—21no—
49C2×C3×C4×D3—21no—
50C2×C3×Dic6—21no—
51C2×C3×Dic6D411yescharacteristic
52C2×C3×Dic6D411yescharacteristic
53C32×C4⋊C4—21no—
54C32×C4⋊C4—21no—
55C32×C4⋊C4D411yescharacteristic
56C32×C4⋊C4D411yescharacteristic
57C32×C42C2311yescharacteristic
58C2×C3⋊(C3⋊C8)—21no—
59C2×C3⋊(C3⋊C8)—21no—
60C3×(C2×D3)⋊C4—21no—
61C3×(C2×D3)⋊C4—21no—
62C3×(C2×D3)⋊C4—21no—
63C3×(C2×D3)⋊C4—21no—
64C3×(C2×D3)⋊C4—21no—
65C3×(C2×D3)⋊C4—21no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3—21no—
68C3×Dic3⋊C4—21no—
69C3×Dic3⋊C4—21no—
70C2×C3×C3⋊C8—21no—
71#3718—42no—
72#3715—42no—
73#1661—91noSylow
74C3×C4⋊Q8—31no—
75C3×#125—21no—
76C3×#125C2×D311yescharacteristic
77C3×#125C2×D311yescharacteristic
78C3×#124—31no—
79C3×#124—31no—
80C3×#123—61no—
81C3×#123—61no—
82C3×C22⋊Q8—61no—
83C3×C22⋊Q8—61no—
84C3×#121—61no—
85C3×#121—61no—
86C2×#216—61no—
87C2×#216—61no—
88C2×#215—61no—
89C2×#214—61no—
90#896—21no—
91#896—21no—
92#895—21no—
93#895—21no—
94#894—21no—
95#894—21no—
96#875C2×D311yescharacteristic
97#875C2×D311yescharacteristic
98#869C2×D311yescharacteristic
99C3×#107—61no—
100C3×#106—61no—
101C3×#97—31no—
102#835—61no—
103#832—61no—
104#810—61no—
105#809—61no—
106#808—21no—
107#807—21no—
108#803—31no—
109#803—31no—
110#803—31no—
111#803—31no—
112#803—61no—
113#803—61no—
114#803C2×D311yescharacteristic
115C22×C3×D3—21no—
116C22×C3×D3—21no—
117C2×C32×C4—21no—
118C2×C32×C4—21no—
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C3×Dic3—21no—
123C2×C3×Dic3—21no—
124C2×C3×Dic3—41no—
125C3×D12—41no—
126C3×D12—41no—
127C3×D12—81no—
128C3×D12—81no—
129C3×C4×D3—81no—
130C3×C4×D3—81no—
131C3×Dic6—41no—
132C3×Dic6—41no—
133C3×Dic6—41no—
134C3×Dic6—41no—
135C3⋊(C3⋊C8)—22no—
136C3⋊(C3⋊C8)—41no—
137C3×C3⋊C8—22no—
138#562—31no—
139#501—31no—
140#497—91no—
141#489—91no—
142#330—91no—
143#329—91no—
144#329—91no—
145C2×C3×Q8—31no—
146C2×C3×Q8—31no—
147C2×C3×Q8—61no—
148C2×C3×D4—31no—
149C2×C3×D4—31no—
150C2×C3×D4—61no—
151C2×C3×D4—61no—
152C22×C3×C4—61no—
153C22×C3×C4—61no—
154C2×D12—21no—
155C2×D12—21no—
156C2×D12#6711yescharacteristic
157C2×D12#6711yescharacteristic
158C2×C4×D3—21no—
159C2×C4×D3—21no—
160C2×Dic6—21no—
161C2×Dic6#6711yescharacteristic
162C2×Dic6#6711yescharacteristic
163C2×C3×C8—61no—
164C3×C4⋊C4—21no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—21no—
170C3×C4⋊C4—41no—
171C3×C4⋊C4—41no—
172C3×C4⋊C4—61no—
173C3×C4⋊C4—61no—
174C3×C4⋊C4—61no—
175C3×C4⋊C4—61no—
176C3×C4⋊C4#6711yescharacteristic
177C3×C4⋊C4#6711yescharacteristic
178C3×C4⋊C4#6711yescharacteristic
179C3×C4⋊C4#6711yescharacteristic
180C3×C22⋊C4—61no—
181C3×C22⋊C4—61no—
182C3×C22⋊C4—61no—
183C3×C22⋊C4—61no—
184C3×C22⋊C4—61no—
185C3×C22⋊C4—61no—
186C3×C42—21no—
187C3×C42C22×D311yescharacteristic
188C3×C42C22×D311yescharacteristic
189#216—122no—
190#216—122no—
191#215—122no—
192#214—122no—
193(C2×D3)⋊C4—21no—
194(C2×D3)⋊C4—21no—
195(C2×D3)⋊C4—21no—
196(C2×D3)⋊C4—21no—
197(C2×D3)⋊C4—21no—
198(C2×D3)⋊C4—21no—
199C4⋊Dic3—21no—
200C4⋊Dic3—21no—
201Dic3⋊C4—21no—
202Dic3⋊C4—21no—
203C2×C3⋊C8—21no—
204C2×C3⋊C8—61no—
205C2×C3⋊C8—61no—
206C2×C3⋊C8—61no—
207C2×C3⋊C8—61no—
208C2×C3⋊C8—121no—
209C2×C3⋊C8—121no—
210C22×C32#12011yescharacteristic
211C2×C3×D3—41no—
212C2×C3×D3—41no—
213C2×C3×D3—41no—
214C2×C3×D3—41no—
215C2×C3×D3—41no—
216C2×C3×D3—41no—
217C32×C4—21no—
218C32×C4—21no—
219C32×C4—21no—
220C32×C4—41no—
221C32×C4—41no—
222C3×Dic3—41no—
223C3×Dic3—41no—
224C3×Dic3—81no—
225C2×Q16—181no—
226C2×SD16—181no—
227C4⋊Q8—31no—
228#125D3211yescharacteristic
229#124—31no—
230#124—31no—
231#123—61no—
232#123—61no—
233C22⋊Q8—61no—
234C22⋊Q8—61no—
235#121—61no—
236#121—61no—
237#107—61no—
238#106—61no—
239#103—181no—
240#102—181no—
241#97—31no—
242#97—91no—
243#97—91no—
244C23×C3—61no—
245C23×C3—61no—
246C22×D3—21no—
247C22×D3—21no—
248C3×Q8—121no—
249C3×Q8—121no—
250C3×Q8—121no—
251C3×Q8—121no—
252C3×D4—121no—
253C3×D4—121no—
254C3×D4—241no—
255C3×D4—241no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—61no—
266C2×C3×C4—61no—
267C2×C3×C4—121no—
268C2×C3×C4—241no—
269C2×C3×C4—241no—
270C2×C3×C4D3×D411yescharacteristic
271C2×C3×C4D3×D411yescharacteristic
272C2×C3×C4C2×#6711yescharacteristic
273C2×C3×C4C2×#6711yescharacteristic
274C2×C3×C4C2×#6711yescharacteristic
275C2×C3×C4C2×#6711yescharacteristic
276C2×Dic3—21no—
277C2×Dic3—21no—
278C2×Dic3—41no—
279D12—41no—
280D12—41no—
281D12—81no—
282D12—81no—
283C4×D3—81no—
284C4×D3—81no—
285Dic6—41no—
286Dic6—41no—
287Dic6—41no—
288Dic6—41no—
289C3×C8—62no—
290C3⋊C8—22no—
291C3⋊C8—62no—
292C3⋊C8—62no—
293C3⋊C8—122no—
294C3⋊C8—121no—
295C3⋊C8—121no—
296C3⋊C8—122no—
297C2×C32#45311yescharacteristic
298C2×C32#45511yescharacteristic
299C2×C32#45611yescharacteristic
300C3×D3—81no—
301C3×D3—81no—
302C2×Q8—31no—
303C2×Q8—31no—
304C2×Q8—61no—
305C2×D4—31no—
306C2×D4—31no—
307C2×D4—61no—
308C2×D4—61no—
309C22×C4—61no—
310C22×C4—61no—
311Q16—362no—
312SD16—362no—
313C2×C8—61no—
314C2×C8—181no—
315C2×C8—181no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C4⋊C4#62511yescharacteristic
323C4⋊C4#62511yescharacteristic
324C22⋊C4—61no—
325C22⋊C4—61no—
326C22⋊C4—61no—
327C22⋊C4—61no—
328C22⋊C4—61no—
329C22⋊C4—61no—
330C42C2×D3211yescharacteristic
331C22×C3—21no—
332C22×C3—121no—
333C22×C3—121no—
334C22×C3—121no—
335C22×C3—121no—
336C22×C3—121no—
337C22×C3—121no—
338C22×C3#93711yescharacteristic
339C22×C3#95311yescharacteristic
340C2×D3—41no—
341C2×D3—41no—
342C2×D3—41no—
343C2×D3—41no—
344C2×D3—41no—
345C2×D3—41no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—41no—
353C3×C4—41no—
354C3×C4—41no—
355C3×C4—41no—
356C3×C4—41no—
357C3×C4—41no—
358C3×C4—41no—
359C3×C4—81no—
360C3×C4—81no—
361C3×C4—121no—
362C3×C4—121no—
363C3×C4—241no—
364Dic3—41no—
365Dic3—41no—
366Dic3—81no—
367C32#166111yescharacteristic, complemented, Sylow
368C23—61no—
369C23—61no—
370Q8—121no—
371Q8—121no—
372Q8—121no—
373Q8—121no—
374D4—121no—
375D4—121no—
376D4—241no—
377D4—241no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—121no—
383C2×C4—241no—
384C2×C4—241no—
385C2×C4C2×#62511yescharacteristic
386C2×C4D3×#6711yescharacteristic
387C2×C4D3×#6711yescharacteristic
388C8—62no—
389C8—182no—
390C8—361no—
391C2×C3—21no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—241no—
395C2×C3—241no—
396C2×C3#509511yescharacteristic
397C2×C3#519811yescharacteristic
398C2×C3#520211yescharacteristic
399C2×C3#524111yescharacteristic
400C2×C3#524211yescharacteristic
401C2×C3#524611yescharacteristic
402D3—81no—
403D3—81no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—121no—
408C22—121no—
409C22—121no—
410C22#6397611yescharacteristic
411C4—21no—
412C4—21no—
413C4—21no—
414C4—41no—
415C4—41no—
416C4—121no—
417C4—121no—
418C4—241no—
419C3—21no—
420C3#7139211yescharacteristic, complemented
421C3#7195311yescharacteristic, complemented
422C2—241no—
423C2—241no—
424C2#1059164811yescharacteristic
425C2#1059166411yescharacteristic
426C2#1059166511yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×#125C22
socleC22×C32#120
prime radicalC3×#5700C2
hypercenter#125D32
hypocenterC32#1661
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ #125

Complements

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

NH ≅ G/Ntype
2. C3×#5064C2semidirect
3. #10590376C2semidirect
4. #10590280C2semidirect
5. #10589409C2semidirect
6. #10589086C2semidirect
7. #10589086C2semidirect
34. #5064D3semidirect
367. C32#1661semidirect
420. C3#71392semidirect
421. C3#71953semidirect

Maximal subgroups

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

subgroupindex
C3×#57002
C3×#50642
#105903762
#105902802
#105894092
#105890862
#719533
#713923