Finite Group Databasebeta

← #11861518

Subgroups — #11861518

Order 1152.

827 conjugacy classes of subgroups.

The subgroup lattice has 514 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×D3×Dic6C211yesmaximal, characteristic
2C8×#634C211yesmaximal, characteristic, complemented
3C3×C4×ΓL(1,𝔽25)C211yesmaximal, characteristic
4C3×#4679C211yesmaximal, characteristic, complemented
5#10590789C212yesmaximal, complemented
6#10590787C212yesmaximal, complemented
7#10590667C212yesmaximal
8#10590663C212yesmaximal
9#10590336C211yesmaximal, characteristic
10#10590258C211yesmaximal, characteristic, complemented
11#10590254C211yesmaximal, characteristic, complemented
12#79962—31nomaximal
13M16×Dic6—31nomaximal
14C2×D3×Dic6C411yescharacteristic
15C4×#634C2211yescharacteristic
16C2×C3×ΓL(1,𝔽25)C2211yescharacteristic
17C2×C3×ΓL(1,𝔽25)C2211yescharacteristic
18C3×C42×D3C2211yescharacteristic
19C3×C4×Dic6C2211yescharacteristic
20C4×#627C2212yes—
21D3×C4⋊Dic3C411yescharacteristic
22D3×Dic3⋊C4C412yes—
23C4×D3×Dic3C2212yes—
24#63844C411yescharacteristic
25Dic3×Dic6C411yescharacteristic
26#63837C412yes—
27C2×#3711C2212yes—
28C32×C4×C8C2211yescharacteristic
29#63641C2212yes—
30C8×C3⋊Dic3C2212yes—
31C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
32C3×#815C2212yes—
33C3×#815C2211yescharacteristic
34C3×#815C2211yescharacteristic
35C3×Dic3⋊C8C2212yes—
36C3×C4⋊(C3⋊C8)C2211yescharacteristic
37C3×C4×C3⋊C8C2211yescharacteristic
38#63572C2212yes—
39Dic3⋊(C3⋊C8)C2212yes—
40Dic3⋊(C3⋊C8)C2212yes—
41#63554C2212yes—
42#63553C2212yes—
43C4×D3×Q8—31no—
44C2×C4×Dic6—31no—
45C3×#446—31no—
46C3×C4×M16—31no—
47#5118—32no—
48Dic3×M16—32no—
49#5024—31no—
50#4924—31no—
51#4837—32no—
52#4837—31no—
53#4830—31no—
54#4828—32no—
55#4707—32no—
56#4701—32no—
57#4701—32no—
58#4701—32no—
59C4×ΓL(1,𝔽25)D311yescharacteristic
60#4679—31no—
61#4679D311yescharacteristic, complemented
62C8×Dic6—31no—
63C8×Dic6—31no—
64C8×Dic6—31no—
65C8×Dic6—61no—
66C2×#634C2×C411yescharacteristic
67C2×C3×C4×D3C2×C411yescharacteristic
68C2×C3×C4×D3C2×C411yescharacteristic
69C2×C3×C4×D3C2311yescharacteristic
70C2×C3×Dic6C2×C411yescharacteristic
71C2×#627C2×C412yes—
72C2×D3×Dic3C2×C412yes—
73C2×D3×Dic3C2×C412yes—
74D3×Dic6—24no—
75C2×C32×C8C2311yescharacteristic
76C2×C32×C8C2311yescharacteristic
77C32×C42C2311yescharacteristic
78C4⋊(C3⋊Dic3)C2×C411yescharacteristic
79#3750C2×C412yes—
80C4×C3⋊Dic3C2312yes—
81C2×C3⋊(C3⋊C8)C2312yes—
82C3×C4⋊Dic3C2×C411yescharacteristic
83C3×Dic3⋊C4C2×C412yes—
84C3×C4×Dic3C2×C411yescharacteristic
85C3×C4×Dic3C2×C411yescharacteristic
86C3×C4×Dic3C2312yes—
87C3×C4×Dic3C2311yescharacteristic
88C2×C3×C3⋊C8C2311yescharacteristic
89C2×C3×C3⋊C8C2312yes—
90C2×C3×C3⋊C8C2311yescharacteristic
91C3×ΓL(1,𝔽25)—22no—
92C3×ΓL(1,𝔽25)Q814yescomplemented
93#3724C2×C412yes—
94Dic3⋊Dic3C2×C412yes—
95Dic3⋊Dic3C2×C412yes—
96Dic32C2×C412yes—
97Dic32C2×C412yes—
98#3711—24no—
99Q8×M16—91noSylow
100C2×D3×Q8—31no—
101C22×Dic6—31no—
102C2×C3×M16—31no—
103C2×C3×M16—31no—
104C3×C4×Q8—31no—
105C2×C3×C42—31no—
106Q8×Dic3—31no—
107C2×C4⋊Dic3—31no—
108C2×Dic3⋊C4—32no—
109C2×C4×Dic3—32no—
110C2×#208—32no—
111C2×ΓL(1,𝔽25)—32no—
112C2×ΓL(1,𝔽25)C2×D311yescharacteristic
113C2×ΓL(1,𝔽25)C2×D311yescharacteristic
114D3×C4⋊C4—32no—
115D3×C4⋊C4—31no—
116#887—32no—
117#887—31no—
118C42×D3—32no—
119C42×D3C2×D311yescharacteristic
120C4×Dic6—31no—
121C4×Dic6—31no—
122C4×Dic6—31no—
123C4×Dic6—31no—
124C4×Dic6—32no—
125C4×Dic6—32no—
126C4×Dic6—32no—
127C4×Dic6—61no—
128C4×Dic6C2×D311yescharacteristic
129C3×C4⋊C8—32no—
130C3×C4⋊C8—31no—
131C3×#97—31no—
132C3×#97—31no—
133C3×#97—32no—
134C3×C4×C8—21no—
135C3×C4×C8—31no—
136C3×C4×C8C2×D311yescharacteristic
137C3×C4×C8C2×D311yescharacteristic
138#815—32no—
139#815C2×D312yes—
140#815C2×D311yescharacteristic
141#815C2×D311yescharacteristic
142Dic3⋊C8—32no—
143Dic3⋊C8—32no—
144Dic3⋊C8—32no—
145Dic3⋊C8—32no—
146Dic3⋊C8—32no—
147Dic3⋊C8—32no—
148Dic3⋊C8—62no—
149Dic3⋊C8C2×D312yes—
150C8×Dic3—32no—
151C8×Dic3—32no—
152C8×Dic3—32no—
153C8×Dic3—32no—
154C8×Dic3—62no—
155C4⋊(C3⋊C8)—31no—
156C4⋊(C3⋊C8)—31no—
157C4⋊(C3⋊C8)—32no—
158C4⋊(C3⋊C8)—32no—
159C4⋊(C3⋊C8)—61no—
160C4⋊(C3⋊C8)C2×D311yescharacteristic
161#803—32no—
162C4×C3⋊C8C2×D311yescharacteristic
163C22×C3×D3C22×C411yescharacteristic
164C2×C32×C4C22×C411yescharacteristic
165C2×C32×C4C22×C411yescharacteristic
166C2×C32×C4C2411yescharacteristic
167C2×C3⋊Dic3C22×C412yes—
168C2×C3⋊Dic3C22×C412yes—
169#634M1614yescomplemented
170C2×C3×Dic3C22×C412yes—
171C2×C3×Dic3C22×C412yes—
172C2×C3×Dic3C22×C411yescharacteristic
173C2×C3×Dic3C22×C411yescharacteristic
174C2×C3×Dic3C22×C411yescharacteristic
175C3×C4×D3—22no—
176C3×C4×D3—41no—
177C3×C4×D3C2×Q812yes—
178C3×C4×D3#4112yes—
179C3×Dic6—22no—
180#627—24no—
181D3×Dic3—24no—
182D3×Dic3—42no—
183C32×C8—21no—
184C32×C8C2×Q812yescomplemented
185C3⋊(C3⋊C8)—22no—
186C3×C3⋊C8—22no—
187C3×C3⋊C8—21no—
188C3×C3⋊C8C2×Q812yescomplemented
189C2×C4×Q8—91no—
190#446—31no—
191#446—91no—
192#446—92no—
193#446—92no—
194C8×Q8—91no—
195C8×Q8—91no—
196#423—91no—
197#423—92no—
198C4×M16—31no—
199C4×M16—92no—
200C2×C3×Q8—31no—
201C22×C3×C4—31no—
202C22×C3×C4—31no—
203C22×C3×C4—31no—
204C22×Dic3—32no—
205C22×Dic3—32no—
206D3×Q8—64no—
207C2×C4×D3—32no—
208C2×C4×D3—32no—
209C2×C4×D3C4×D311yescharacteristic
210C2×C4×D3C4×D311yescharacteristic
211C2×C4×D3C22×D311yescharacteristic
212C2×Dic6—31no—
213C2×Dic6—31no—
214C2×Dic6—32no—
215C2×Dic6—32no—
216C2×Dic6—61no—
217C2×Dic6—64no—
218C2×Dic6C4×D311yescharacteristic
219C3×M16—34no—
220C3×M16—62no—
221C2×C3×C8—21no—
222C2×C3×C8—21no—
223C2×C3×C8—31no—
224C2×C3×C8—32no—
225C2×C3×C8—31no—
226C2×C3×C8C22×D311yescharacteristic
227C2×C3×C8C22×D311yescharacteristic
228C2×C3×C8C22×D311yescharacteristic
229C2×C3×C8C22×D311yescharacteristic
230C3×C4⋊C4—32no—
231C3×C4⋊C4—31no—
232C3×C42—21no—
233C3×C42—31no—
234C3×C42—31no—
235C3×C42—32no—
236C3×C42—31no—
237C3×C42C22×D311yescharacteristic
238C3×C42C22×D311yescharacteristic
239C4⋊Dic3—31no—
240C4⋊Dic3—31no—
241C4⋊Dic3—32no—
242C4⋊Dic3—32no—
243C4⋊Dic3—61no—
244C4⋊Dic3C4×D311yescharacteristic
245Dic3⋊C4—32no—
246Dic3⋊C4—32no—
247Dic3⋊C4—32no—
248Dic3⋊C4—32no—
249Dic3⋊C4—32no—
250Dic3⋊C4—32no—
251Dic3⋊C4—62no—
252Dic3⋊C4C4×D312yes—
253C4×Dic3—32no—
254C4×Dic3—32no—
255C4×Dic3—32no—
256C4×Dic3—32no—
257C4×Dic3—32no—
258C4×Dic3—32no—
259C4×Dic3—62no—
260C4×Dic3C4×D311yescharacteristic
261C4×Dic3C4×D311yescharacteristic
262C4×Dic3C22×D311yescharacteristic
263C4×Dic3C22×D312yes—
264#208—64no—
265C2×C3⋊C8—32no—
266C2×C3⋊C8—32no—
267C2×C3⋊C8—62no—
268C2×C3⋊C8C22×D311yescharacteristic
269C2×C3⋊C8C22×D312yes—
270C2×C3⋊C8C22×D311yescharacteristic
271ΓL(1,𝔽25)—22no—
272ΓL(1,𝔽25)—64no—
273ΓL(1,𝔽25)Dic614yescomplemented
274C22×C32C23×C411yescharacteristic
275C2×C3×D3—21no—
276C2×C3×D3—21no—
277C2×C3×D3C4×Q812yes—
278C32×C4—21no—
279C32×C4C2×M1612yes—
280C32×C4C22×Q811yescharacteristic
281C32×C4C2×#4111yescharacteristic
282C3⋊Dic3—22no—
283C3⋊Dic3C2×M1614yes—
284C3×Dic3—22no—
285C3×Dic3—21no—
286C3×Dic3—22no—
287C3×Dic3—21no—
288C3×Dic3C4×Q812yes—
289C22×Q8—91no—
290C2×M16—31no—
291C2×M16—31no—
292C2×M16—92no—
293C4×Q8—31no—
294C4×Q8—92no—
295C4×Q8—92no—
296C4×Q8—91no—
297C4×Q8—91no—
298C4×Q8—91no—
299C2×C4⋊C4—92no—
300C2×C4⋊C4—91no—
301C2×C42—31no—
302C2×C42—92no—
303C4⋊C8—32no—
304C4⋊C8—31no—
305C4⋊C8—92no—
306C4⋊C8—92no—
307C4⋊C8—92no—
308C4⋊C8—92no—
309C4⋊C8—91no—
310#97—31no—
311#97—31no—
312#97—32no—
313#97—92no—
314C4×C8—31no—
315C4×C8—92no—
316C4×C8—92no—
317C4×C8D3211yescharacteristic
318C23×C3—31no—
319C22×D3C2×C4×D311yescharacteristic
320C3×Q8—62no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—21no—
324C2×C3×C4—31no—
325C2×C3×C4—32no—
326C2×C3×C4—32no—
327C2×C3×C4—32no—
328C2×C3×C4—32no—
329C2×C3×C4—31no—
330C2×C3×C4—31no—
331C2×C3×C4—62no—
332C2×C3×C4—121no—
333C2×C3×C4C2×C4×D311yescharacteristic
334C2×C3×C4C2×C4×D311yescharacteristic
335C2×C3×C4C2×C4×D311yescharacteristic
336C2×C3×C4C2×C4×D311yescharacteristic
337C2×C3×C4C23×D311yescharacteristic
338C2×C3×C4C23×D311yescharacteristic
339C2×Dic3—32no—
340C2×Dic3—32no—
341C2×Dic3—32no—
342C2×Dic3—32no—
343C2×Dic3—62no—
344C2×Dic3—62no—
345C2×Dic3—64no—
346C2×Dic3—122no—
347C2×Dic3C2×C4×D312yes—
348C2×Dic3C2×C4×D312yes—
349C2×Dic3C2×C4×D311yescharacteristic
350C2×Dic3C2×C4×D311yescharacteristic
351C2×Dic3C2×C4×D311yescharacteristic
352C4×D3—22no—
353C4×D3—41no—
354C4×D3—64no—
355C4×D3—122no—
356C4×D3C2×Dic612yes—
357C4×D3#23512yes—
358Dic6—22no—
359Dic6—34no—
360Dic6—34no—
361Dic6—64no—
362Dic6—64no—
363Dic6—64no—
364C3×C8—21no—
365C3×C8—21no—
366C3×C8—22no—
367C3×C8—32no—
368C3×C8—41no—
369C3×C8—62no—
370C3×C8—61no—
371C3×C8C2×Dic612yescomplemented
372C3×C8D3×Q812yescomplemented
373C3⋊C8—21no—
374C3⋊C8—22no—
375C3⋊C8—62no—
376C3⋊C8—62no—
377C3⋊C8—122no—
378C3⋊C8C2×Dic612yescomplemented
379C2×C32C2×C4×Q811yescharacteristic
380C2×C32C22×M1611yescharacteristic
381C2×C32#56811yescharacteristic
382C3×D3—22no—
383C2×Q8—31no—
384C2×Q8—92no—
385C2×Q8—91no—
386C2×Q8—184no—
387C22×C4—31no—
388C22×C4—31no—
389C22×C4—31no—
390C22×C4—92no—
391C22×C4—92no—
392M16—34no—
393M16—62no—
394M16—184no—
395C2×C8—31no—
396C2×C8—31no—
397C2×C8—32no—
398C2×C8—92no—
399C2×C8C2×D3211yescharacteristic
400C2×C8C2×D3211yescharacteristic
401C4⋊C4—32no—
402C4⋊C4—31no—
403C4⋊C4—92no—
404C4⋊C4—92no—
405C4⋊C4—92no—
406C4⋊C4—92no—
407C4⋊C4—91no—
408C42—31no—
409C42—32no—
410C42—31no—
411C42—31no—
412C42—92no—
413C42—92no—
414C42—92no—
415C42C2×D3211yescharacteristic
416C22×C3—21no—
417C22×C3—32no—
418C22×C3—61no—
419C22×C3—61no—
420C22×C3C22×C4×D311yescharacteristic
421C22×C3C22×C4×D311yescharacteristic
422C2×D3—21no—
423C2×D3—21no—
424C2×D3C4×Dic612yes—
425C3×C4—21no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—22no—
429C3×C4—21no—
430C3×C4—32no—
431C3×C4—41no—
432C3×C4—61no—
433C3×C4—62no—
434C3×C4—62no—
435C3×C4—61no—
436C3×C4D3×M1612yes—
437C3×C4D3×M1612yes—
438C3×C4C22×Dic611yescharacteristic
439C3×C4C2×#23511yescharacteristic
440C3×C4C2×D3×Q811yescharacteristic
441C3×C4D3×#4111yescharacteristic
442Dic3—22no—
443Dic3—21no—
444Dic3—21no—
445Dic3—22no—
446Dic3—34no—
447Dic3—34no—
448Dic3—64no—
449Dic3—62no—
450Dic3—62no—
451Dic3—122no—
452Dic3C4×Dic612yes—
453C32Q8×M1611yescharacteristic, complemented, Sylow
454C23—31no—
455Q8—62no—
456Q8—94no—
457Q8—184no—
458C2×C4—31no—
459C2×C4—32no—
460C2×C4—32no—
461C2×C4—31no—
462C2×C4—31no—
463C2×C4—32no—
464C2×C4—32no—
465C2×C4—62no—
466C2×C4—92no—
467C2×C4—92no—
468C2×C4—121no—
469C2×C4—184no—
470C2×C4—362no—
471C2×C4C4×D3211yescharacteristic
472C2×C4C4×D3211yescharacteristic
473C2×C4C22×D3211yescharacteristic
474C8—21no—
475C8—32no—
476C8—61no—
477C8—62no—
478C8—182no—
479C8D3×Dic612yescomplemented
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—62no—
484C2×C3C2×C4×Dic611yescharacteristic
485C2×C3C4×D3×Q811yescharacteristic
486C2×C3C2×D3×M1611yescharacteristic
487C2×C3C2×D3×M1611yescharacteristic
488C2×C3#574511yescharacteristic
489C2×C3#575011yescharacteristic
490D3—22no—
491C22—32no—
492C22—61no—
493C22—61no—
494C22C2×C4×D3211yescharacteristic
495C4—21no—
496C4—32no—
497C4—61no—
498C4—62no—
499C4—61no—
500C4—62no—
501C4—94no—
502C4—182no—
503C4#6379112yes—
504C4C2×D3×Dic611yescharacteristic
505C4D3×#23511yescharacteristic
506C3—21no—
507C3M16×Dic611yescharacteristic, complemented
508C3#7996211yescharacteristic, complemented
509C2—62no—
510C2C4×D3×Dic611yescharacteristic
511C2C2×#6379111yescharacteristic
512C2#1059466811yescharacteristic
subgroupG/H
center, FrattiniC2×C4C22×D32
commutator, socleC22×C32C23×C4
FittingC32×C4×C8C22
prime radicalC22×C3×D3C22×C4
cosocleC2×C32×C4C24
hypercenterC4×C8D32
hypocenterC32Q8×M16
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C4×C8

Complements

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

NH ≅ G/Ntype
2. C8×#634C2semidirect
4. C3×#4679C2semidirect
5. #10590789C2semidirect
6. #10590787C2semidirect
10. #10590258C2semidirect
11. #10590254C2semidirect
61. #4679D3semidirect
92. C3×ΓL(1,𝔽25)Q8semidirect
169. #634M16semidirect
184. C32×C8C2×Q8semidirect
188. C3×C3⋊C8C2×Q8semidirect
273. ΓL(1,𝔽25)Dic6semidirect
371. C3×C8C2×Dic6semidirect
372. C3×C8D3×Q8semidirect
378. C3⋊C8C2×Dic6semidirect
453. C32Q8×M16semidirect
479. C8D3×Dic6semidirect
507. C3M16×Dic6semidirect
508. C3#79962semidirect

Maximal subgroups

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

subgroupindex
C4×D3×Dic62
C8×#6342
C3×C4×ΓL(1,𝔽25)2
C3×#46792
#105907892
#105907872
#105906672
#105906632
#105903362
#105902582
#105902542
#799623
M16×Dic63