Finite Group Databasebeta

← #11858674

Subgroups — #11858674

Order 1152.

933 conjugacy classes of subgroups.

The subgroup lattice has 473 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5745C211yesmaximal, characteristic
2#10594776C211yesmaximal, characteristic
3C2×#63815C211yesmaximal, characteristic
4C2×#63649C211yesmaximal, characteristic, complemented
5C3×#5124C211yesmaximal, characteristic, complemented
6C2×#63578C211yesmaximal, characteristic, complemented
7#10591141C211yesmaximal, characteristic, complemented
8#10590973C214yesmaximal, complemented
9D3×#822C214yesmaximal, complemented
10#83810—31nomaximal
11#79260—31nomaximal
12C2×C3×#235C2211yescharacteristic
13C2×C32×M16C2211yescharacteristic
14C2×#3748C2211yescharacteristic
15C22×C3⋊(C3⋊C8)C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C2×C3×#208C2211yescharacteristic
18C2×C3×#208C2211yescharacteristic
19C3×#907—22no—
20C3×D3×M16C2214yes—
21C3×#901—22no—
22C2×#3711C2212yes—
23#63815—22no—
24#63815C414yes—
25#63815C414yes—
26#63814—22no—
27D3×#208C2214yes—
28C2×D3×C3⋊C8C2212yes—
29#63649C2212yes—
30#63649C2212yes—
31C3×#822C2214yes—
32#63578C2212yes—
33#63578C2212yes—
34#63577C2214yes—
35C3×#568—31no—
36#5819—31no—
37C2×#948—31no—
38#5745—31no—
39#5745D311yescharacteristic
40C2×#907—31no—
41C3×#430—31no—
42#5125—31no—
43#5124—31no—
44#5124—34no—
45#5124D311yescharacteristic, complemented
46C2×#822—31no—
47C2×#822—31no—
48C2×#822—31no—
49C2×#822—31no—
50C2×#822—34no—
51C2×#822—61no—
52#4893—34no—
53D3×#108—34no—
54C22×C32×C4C2311yescharacteristic
55C2×C3×#67Q812yes—
56C3×#235C2×C412yes—
57C3×#235C2×C412yes—
58C3×#235D412yes—
59C3×#235D412yes—
60C2×C3×D12Q811yescharacteristic
61C2×C3×C4×D3C2312yes—
62C2×C3×Dic6Q811yescharacteristic
63C32×M16—21no—
64C32×M16C2312yes—
65C2×C32×C8—21no—
66#3748—21no—
67#3748C2312yes—
68C2×C3⋊(C3⋊C8)—21no—
69C2×C3⋊(C3⋊C8)C2×C412yes—
70C2×C3⋊(C3⋊C8)C2×C412yes—
71C2×C3⋊(C3⋊C8)C2311yescharacteristic
72C2×C3⋊(C3⋊C8)C2311yescharacteristic
73C3×#208—21no—
74C3×#208—21no—
75C3×#208C2×C412yes—
76C3×#208C2×C412yes—
77C3×#208C2312yes—
78C3×#208C2312yes—
79C2×C3×C3⋊C8—21no—
80C2×C3×C3⋊C8—21no—
81C2×C3×C3⋊C8C2312yes—
82C3×ΓL(1,𝔽25)—24no—
83C3×D3×C8—24no—
84#3711—24no—
85#3711C2×C418yes—
86D3×C3⋊C8—24no—
87D3×C3⋊C8C2×C418yes—
88#2916—91noSylow
89C2×C3×#41—31no—
90C2×#235C2×D311yescharacteristic
91C3×#131—62no—
92C3×#131—62no—
93C2×C3×M16—21no—
94C2×C3×M16—31no—
95C2×C3×M16—31no—
96C2×C3×M16—31no—
97C2×C3×M16—34no—
98C2×C3×M16C2×D311yescharacteristic
99C2×C3×M16C2×D311yescharacteristic
100#948—34no—
101#948—34no—
102#948—62no—
103#948—62no—
104C2×#208—32no—
105C2×#208—31no—
106C2×#208—31no—
107C2×#208—34no—
108C2×#208—61no—
109C2×#208C2×D311yescharacteristic
110C2×#208C2×D311yescharacteristic
111C2×#208C2×D311yescharacteristic
112C22×C3⋊C8—31no—
113C22×C3⋊C8—31no—
114C22×C3⋊C8—32no—
115C22×C3⋊C8—61no—
116#907—22no—
117#907—34no—
118#907—34no—
119#907—62no—
120D3×M16—34no—
121D3×M16C2×D314yes—
122#901—22no—
123#901—62no—
124C2×ΓL(1,𝔽25)—32no—
125C2×D3×C8—32no—
126C3×#108—34no—
127#822—32no—
128#822—32no—
129#822—32no—
130#822—32no—
131#822—32no—
132#822—32no—
133#822—32no—
134#822—32no—
135#822—34no—
136#822—62no—
137#822—62no—
138#822C2×D314yes—
139#819—34no—
140C23×C32C2×Q811yescharacteristic
141C22×C3×D3C2×Q812yes—
142C2×C32×C4C22×C411yescharacteristic
143C2×C32×C4C22×C411yescharacteristic
144C2×C32×C4C2×D411yescharacteristic
145C2×C32×C4C2×D411yescharacteristic
146C2×C32×C4C2×Q811yescharacteristic
147C2×C32×C4C2411yescharacteristic
148C3×#67C4⋊C414yes—
149C3×#67C4⋊C414yes—
150C2×C3×Dic3C2×Q812yes—
151C3×D12C4⋊C412yes—
152C3×D12C4⋊C412yes—
153C3×C4×D3C22×C414yes—
154C3×C4×D3C2×D414yes—
155C3×Dic6C4⋊C412yes—
156C3×Dic6C4⋊C412yes—
157C32×C8—22no—
158C3⋊(C3⋊C8)—22no—
159C3⋊(C3⋊C8)C22×C412yes—
160C3⋊(C3⋊C8)C22×C412yes—
161C3×C3⋊C8—22no—
162C3×C3⋊C8—22no—
163C3×C3⋊C8C22×C414yes—
164#568—31no—
165#568—91no—
166C2×#131—91no—
167#430—31no—
168#430—94no—
169#430—91no—
170C2×#108—94no—
171C2×#108—91no—
172C2×#108—91no—
173C3×#41—32no—
174C3×#41—32no—
175C3×#41—32no—
176C3×#41—32no—
177C2×C3×Q8—31no—
178C2×C3×D4—32no—
179C2×C3×D4—31no—
180C22×C3×C4—21no—
181C22×C3×C4—32no—
182C22×C3×C4C22×D311yescharacteristic
183C22×C3×C4C22×D311yescharacteristic
184C2×#67Dic612yes—
185#235C4×D312yes—
186#235C4×D312yes—
187#235#6712yes—
188#235#6712yes—
189C2×D12Dic611yescharacteristic
190C2×C4×D3C22×D312yes—
191C2×Dic6Dic611yescharacteristic
192C3×M16—21no—
193C3×M16—21no—
194C3×M16—22no—
195C3×M16—32no—
196C3×M16—32no—
197C3×M16—32no—
198C3×M16—32no—
199C3×M16—41no—
200C3×M16—64no—
201C3×M16—61no—
202C3×M16—61no—
203C3×M16C22×D312yes—
204C3×M16C22×D312yes—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—32no—
208C2×C3×C8—41no—
209C2×C3×C8—64no—
210C2×C3×C8—61no—
211C2×C3×C8—61no—
212#208—21no—
213#208—21no—
214#208—38no—
215#208—32no—
216#208—32no—
217#208—62no—
218#208—64no—
219#208—61no—
220#208—61no—
221#208—121no—
222#208C4×D312yes—
223#208C4×D312yes—
224#208C22×D312yes—
225#208C22×D312yes—
226C2×C3⋊C8—21no—
227C2×C3⋊C8—21no—
228C2×C3⋊C8—32no—
229C2×C3⋊C8—32no—
230C2×C3⋊C8—32no—
231C2×C3⋊C8—32no—
232C2×C3⋊C8—31no—
233C2×C3⋊C8—31no—
234C2×C3⋊C8—31no—
235C2×C3⋊C8—31no—
236C2×C3⋊C8—38no—
237C2×C3⋊C8—61no—
238C2×C3⋊C8—61no—
239C2×C3⋊C8—62no—
240C2×C3⋊C8—62no—
241C2×C3⋊C8—64no—
242C2×C3⋊C8—61no—
243C2×C3⋊C8—61no—
244C2×C3⋊C8—121no—
245C2×C3⋊C8C22×D312yes—
246ΓL(1,𝔽25)—24no—
247ΓL(1,𝔽25)—38no—
248ΓL(1,𝔽25)—64no—
249D3×C8—24no—
250D3×C8—38no—
251D3×C8—64no—
252C22×C32—41no—
253C22×C32C2×C4⋊C411yescharacteristic
254C22×C32C2×C4⋊C411yescharacteristic
255C22×C32C22×Q811yescharacteristic
256C2×C3×D3—42no—
257C2×C3×D3C2×C4⋊C414yes—
258C32×C4C2×C4⋊C411yescharacteristic
259C32×C4C2×C4⋊C411yescharacteristic
260C32×C4C23×C411yescharacteristic
261C32×C4C22×D411yescharacteristic
262C3×Dic3C2×C4⋊C414yes—
263C2×#41—31no—
264#131—62no—
265#131—62no—
266#131—94no—
267#131—94no—
268#131—182no—
269#131—182no—
270C2×M16—31no—
271C2×M16—31no—
272C2×M16—31no—
273C2×M16—34no—
274C2×M16—92no—
275C2×M16—94no—
276C2×M16—91no—
277C2×M16D3211yescharacteristic
278C22×C8—92no—
279C22×C8—91no—
280#108—34no—
281#108—94no—
282#108—92no—
283#108—92no—
284#108—92no—
285#108—92no—
286C23×C3—21no—
287C23×C3—32no—
288C23×C3C2×Dic611yescharacteristic
289C23×C3D3×Q811yescharacteristic
290C22×D3C2×Dic612yes—
291C3×Q8—32no—
292C3×Q8—32no—
293C3×D4—34no—
294C3×D4—34no—
295C3×D4—32no—
296C3×D4—32no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—32no—
304C2×C3×C4—34no—
305C2×C3×C4—34no—
306C2×C3×C4C2×Dic611yescharacteristic
307C2×C3×C4C2×C4×D311yescharacteristic
308C2×C3×C4C2×C4×D311yescharacteristic
309C2×C3×C4C2×C4×D311yescharacteristic
310C2×C3×C4C2×C4×D311yescharacteristic
311C2×C3×C4D3×D411yescharacteristic
312C2×C3×C4D3×D411yescharacteristic
313C2×C3×C4D3×Q811yescharacteristic
314C2×C3×C4C2×#6711yescharacteristic
315C2×C3×C4C2×#6711yescharacteristic
316C2×C3×C4C23×D311yescharacteristic
317C2×C3×C4C23×D311yescharacteristic
318#67Dic3⋊C414yes—
319#67Dic3⋊C414yes—
320C2×Dic3C2×Dic612yes—
321D12Dic3⋊C412yes—
322D12Dic3⋊C412yes—
323C4×D3C2×C4×D314yes—
324C4×D3C2×#6714yes—
325Dic6Dic3⋊C412yes—
326Dic6Dic3⋊C412yes—
327C3×C8—22no—
328C3×C8—22no—
329C3×C8—34no—
330C3×C8—42no—
331C3×C8—62no—
332C3×C8—62no—
333C3⋊C8—22no—
334C3⋊C8—22no—
335C3⋊C8—32no—
336C3⋊C8—32no—
337C3⋊C8—32no—
338C3⋊C8—32no—
339C3⋊C8—62no—
340C3⋊C8—62no—
341C3⋊C8—62no—
342C3⋊C8—62no—
343C3⋊C8—122no—
344C3⋊C8C2×C4×D314yes—
345C2×C32—21no—
346C2×C32—21no—
347C2×C32—21no—
348C2×C32C22×C4⋊C411yescharacteristic
349C3×D3—24no—
350#41—32no—
351#41—32no—
352#41—32no—
353#41—32no—
354C2×Q8—31no—
355C2×D4—32no—
356C2×D4—31no—
357C22×C4—32no—
358C22×C4C2×D3211yescharacteristic
359M16—21no—
360M16—32no—
361M16—32no—
362M16—32no—
363M16—32no—
364M16—64no—
365M16—61no—
366M16—61no—
367M16—98no—
368M16—92no—
369M16—184no—
370M16—181no—
371M16C2×D3212yes—
372C2×C8—21no—
373C2×C8—32no—
374C2×C8—64no—
375C2×C8—61no—
376C2×C8—61no—
377C2×C8—98no—
378C2×C8—91no—
379C2×C8—91no—
380C2×C8—92no—
381C2×C8—92no—
382C2×C8—184no—
383C2×C8—181no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—34no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—42no—
391C22×C3—122no—
392C22×C3D3×C4⋊C411yescharacteristic
393C22×C3D3×C4⋊C411yescharacteristic
394C22×C3C2×Dic3⋊C411yescharacteristic
395C22×C3C2×Dic3⋊C411yescharacteristic
396C22×C3C22×Dic611yescharacteristic
397C22×C3C2×D3×Q811yescharacteristic
398C2×D3—42no—
399C2×D3C2×Dic3⋊C414yes—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—21no—
404C3×C4—34no—
405C3×C4D3×C4⋊C411yescharacteristic
406C3×C4D3×C4⋊C411yescharacteristic
407C3×C4C2×Dic3⋊C411yescharacteristic
408C3×C4C2×Dic3⋊C411yescharacteristic
409C3×C4C22×C4×D311yescharacteristic
410C3×C4C22×C4×D311yescharacteristic
411C3×C4C2×D3×D411yescharacteristic
412C3×C4C22×#6711yescharacteristic
413Dic3C2×Dic3⋊C414yes—
414C32#291611yescharacteristic, complemented, Sylow
415C23—32no—
416C23D3×Dic611yescharacteristic
417Q8—32no—
418Q8—32no—
419D4—34no—
420D4—34no—
421D4—32no—
422D4—32no—
423C2×C4—32no—
424C2×C4—34no—
425C2×C4—34no—
426C2×C4D3×Dic611yescharacteristic
427C2×C4C4×D3211yescharacteristic
428C2×C4C4×D3211yescharacteristic
429C2×C4D3×#6711yescharacteristic
430C2×C4D3×#6711yescharacteristic
431C2×C4C22×D3211yescharacteristic
432C8—22no—
433C8—34no—
434C8—62no—
435C8—62no—
436C8—92no—
437C8—92no—
438C8—182no—
439C2×C3—22no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—41no—
448C2×C3—41no—
449C2×C3—64no—
450C2×C3C2×D3×C4⋊C411yescharacteristic
451C2×C3C22×Dic3⋊C411yescharacteristic
452D3—24no—
453C22—34no—
454C22—41no—
455C22—122no—
456C22D3×Dic3⋊C411yescharacteristic
457C22D3×Dic3⋊C411yescharacteristic
458C22C2×D3×Dic611yescharacteristic
459C4—34no—
460C4D3×Dic3⋊C411yescharacteristic
461C4D3×Dic3⋊C411yescharacteristic
462C4C2×C4×D3211yescharacteristic
463C4C2×D3×#6711yescharacteristic
464C3—21no—
465C3#7926011yescharacteristic, complemented
466C3#8381011yescharacteristic, complemented
467C2—21no—
468C2—21no—
469C2—21no—
470C2—64no—
471C2C2×D3×Dic3⋊C411yescharacteristic
subgroupG/H
centerC4C2×D3×#67
commutatorC32×C4C23×C4
FrattiniC2×C4C22×D32
FittingC2×C32×M16C22
socleC2×C32C22×C4⋊C4
prime radicalC2×C3×#235C22
cosocleC2×C32×C4C24
hypercenterC2×M16D32
hypocenterC32#2916
LCSG ▷ C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. C2×#63649C2semidirect
5. C3×#5124C2semidirect
6. C2×#63578C2semidirect
7. #10591141C2semidirect
8. #10590973C2semidirect
9. D3×#822C2semidirect
45. #5124D3semidirect
414. C32#2916semidirect
465. C3#79260semidirect
466. C3#83810semidirect

Maximal subgroups

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

subgroupindex
C3×#57452
#105947762
C2×#638152
C2×#636492
C3×#51242
C2×#635782
#105911412
#105909732
D3×#8222
#838103
#792603