Finite Group Databasebeta

← #11878709

Subgroups — #11878709

Order 1152.

905 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10593897C211yesmaximal, characteristic
2D3×#869C211yesmaximal, characteristic, complemented
3#10592380C211yesmaximal, characteristic, complemented
4C3×#4693C211yesmaximal, characteristic, complemented
5C3×#4685C211yesmaximal, characteristic, complemented
6#10590822C214yesmaximal, complemented
7#10590752C214yesmaximal, complemented
8#10590369C211yesmaximal, characteristic, complemented
9#10590287C211yesmaximal, characteristic, complemented
10#81764—31nomaximal
11#79312—31nomaximal
12C2×#3795—21no—
13C2×D3×Dic6C2212yes—
14#64078C2211yescharacteristic
15#64078C2211yescharacteristic
16C2×C3×#204C2214yes—
17C3×#874C2211yescharacteristic
18C3×C42×D3C2211yescharacteristic
19C3×#869C2211yescharacteristic
20#63918—21no—
21#63918—21no—
22#63908—21no—
23#63908C2212yes—
24Dic3×D12—21no—
25Dic3×D12C2212yes—
26D3×C4⋊Dic3—21no—
27D3×C4⋊Dic3—21no—
28#63866C2212yes—
29#63860—21no—
30C2×#3716C2214yes—
31C32×C4×C8C2211yescharacteristic
32#63643C2214yes—
33C3×(C2×D3)⋊C8C2212yes—
34C3×Dic3⋊C8C2212yes—
35C3×C4×C3⋊C8C2211yescharacteristic
36#63566C2214yes—
37D3×C4⋊Q8—31no—
38#5716—31no—
39#5477—31no—
40C2×#869—31no—
41C3×#492—31no—
42C3×#431—31no—
43#5170—34no—
44#5162—34no—
45#5079—31no—
46#5076—31no—
47#4841—34no—
48#4738—34no—
49#4693—31no—
50#4693D311yescharacteristic, complemented
51#4685D311yescharacteristic, complemented
52#4681—31no—
53#4681—31no—
54#4681—31no—
55#4681—61no—
56C2×#634—21no—
57C2×#634C2312yes—
58C2×C3×D12—21no—
59C2×C3×D12—21no—
60C2×C3×D12C2312yes—
61C2×C3×C4×D3D411yescharacteristic
62C2×C3×C4×D3D411yescharacteristic
63C2×C3×C4×D3C2311yescharacteristic
64C2×C3×Dic6C2312yes—
65C2×#627—22no—
66C2×#625—22no—
67C2×D3×Dic3—22no—
68C2×D3×Dic3—22no—
69#3795—44no—
70D3×Dic6—44no—
71C2×C32×C8C2312yes—
72C32×C42C2311yescharacteristic
73C4⋊(C3⋊Dic3)—21no—
74C4⋊(C3⋊Dic3)—21no—
75C4⋊(C3⋊Dic3)—21no—
76C4⋊(C3⋊Dic3)C2312yes—
77C3×C4⋊Dic3—21no—
78C3×C4⋊Dic3—21no—
79C3×C4×Dic3D411yescharacteristic
80C3×C4×Dic3D411yescharacteristic
81C3×C4×Dic3C2311yescharacteristic
82C2×C3×C3⋊C8C2312yes—
83C3×#204—44no—
84Dic3⋊Dic3—22no—
85Dic3⋊Dic3—22no—
86(C2×D3)⋊Dic3—22no—
87(C2×D3)⋊Dic3—22no—
88#3716—44no—
89#3206—91noSylow
90C2×D3×Q8—32no—
91C2×#237—61no—
92C2×#235—61no—
93C22×Dic6—32no—
94C2×C3×SD16—34no—
95C3×C4⋊Q8—31no—
96C3×#127—31no—
97C2×C3×C42—31no—
98#944—32no—
99#938—32no—
100#938—61no—
101#936—61no—
102D4×Dic3—32no—
103D4×Dic3—61no—
104C2×#214—34no—
105#930—32no—
106#930—61no—
107C2×C4⋊Dic3—61no—
108C2×C4⋊Dic3—61no—
109C2×#204—34no—
110C2×#204C2×D314yes—
111D3×C4⋊C4—61no—
112D3×C4⋊C4—61no—
113#888—61no—
114#888—61no—
115#875—61no—
116#874C2×D311yescharacteristic
117C4×D12—32no—
118C4×D12—61no—
119C42×D3C2×D311yescharacteristic
120#869—31no—
121#869—31no—
122#869—32no—
123#869—31no—
124#869—31no—
125#869—61no—
126#869—61no—
127#869—61no—
128#869—61no—
129#869C2×D311yescharacteristic
130C3×C4⋊C8—32no—
131C3×C22⋊C8—32no—
132C3×C4×C8—21no—
133C3×C4×C8—31no—
134C3×C4×C8C2×D311yescharacteristic
135C3×C4×C8C2×D311yescharacteristic
136#835—34no—
137(C2×D3)⋊C8C2×D312yes—
138#816—34no—
139#816—34no—
140#816—34no—
141#816—64no—
142Dic3⋊C8C2×D312yes—
143C4×C3⋊C8C2×D311yescharacteristic
144C22×C3×D3—22no—
145C22×C3×D3C2×D411yescharacteristic
146C2×C32×C4C2×D411yescharacteristic
147C2×C32×C4C2×D411yescharacteristic
148C2×C32×C4C2411yescharacteristic
149C2×C3⋊Dic3—22no—
150C2×C3⋊Dic3—22no—
151#634—24no—
152#634—42no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3C2×D411yescharacteristic
155C2×C3×Dic3C2×D411yescharacteristic
156C2×C3×Dic3C2×D411yescharacteristic
157C3×D12—24no—
158C3×D12—42no—
159C3×D12—42no—
160C3×C4×D3—22no—
161C3×C4×D3—22no—
162C3×C4×D3SD1614yescomplemented
163C3×Dic6—24no—
164#627—44no—
165#625—44no—
166D3×Dic3—44no—
167D3×Dic3—44no—
168C32×C8—22no—
169C3×C3⋊C8—22no—
170#532—91no—
171C2×C4⋊Q8—91no—
172#492—31no—
173#492—91no—
174#487—91no—
175#487—91no—
176#461—94no—
177#448—94no—
178#431—31no—
179C2×C3×Q8—32no—
180C2×C3×D4—32no—
181C2×C3×D4—61no—
182C2×C3×D4—61no—
183C22×C3×C4—31no—
184C22×C3×C4—31no—
185C22×C3×C4—31no—
186C2×#67—62no—
187C2×#67—62no—
188C22×Dic3—62no—
189C22×Dic3—62no—
190D3×Q8—124no—
191#237—124no—
192#235—124no—
193C2×D12—21no—
194C2×D12—21no—
195C2×D12C22×D312yes—
196C2×C4×D3—62no—
197C2×C4×D3—62no—
198C2×C4×D3D1211yescharacteristic
199C2×C4×D3D1211yescharacteristic
200C2×C4×D3C22×D311yescharacteristic
201C2×Dic6—32no—
202C2×Dic6—32no—
203C2×Dic6—62no—
204C2×Dic6—62no—
205C2×Dic6—61no—
206C2×Dic6—61no—
207C2×Dic6—62no—
208C2×Dic6—121no—
209C2×Dic6—124no—
210C2×Dic6C22×D312yes—
211C3×SD16—124no—
212C2×C3×C8—22no—
213C2×C3×C8—32no—
214C2×C3×C8C22×D312yes—
215C2×C3×C8C22×D312yes—
216C3×C4⋊C4—61no—
217C3×C4⋊C4—61no—
218C3×C42—21no—
219C3×C42—31no—
220C3×C42—31no—
221C3×C42—31no—
222C3×C42C22×D311yescharacteristic
223C3×C42C22×D311yescharacteristic
224#217—62no—
225#217—62no—
226#214—124no—
227(C2×D3)⋊C4—62no—
228(C2×D3)⋊C4—62no—
229C4⋊Dic3—21no—
230C4⋊Dic3—21no—
231C4⋊Dic3—32no—
232C4⋊Dic3—32no—
233C4⋊Dic3—62no—
234C4⋊Dic3—62no—
235C4⋊Dic3—62no—
236C4⋊Dic3—61no—
237C4⋊Dic3—61no—
238C4⋊Dic3—61no—
239C4⋊Dic3—61no—
240C4⋊Dic3—61no—
241C4⋊Dic3—61no—
242C4⋊Dic3—121no—
243C4⋊Dic3—121no—
244C4⋊Dic3—121no—
245Dic3⋊C4—62no—
246Dic3⋊C4—62no—
247C4×Dic3D1211yescharacteristic
248C4×Dic3D1211yescharacteristic
249C4×Dic3C22×D311yescharacteristic
250C2×C3⋊C8C22×D312yes—
251#204—44no—
252#204—124no—
253C22×C32C22×D411yescharacteristic
254C2×C3×D3—21no—
255C2×C3×D3—21no—
256C2×C3×D3—42no—
257C2×C3×D3—42no—
258C2×C3×D3—42no—
259C2×C3×D3C2×SD1612yes—
260C32×C4C2×SD1612yes—
261C32×C4#13712yes—
262C32×C4H(5,𝔽2)12yes—
263C3⋊Dic3—42no—
264C3⋊Dic3—42no—
265C3×Dic3—21no—
266C3×Dic3—21no—
267C3×Dic3—42no—
268C3×Dic3C2×SD1612yes—
269C2×#41—181no—
270C22×Q8—92no—
271C2×SD16—34no—
272C2×SD16—94no—
273C4⋊Q8—31no—
274C4⋊Q8—92no—
275C4⋊Q8—91no—
276C4⋊Q8—91no—
277C4⋊Q8—181no—
278C4⋊Q8—181no—
279#127—31no—
280#124—181no—
281#121—92no—
282#121—181no—
283C4×D4—92no—
284C4×D4—181no—
285C2×C4⋊C4—181no—
286C2×C4⋊C4—181no—
287C2×C42—31no—
288C4⋊C8—32no—
289#103—94no—
290#103—94no—
291C22⋊C8—32no—
292C4×C8—31no—
293C4×C8D3211yescharacteristic
294C23×C3—31no—
295C23×C3—62no—
296C22×D3—22no—
297C22×D3C2×D1211yescharacteristic
298C3×Q8—64no—
299C3×D4—64no—
300C3×D4—122no—
301C3×D4—122no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—31no—
306C2×C3×C4—34no—
307C2×C3×C4—31no—
308C2×C3×C4—31no—
309C2×C3×C4—62no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4C2×D1211yescharacteristic
313C2×C3×C4C2×D1211yescharacteristic
314C2×C3×C4D3×D411yescharacteristic
315C2×C3×C4D3×D411yescharacteristic
316C2×C3×C4C23×D311yescharacteristic
317C2×C3×C4C23×D311yescharacteristic
318#67—124no—
319#67—124no—
320C2×Dic3—22no—
321C2×Dic3—62no—
322C2×Dic3—62no—
323C2×Dic3—62no—
324C2×Dic3—62no—
325C2×Dic3—122no—
326C2×Dic3—122no—
327C2×Dic3—124no—
328C2×Dic3—124no—
329C2×Dic3C2×D1211yescharacteristic
330C2×Dic3C2×D1211yescharacteristic
331C2×Dic3C2×D1211yescharacteristic
332D12—24no—
333D12—42no—
334D12—42no—
335C4×D3—22no—
336C4×D3—22no—
337C4×D3—124no—
338C4×D3—124no—
339C4×D3#20414yescomplemented
340Dic6—24no—
341Dic6—64no—
342Dic6—64no—
343Dic6—124no—
344Dic6—124no—
345Dic6—124no—
346Dic6—122no—
347Dic6—122no—
348Dic6—242no—
349C3×C8—22no—
350C3×C8—22no—
351C3×C8—42no—
352C3×C8—62no—
353C3⋊C8—22no—
354C2×C32#53511yescharacteristic
355C2×C32C22×SD1611yescharacteristic
356C2×C32C2×#13711yescharacteristic
357C3×D3—22no—
358C3×D3—82no—
359#41—364no—
360C2×Q8—32no—
361C2×Q8—92no—
362C2×Q8—182no—
363C2×Q8—181no—
364C2×Q8—364no—
365C2×D4—32no—
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—182no—
369C22×C4—31no—
370C22×C4—31no—
371C22×C4—31no—
372C22×C4—182no—
373C22×C4—182no—
374SD16—124no—
375SD16—364no—
376C2×C8—32no—
377C2×C8C2×D3212yes—
378C4⋊C4—61no—
379C4⋊C4—61no—
380C4⋊C4—92no—
381C4⋊C4—182no—
382C4⋊C4—182no—
383C4⋊C4—181no—
384C4⋊C4—181no—
385C4⋊C4—181no—
386C22⋊C4—182no—
387C22⋊C4—182no—
388C42—31no—
389C42—31no—
390C42—31no—
391C42C2×D3211yescharacteristic
392C22×C3—21no—
393C22×C3—32no—
394C22×C3—61no—
395C22×C3—61no—
396C22×C3—122no—
397C22×C3—122no—
398C22×C3—122no—
399C22×C3C22×D1211yescharacteristic
400C22×C3C2×D3×D411yescharacteristic
401C2×D3—21no—
402C2×D3—21no—
403C2×D3—42no—
404C2×D3—42no—
405C2×D3—42no—
406C2×D3C2×#20412yes—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—32no—
411C3×C4—61no—
412C3×C4—61no—
413C3×C4—122no—
414C3×C4C2×#20412yes—
415C3×C4#90912yes—
416C3×C4D3×SD1612yes—
417C3×C4#91512yes—
418C3×C4#100412yes—
419C3×C4#100912yes—
420Dic3—21no—
421Dic3—21no—
422Dic3—42no—
423Dic3—122no—
424Dic3—122no—
425Dic3—122no—
426Dic3—122no—
427Dic3—242no—
428Dic3—242no—
429Dic3C2×#20412yes—
430C32#320611yescharacteristic, complemented, Sylow
431C23—31no—
432C23—62no—
433Q8—64no—
434Q8—184no—
435Q8—364no—
436Q8—362no—
437D4—64no—
438D4—122no—
439D4—122no—
440D4—364no—
441C2×C4—31no—
442C2×C4—34no—
443C2×C4—31no—
444C2×C4—31no—
445C2×C4—62no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—182no—
449C2×C4—182no—
450C2×C4—364no—
451C2×C4—364no—
452C2×C4D3×D1211yescharacteristic
453C2×C4D3×D1211yescharacteristic
454C2×C4C22×D3211yescharacteristic
455C8—22no—
456C8—62no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—62no—
461C2×C3—242no—
462C2×C3#552511yescharacteristic
463C2×C3#571511yescharacteristic
464C2×C3C22×#20411yescharacteristic
465C2×C3C2×#90911yescharacteristic
466C2×C3C2×D3×SD1611yescharacteristic
467C2×C3C2×#91511yescharacteristic
468D3—22no—
469D3—82no—
470C22—32no—
471C22—61no—
472C22—61no—
473C22—122no—
474C22—122no—
475C22—122no—
476C22C2×D3×D1211yescharacteristic
477C4—32no—
478C4—61no—
479C4—61no—
480C4—122no—
481C4—362no—
482C4—362no—
483C4D3×#20412yes—
484C4#6380012yes—
485C4#6430712yes—
486C3—21no—
487C3#7931211yescharacteristic, complemented
488C3#8176411yescharacteristic, complemented
489C2—62no—
490C2—242no—
491C2#1059400011yescharacteristic
492C2C2×D3×#20411yescharacteristic
493C2C2×#6380011yescharacteristic
subgroupG/H
centerC22C2×D3×D12
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×C8C22
socleC22×C32C22×D4
prime radical#10593897C2
hypercenterC4×C8D32
hypocenterC32#3206
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
2. D3×#869C2semidirect
3. #10592380C2semidirect
4. C3×#4693C2semidirect
5. C3×#4685C2semidirect
6. #10590822C2semidirect
7. #10590752C2semidirect
8. #10590369C2semidirect
9. #10590287C2semidirect
50. #4693D3semidirect
51. #4685D3semidirect
162. C3×C4×D3SD16semidirect
339. C4×D3#204semidirect
430. C32#3206semidirect
487. C3#79312semidirect
488. C3#81764semidirect

Maximal subgroups

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

subgroupindex
#105938972
D3×#8692
#105923802
C3×#46932
C3×#46852
#105908222
#105907522
#105903692
#105902872
#817643
#793123