Finite Group Databasebeta

← #11779735

Subgroups — #11779735

Order 1152.

657 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 657 conjugacy classes of subgroups — too many to draw. Only its 92 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#10594004C211yesmaximal, characteristic
2C3×#4841C211yesmaximal, characteristic, complemented
3#10590313C211yesmaximal, characteristic, complemented
4#10590277C211yesmaximal, characteristic, complemented
5#10589349C211yesmaximal, characteristic, complemented
6C3×#4485C211yesmaximal, characteristic, complemented
7#10589080C211yesmaximal, characteristic, complemented
8#71044—31nomaximal
9#68392—31nomaximal
10C2×#3802—21no—
11C2×D3×D12—21no—
12C4×C4⋊(C3⋊D3)C2211yescharacteristic
13C2×C3×#204—21no—
14C3×#874C2211yescharacteristic
15C3×C4×D12C2211yescharacteristic
16C3×#869C2211yescharacteristic, complemented
17#63922—22no—
18#63916—21no—
19#63908—21no—
20C32×C4⋊C8C2211yescharacteristic, complemented
21C4×C3⋊(C3⋊C8)C2211yescharacteristic, complemented
22C3×#816—21no—
23C3×C4⋊(C3⋊C8)C2211yescharacteristic, complemented
24#63575—21no—
25#63564—21no—
26#63562—22no—
27#5717—31no—
28D4×D12—31no—
29C3×#461—31no—
30#5088—31no—
31#5087—31no—
32#5070—31no—
33#4846—31no—
34#4841D311yescharacteristic, complemented
35C3×#331—31no—
36D4⋊(C3⋊C8)—31no—
37#4485D311yescharacteristic, complemented
38#4483—31no—
39#4483—31no—
40#4483—31no—
41#4483—61no—
42C22×D32—22no—
43C2×C4⋊(C3⋊D3)D411yescharacteristic
44C2×C4×C3⋊D3—21no—
45C2×C3×D12—21no—
46C2×C3×D12—21no—
47C2×C3×D12D411yescharacteristic
48C2×C3×D12D412yes—
49C2×C3×C4×D3—21no—
50C2×C3×Dic6—21no—
51C2×#626—22no—
52C2×#625—22no—
53#3802—42no—
54#3802—42no—
55D3×D12—42no—
56D3×D12—42no—
57C2×C32×C8—21no—
58C32×C42C2311yescharacteristic
59#3752—21no—
60C4⋊(C3⋊Dic3)D411yescharacteristic
61C2×C3⋊(C3⋊C8)—21no—
62C3×(C2×D3)⋊C4—21no—
63C3×C4⋊Dic3—21no—
64C3×C4⋊Dic3—21no—
65C3×C4⋊Dic3D411yescharacteristic
66C2×C3×C3⋊C8—21no—
67C3×#204—42no—
68(C2×D3)⋊Dic3—22no—
69#1639—91noSylow
70C2×D3×D4—61no—
71C2×D3×D4—61no—
72C2×D3×D4—61no—
73C22×D12—61no—
74C2×C3×SD16—61no—
75C3×C4⋊Q8—31no—
76C3×#127—31no—
77C3×C4×D4—31no—
78#938—61no—
79#938—61no—
80#937—62no—
81#930—61no—
82C2×#204—21no—
83#895—61no—
84#882—62no—
85#874C2×D311yescharacteristic
86C4×D12—31no—
87C4×D12—31no—
88C4×D12—61no—
89C4×D12C2×D311yescharacteristic
90#869C2×D311yescharacteristic, complemented
91C3×C4⋊C8—21no—
92C3×C4⋊C8—31no—
93C3×C4⋊C8C2×D311yescharacteristic, complemented
94C3×C4⋊C8C2×D311yescharacteristic, complemented
95C3×#103—61no—
96#832—62no—
97#821—61no—
98#818—61no—
99#816—21no—
100#809—61no—
101#809—62no—
102#807—61no—
103C4⋊(C3⋊C8)C2×D311yescharacteristic, complemented
104C4×C3⋊C8—31no—
105C4×C3⋊C8—31no—
106C4×C3⋊C8—61no—
107C22×C3⋊D3—21no—
108C22×C3×D3—21no—
109C22×C3×D3—22no—
110C2×D32—44no—
111C2×D32—44no—
112C2×D32—82no—
113C2×C32×C4C2×D411yescharacteristic
114C2×C32×C4C2×D411yescharacteristic
115C2×C32×C4C2×D411yescharacteristic
116C2×C3⋊Dic3—21no—
117C4⋊(C3⋊D3)—21no—
118C4⋊(C3⋊D3)SD1612yescomplemented
119C4×C3⋊D3—41no—
120C4×C3⋊D3—41no—
121C2×C3×Dic3—21no—
122C2×C3×Dic3—41no—
123C3×D12—21no—
124C3×D12—42no—
125C3×D12—41no—
126C3×D12—41no—
127C3×D12—41no—
128C3×D12—41no—
129C3×D12D812yes—
130C3×C4×D3—41no—
131C3×C4×D3—41no—
132C3×Dic6—41no—
133C3×Dic6—41no—
134#626—44no—
135#625—44no—
136C32×C8—41no—
137C3⋊(C3⋊C8)—41no—
138C3×C3⋊C8—41no—
139D42—91no—
140#486—91no—
141#474—91no—
142#461—31no—
143#331—31no—
144D4⋊C8—91no—
145D4⋊C8—91no—
146C23×D3—62no—
147C23×D3—62no—
148C2×C3×Q8—61no—
149C2×C3×D4—31no—
150C2×C3×D4—32no—
151C2×C3×D4—61no—
152C2×C3×D4—61no—
153C22×C3×C4—61no—
154C2×#67—62no—
155C2×#67—62no—
156C2×#67—62no—
157D3×D4—122no—
158D3×D4—122no—
159D3×D4—122no—
160D3×D4—122no—
161D3×D4—122no—
162D3×D4—122no—
163C2×D12—21no—
164C2×D12—21no—
165C2×D12—31no—
166C2×D12—31no—
167C2×D12—61no—
168C2×D12—62no—
169C2×D12—122no—
170C2×D12—122no—
171C2×D12D1212yes—
172C2×D12#6711yescharacteristic
173C2×C4×D3—21no—
174C2×C4×D3—61no—
175C2×C4×D3—61no—
176C2×C4×D3—121no—
177C2×Dic6—21no—
178C3×SD16—122no—
179C2×C3×C8—21no—
180C2×C3×C8—21no—
181C2×C3×C8—41no—
182C2×C3×C8—61no—
183C3×C4⋊C4—31no—
184C3×C4⋊C4—61no—
185C3×C4⋊C4—61no—
186C3×C22⋊C4—61no—
187C3×C42—21no—
188C3×C42C22×D311yescharacteristic
189C3×C42C22×D311yescharacteristic
190#217—62no—
191(C2×D3)⋊C4—21no—
192(C2×D3)⋊C4—62no—
193(C2×D3)⋊C4—61no—
194(C2×D3)⋊C4—61no—
195(C2×D3)⋊C4—121no—
196C4⋊Dic3—21no—
197C4⋊Dic3—21no—
198C4⋊Dic3—31no—
199C4⋊Dic3—31no—
200C4⋊Dic3—61no—
201C4⋊Dic3#6711yescharacteristic
202C2×C3⋊C8—21no—
203C2×C3⋊C8—61no—
204C2×C3⋊C8—61no—
205C2×C3⋊C8—121no—
206#204—42no—
207C22×C32#12011yescharacteristic
208C2×C3⋊D3—22no—
209C2×C3⋊D3—41no—
210C2×C3⋊D3—41no—
211C2×C3×D3—22no—
212C2×C3×D3—41no—
213C2×C3×D3—41no—
214C2×C3×D3—42no—
215C2×C3×D3—42no—
216C2×C3×D3—42no—
217D32—88no—
218C32×C4—21no—
219C32×C4C2×D811yescharacteristic
220C32×C4C2×SD1611yescharacteristic
221C32×C4AGL(1,ℤ/8)11yescharacteristic
222C32×C4AGL(1,ℤ/8)11yescharacteristic
223C3⋊Dic3—41no—
224C3×Dic3—41no—
225C3×Dic3—81no—
226C22×D4—181no—
227C22×D4—181no—
228C2×SD16—61no—
229C4⋊Q8—31no—
230#127—31no—
231#121—181no—
232#121—181no—
233#120—182no—
234C4×D4—31no—
235C4×D4—91no—
236#107—181no—
237C4⋊C8—31no—
238C4⋊C8D3211yescharacteristic, complemented
239#103—61no—
240#102—181no—
241#102—182no—
242C4×C8—91no—
243C23×C3—61no—
244C23×C3—62no—
245C22×D3—21no—
246C22×D3—22no—
247C22×D3—61no—
248C22×D3—61no—
249C22×D3—121no—
250C22×D3—124no—
251C22×D3—124no—
252C22×D3—124no—
253C22×D3—124no—
254C22×D3—242no—
255C22×D3—242no—
256C3×Q8—121no—
257C3×Q8—121no—
258C3×D4—32no—
259C3×D4—61no—
260C3×D4—122no—
261C3×D4—121no—
262C3×D4—121no—
263C3×D4—121no—
264C3×D4—121no—
265C2×C3×C4—21no—
266C2×C3×C4—21no—
267C2×C3×C4—21no—
268C2×C3×C4—61no—
269C2×C3×C4—121no—
270C2×C3×C4—121no—
271C2×C3×C4—121no—
272C2×C3×C4C2×D1211yescharacteristic
273C2×C3×C4D3×D411yescharacteristic
274C2×C3×C4D3×D411yescharacteristic
275C2×C3×C4D3×D411yescharacteristic
276C2×C3×C4D3×D411yescharacteristic
277C2×C3×C4C2×#6711yescharacteristic
278#67—124no—
279#67—124no—
280#67—124no—
281C2×Dic3—21no—
282C2×Dic3—41no—
283C2×Dic3—61no—
284C2×Dic3—61no—
285C2×Dic3—121no—
286D12—21no—
287D12—32no—
288D12—32no—
289D12—42no—
290D12—41no—
291D12—41no—
292D12—41no—
293D12—41no—
294D12—62no—
295D12—61no—
296D12—61no—
297D12—121no—
298D12—124no—
299D12#21312yes—
300C4×D3—41no—
301C4×D3—41no—
302C4×D3—121no—
303C4×D3—121no—
304C4×D3—121no—
305C4×D3—121no—
306C4×D3—241no—
307C4×D3—241no—
308Dic6—41no—
309Dic6—41no—
310C3×C8—41no—
311C3×C8—41no—
312C3×C8—81no—
313C3×C8—121no—
314C3⋊C8—41no—
315C3⋊C8—121no—
316C3⋊C8—121no—
317C3⋊C8—122no—
318C2×C32#44711yescharacteristic
319C2×C32#45011yescharacteristic
320C2×C32#45511yescharacteristic
321C3⋊D3—42no—
322C3×D3—42no—
323C3×D3—82no—
324C24—182no—
325C2×Q8—61no—
326C2×D4—32no—
327C2×D4—31no—
328C2×D4—61no—
329C2×D4—61no—
330C2×D4—91no—
331C2×D4—182no—
332C2×D4—182no—
333C2×D4—362no—
334C2×D4—362no—
335C2×D4—362no—
336C2×D4—362no—
337C22×C4—61no—
338C22×C4—181no—
339SD16—122no—
340C2×C8—21no—
341C2×C8—61no—
342C2×C8—181no—
343C4⋊C4—31no—
344C4⋊C4—61no—
345C4⋊C4—61no—
346C4⋊C4—91no—
347C22⋊C4—61no—
348C22⋊C4—182no—
349C22⋊C4—181no—
350C42C2×D3211yescharacteristic
351C22×C3—21no—
352C22×C3—62no—
353C22×C3—121no—
354C22×C3—121no—
355C22×C3—122no—
356C22×C3—122no—
357C22×C3—122no—
358C22×C3#88211yescharacteristic
359C22×C3#93711yescharacteristic
360C2×D3—22no—
361C2×D3—41no—
362C2×D3—41no—
363C2×D3—42no—
364C2×D3—42no—
365C2×D3—42no—
366C2×D3—62no—
367C2×D3—62no—
368C2×D3—122no—
369C2×D3—121no—
370C2×D3—121no—
371C2×D3—121no—
372C2×D3—121no—
373C2×D3—241no—
374C2×D3—241no—
375C2×D3—248no—
376C2×D3—248no—
377C3×C4—21no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—21no—
383C3×C4—41no—
384C3×C4—121no—
385C3×C4—241no—
386C3×C4D3×D811yescharacteristic
387C3×C4#91111yescharacteristic
388C3×C4D3×SD1611yescharacteristic
389C3×C4D3×SD1611yescharacteristic
390C3×C4#91411yescharacteristic
391C3×C4#91411yescharacteristic
392C3×C4C2×#21311yescharacteristic
393C3×C4#94911yescharacteristic
394Dic3—41no—
395Dic3—81no—
396Dic3—121no—
397Dic3—121no—
398Dic3—241no—
399C32#163911yescharacteristic, complemented, Sylow
400C23—61no—
401C23—62no—
402C23—181no—
403C23—364no—
404C23—364no—
405C23—722no—
406Q8—121no—
407Q8—121no—
408D4—32no—
409D4—61no—
410D4—92no—
411D4—122no—
412D4—121no—
413D4—121no—
414D4—121no—
415D4—121no—
416D4—181no—
417D4—364no—
418D4—364no—
419C2×C4—61no—
420C2×C4—121no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—181no—
424C2×C4—361no—
425C2×C4—361no—
426C2×C4D3×D1211yescharacteristic
427C2×C4#380211yescharacteristic
428C2×C4D3×#6711yescharacteristic
429C8—41no—
430C8—121no—
431C8—361no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—122no—
436C2×C3—242no—
437C2×C3#474611yescharacteristic
438C2×C3#477311yescharacteristic
439C2×C3#477611yescharacteristic
440C2×C3#503611yescharacteristic
441C2×C3#516911yescharacteristic
442C2×C3#520211yescharacteristic
443D3—42no—
444D3—82no—
445D3—122no—
446D3—122no—
447D3—242no—
448C22—62no—
449C22—121no—
450C22—121no—
451C22—122no—
452C22—122no—
453C22—122no—
454C22—182no—
455C22—361no—
456C22—361no—
457C22—728no—
458C22#6392211yescharacteristic
459C4—21no—
460C4—121no—
461C4—241no—
462C4—361no—
463C4#6379611yescharacteristic
464C4#6379811yescharacteristic
465C4D3×#21311yescharacteristic
466C4#6393911yescharacteristic
467C3—21no—
468C3#6839211yescharacteristic, complemented
469C3#7104411yescharacteristic, complemented
470C2—122no—
471C2—242no—
472C2—362no—
473C2#1059059511yescharacteristic
474C2#1059094311yescharacteristic
475C2#1059101211yescharacteristic
subgroupG/H
centerC22#63922
commutator, cosocleC32×C42C23
FrattiniC42C2×D32
FittingC32×C4⋊C8C22
socleC22×C32#120
prime radical#10594004C2
hypercenterC4⋊C8D32
hypocenterC32#1639
LCSG ▷ C32×C42 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4⋊C8

Complements

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

NH ≅ G/Ntype
2. C3×#4841C2semidirect
3. #10590313C2semidirect
4. #10590277C2semidirect
5. #10589349C2semidirect
6. C3×#4485C2semidirect
7. #10589080C2semidirect
16. C3×#869C22semidirect
20. C32×C4⋊C8C22semidirect
21. C4×C3⋊(C3⋊C8)C22semidirect
23. C3×C4⋊(C3⋊C8)C22semidirect
34. #4841D3semidirect
37. #4485D3semidirect
90. #869C2×D3semidirect
93. C3×C4⋊C8C2×D3semidirect
94. C3×C4⋊C8C2×D3semidirect
103. C4⋊(C3⋊C8)C2×D3semidirect
118. C4⋊(C3⋊D3)SD16semidirect
238. C4⋊C8D32semidirect
399. C32#1639semidirect
468. C3#68392semidirect
469. C3#71044semidirect

Maximal subgroups

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

subgroupindex
#105940042
C3×#48412
#105903132
#105902772
#105893492
C3×#44852
#105890802
#710443
#683923