Finite Group Databasebeta

← #11780434

Subgroups — #11780434

Order 1152.

540 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 540 conjugacy classes of subgroups — too many to draw. Only its 90 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
1#10593896C211yesmaximal, characteristic
2#10592782C211yesmaximal, characteristic, complemented
3C3×#5087C211yesmaximal, characteristic, complemented
4C3×#5080C211yesmaximal, characteristic, complemented
5#10589093C211yesmaximal, characteristic, complemented
6#10589082C211yesmaximal, characteristic, complemented
7#10589081C211yesmaximal, characteristic, complemented
8#71343—31nomaximal
9#71111—31nomaximal
10C2×#3796—21no—
11C32×C4⋊Q8C2211yescharacteristic
12C2×#3755—21no—
13C4×C4⋊(C3⋊D3)C2211yescharacteristic
14C3×C4×D12C2211yescharacteristic
15C3×#869C2211yescharacteristic
16#63909—21no—
17#63890—22no—
18D3×C4⋊Dic3—21no—
19#63871—21no—
20#63662—21no—
21C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
22C3×#809—21no—
23C3×#808—22no—
24C3×#807—21no—
25C3×C4⋊(C3⋊C8)C2211yescharacteristic
26C3×C4×C3⋊C8C2211yescharacteristic
27#5726—31no—
28#5555—31no—
29C3×#498—31no—
30C3×#474—31no—
31#5087D311yescharacteristic, complemented
32#5083—31no—
33#5083—31no—
34#5083—61no—
35#5080D311yescharacteristic, complemented
36#4547—31no—
37D4⋊(C3⋊C8)—31no—
38#4483—31no—
39#4482—31no—
40#4459—31no—
41#4459—31no—
42C2×C32×Q8—21no—
43C2×C4⋊(C3⋊D3)D411yescharacteristic
44C2×C4×C3⋊D3—21no—
45C2×C3×D12D411yescharacteristic
46C2×C3×C4×D3—21no—
47C2×C3×Dic6—21no—
48C2×#626—22no—
49C2×D3×Dic3—22no—
50#3796—42no—
51#3796—42no—
52C32×C4⋊C4—21no—
53C32×C4⋊C4—21no—
54C32×C42C2311yescharacteristic
55#3755—42no—
56#3752—21no—
57C4⋊(C3⋊Dic3)D411yescharacteristic
58C2×C3⋊(C3⋊C8)—21no—
59C3×(C2×D3)⋊C4—21no—
60C3×C4⋊Dic3—21no—
61C3×C4⋊Dic3D411yescharacteristic
62C3×C4⋊Dic3D412yes—
63C2×C3×C3⋊C8—21no—
64C2×C3×C3⋊C8—21no—
65Dic3⋊Dic3—22no—
66#3722—22no—
67#1654—91noSylow
68C2×#239—61no—
69C2×#237—61no—
70C3×C4⋊Q8—21no—
71C3×C4⋊Q8—31no—
72C3×C4⋊Q8C2×D311yescharacteristic
73C3×C4⋊Q8C2×D311yescharacteristic
74C3×C4×D4—31no—
75#946—61no—
76C2×#215—61no—
77C2×#215—61no—
78C2×#215—62no—
79#930—61no—
80C2×C4⋊Dic3—61no—
81#897—61no—
82#895—61no—
83#894—62no—
84D3×C4⋊C4—61no—
85#886—62no—
86C4×D12—31no—
87C4×D12—31no—
88C4×D12—61no—
89C4×D12C2×D311yescharacteristic
90#869C2×D311yescharacteristic
91C3×#107—61no—
92C3×#106—62no—
93C3×C4⋊C8—31no—
94C3×#102—61no—
95C3×C4×C8—31no—
96#835—61no—
97#835—61no—
98#835—62no—
99#809—21no—
100#808—22no—
101#807—21no—
102C4⋊(C3⋊C8)—31no—
103C4⋊(C3⋊C8)—31no—
104C4⋊(C3⋊C8)—61no—
105C4⋊(C3⋊C8)C2×D311yescharacteristic
106C4×C3⋊C8C2×D311yescharacteristic
107C22×C3⋊D3—21no—
108C22×C3×D3—21no—
109C32×Q8—41no—
110C32×Q8—41no—
111C2×C32×C4—41no—
112C2×C32×C4C2×D411yescharacteristic
113C2×C32×C4C2×D411yescharacteristic
114C2×C32×C4C2×D411yescharacteristic
115C2×C3⋊Dic3—21no—
116C4⋊(C3⋊D3)—21no—
117C4⋊(C3⋊D3)—21no—
118C4×C3⋊D3—41no—
119C4×C3⋊D3—41no—
120C2×C3×Dic3—21no—
121C2×C3×Dic3—22no—
122C3×D12—21no—
123C3×D12SD1612yescomplemented
124C3×C4×D3—41no—
125C3×C4×D3—41no—
126C3×Dic6—41no—
127C3×Dic6—41no—
128#626—44no—
129D3×Dic3—44no—
130C3⋊(C3⋊C8)—41no—
131C3×C3⋊C8—22no—
132C3×C3⋊C8—41no—
133#547—91no—
134#498—31no—
135#474—31no—
136#461—91no—
137#332—91no—
138D4⋊C8—91no—
139D4⋊C8—91no—
140C2×C3×Q8—22no—
141C2×C3×Q8—21no—
142C2×C3×Q8—21no—
143C2×C3×Q8—61no—
144C2×C3×D4—31no—
145C22×C3×C4—61no—
146C2×#67—62no—
147C22×Dic3—62no—
148#239—122no—
149#239—122no—
150#237—122no—
151#237—122no—
152C2×D12—31no—
153C2×D12—31no—
154C2×D12—61no—
155C2×D12—62no—
156C2×D12#6711yescharacteristic
157C2×C4×D3—21no—
158C2×C4×D3—62no—
159C2×C4×D3—61no—
160C2×C4×D3—61no—
161C2×C4×D3—121no—
162C2×Dic6—21no—
163C2×C3×C8—61no—
164C2×C3×C8—61no—
165C3×C4⋊C4—21no—
166C3×C4⋊C4—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4—21no—
169C3×C4⋊C4—32no—
170C3×C4⋊C4—31no—
171C3×C4⋊C4—41no—
172C3×C4⋊C4—41no—
173C3×C4⋊C4—61no—
174C3×C22⋊C4—61no—
175C3×C42—21no—
176C3×C42C22×D311yescharacteristic
177C3×C42C22×D311yescharacteristic
178#215—124no—
179#215—122no—
180#215—122no—
181(C2×D3)⋊C4—21no—
182(C2×D3)⋊C4—62no—
183(C2×D3)⋊C4—62no—
184(C2×D3)⋊C4—61no—
185(C2×D3)⋊C4—61no—
186(C2×D3)⋊C4—121no—
187C4⋊Dic3—21no—
188C4⋊Dic3—31no—
189C4⋊Dic3—31no—
190C4⋊Dic3—61no—
191C4⋊Dic3—62no—
192C4⋊Dic3#6711yescharacteristic
193C4⋊Dic3#6712yes—
194Dic3⋊C4—62no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—62no—
198C2×C3⋊C8—61no—
199C2×C3⋊C8—61no—
200C22×C32#12011yescharacteristic
201C2×C3⋊D3—41no—
202C2×C3⋊D3—41no—
203C2×C3⋊D3—41no—
204C2×C3×D3—22no—
205C2×C3×D3—41no—
206C2×C3×D3—41no—
207C32×C4—21no—
208C32×C4—81no—
209C32×C4C2×SD1611yescharacteristic
210C32×C4#13511yescharacteristic
211C32×C4AGL(1,ℤ/8)11yescharacteristic
212C32×C4AGL(1,ℤ/8)11yescharacteristic
213C3⋊Dic3—41no—
214C3×Dic3—41no—
215C3×Dic3—42no—
216C2×#41—181no—
217C2×SD16—181no—
218C4⋊Q8—31no—
219C4⋊Q8D3211yescharacteristic
220#123—182no—
221C22⋊Q8—181no—
222#121—181no—
223C4×D4—31no—
224C4×D4—91no—
225C2×C4⋊C4—181no—
226#107—61no—
227#106—62no—
228C4⋊C8—31no—
229C4⋊C8—91no—
230#103—181no—
231#102—61no—
232C4×C8—31no—
233C23×C3—61no—
234C22×D3—21no—
235C22×D3—61no—
236C22×D3—61no—
237C22×D3—121no—
238C3×Q8—42no—
239C3×Q8—42no—
240C3×Q8—41no—
241C3×Q8—41no—
242C3×Q8—41no—
243C3×Q8—41no—
244C3×Q8—121no—
245C3×Q8—121no—
246C3×D4—32no—
247C3×D4—61no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—42no—
252C2×C3×C4—41no—
253C2×C3×C4—41no—
254C2×C3×C4—62no—
255C2×C3×C4—61no—
256C2×C3×C4—121no—
257C2×C3×C4—121no—
258C2×C3×C4D3×D411yescharacteristic
259C2×C3×C4D3×D411yescharacteristic
260C2×C3×C4D3×D411yescharacteristic
261C2×C3×C4D3×D411yescharacteristic
262C2×C3×C4C2×#6711yescharacteristic
263C2×C3×C4C2×#6711yescharacteristic
264#67—124no—
265C2×Dic3—21no—
266C2×Dic3—22no—
267C2×Dic3—61no—
268C2×Dic3—61no—
269C2×Dic3—121no—
270C2×Dic3—124no—
271D12—21no—
272D12—62no—
273D12—61no—
274D12—61no—
275D12—61no—
276D12—61no—
277D12—121no—
278D12—124no—
279D12#21512yescomplemented
280C4×D3—41no—
281C4×D3—41no—
282C4×D3—124no—
283C4×D3—121no—
284C4×D3—121no—
285C4×D3—121no—
286C4×D3—121no—
287C4×D3—241no—
288C4×D3—241no—
289Dic6—41no—
290Dic6—41no—
291C3×C8—62no—
292C3×C8—121no—
293C3⋊C8—22no—
294C3⋊C8—41no—
295C3⋊C8—122no—
296C3⋊C8—121no—
297C3⋊C8—121no—
298C2×C32#44911yescharacteristic
299C2×C32#45011yescharacteristic
300C2×C32#45611yescharacteristic
301C3⋊D3—81no—
302C3×D3—42no—
303#41—362no—
304#41—362no—
305C2×Q8—21no—
306C2×Q8—61no—
307C2×D4—31no—
308C2×D4—91no—
309C2×D4—182no—
310C22×C4—61no—
311C22×C4—182no—
312C22×C4—181no—
313SD16—362no—
314C2×C8—61no—
315C2×C8—61no—
316C2×C8—181no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—32no—
320C4⋊C4—31no—
321C4⋊C4—61no—
322C4⋊C4—91no—
323C4⋊C4—182no—
324C22⋊C4—61no—
325C22⋊C4—182no—
326C22⋊C4—181no—
327C42C2×D3211yescharacteristic
328C22×C3—21no—
329C22×C3—62no—
330C22×C3—121no—
331C22×C3—121no—
332C22×C3#93711yescharacteristic
333C22×C3#93711yescharacteristic
334C2×D3—22no—
335C2×D3—41no—
336C2×D3—41no—
337C2×D3—122no—
338C2×D3—121no—
339C2×D3—121no—
340C2×D3—121no—
341C2×D3—121no—
342C2×D3—121no—
343C2×D3—121no—
344C2×D3—241no—
345C2×D3—241no—
346C3×C4—21no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—21no—
350C3×C4—21no—
351C3×C4—21no—
352C3×C4—41no—
353C3×C4—82no—
354C3×C4—81no—
355C3×C4—81no—
356C3×C4—121no—
357C3×C4—122no—
358C3×C4#91111yescharacteristic
359C3×C4#91211yescharacteristic
360C3×C4D3×SD1611yescharacteristic
361C3×C4#91411yescharacteristic
362C3×C4#91411yescharacteristic
363C3×C4#91611yescharacteristic
364C3×C4C2×#21511yescharacteristic
365C3×C4#94911yescharacteristic
366Dic3—41no—
367Dic3—42no—
368Dic3—121no—
369Dic3—121no—
370Dic3—241no—
371C32#165411yescharacteristic, complemented, Sylow
372C23—61no—
373C23—181no—
374Q8—41no—
375Q8—41no—
376Q8—121no—
377Q8—121no—
378D4—32no—
379D4—61no—
380D4—181no—
381D4—181no—
382D4—364no—
383C2×C4—41no—
384C2×C4—61no—
385C2×C4—62no—
386C2×C4—121no—
387C2×C4—121no—
388C2×C4—181no—
389C2×C4—364no—
390C2×C4—361no—
391C2×C4—361no—
392C2×C4D3×#6711yescharacteristic
393C2×C4D3×#6711yescharacteristic
394C2×C4#381111yescharacteristic
395C8—62no—
396C8—121no—
397C8—361no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—21no—
401C2×C3—122no—
402C2×C3#504611yescharacteristic
403C2×C3#509511yescharacteristic
404C2×C3#515811yescharacteristic
405C2×C3#516911yescharacteristic
406C2×C3#517211yescharacteristic
407C2×C3#520511yescharacteristic
408D3—42no—
409D3—242no—
410D3—241no—
411D3—241no—
412C22—62no—
413C22—121no—
414C22—121no—
415C22—361no—
416C22—361no—
417C22—361no—
418C22#6398311yescharacteristic
419C4—21no—
420C4—81no—
421C4—121no—
422C4—122no—
423C4—361no—
424C4#6393611yescharacteristic
425C4#6393711yescharacteristic
426C4D3×#21511yescharacteristic
427C4#6393911yescharacteristic
428C3—21no—
429C3#7111111yescharacteristic, complemented
430C3#7134311yescharacteristic, complemented
431C2—122no—
432C2—721no—
433C2#1059152011yescharacteristic
434C2#1059165111yescharacteristic
435C2#1059166111yescharacteristic
subgroupG/H
centerC22#63983
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊Q8C22
socleC22×C32#120
prime radical#10593896C2
hypercenterC4⋊Q8D32
hypocenterC32#1654
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊Q8

Complements

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

NH ≅ G/Ntype
2. #10592782C2semidirect
3. C3×#5087C2semidirect
4. C3×#5080C2semidirect
5. #10589093C2semidirect
6. #10589082C2semidirect
7. #10589081C2semidirect
31. #5087D3semidirect
35. #5080D3semidirect
123. C3×D12SD16semidirect
279. D12#215semidirect
371. C32#1654semidirect
429. C3#71111semidirect
430. C3#71343semidirect

Maximal subgroups

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

subgroupindex
#105938962
#105927822
C3×#50872
C3×#50802
#105890932
#105890822
#105890812
#713433
#711113