Finite Group Databasebeta

← #11861519

Subgroups — #11861519

Order 1152.

847 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#3797C211yesmaximal, characteristic
2C4×#3743C211yesmaximal, characteristic
3C3×#4679C211yesmaximal, characteristic, complemented
4C3×C8×Dic6C211yesmaximal, characteristic, complemented
5#10590784C212yesmaximal, complemented
6#10590666C212yesmaximal
7#10590664C212yesmaximal
8#10590646C212yesmaximal, complemented
9#10590340C211yesmaximal, characteristic
10#10590259C211yesmaximal, characteristic, complemented
11#10590250C211yesmaximal, characteristic, complemented
12#79962—31nomaximal
13Q8×ΓL(1,𝔽25)—31nomaximal
14C2×#3797C411yescharacteristic
15C2×#3743C2211yescharacteristic
16C2×#3743C2211yescharacteristic
17C42×C3⋊D3C2211yescharacteristic
18C3×C4×Dic6C2211yescharacteristic
19C3×C4×Dic6C2211yescharacteristic
20C4×#627C2212yes—
21#63900C411yescharacteristic
22#63883C412yes—
23C4×#624C2212yes—
24#63843C411yescharacteristic
25#63843C411yescharacteristic
26#63838C412yes—
27C2×#3712C2212yes—
28C32×C4×C8C2211yescharacteristic
29#63642C2211yescharacteristic
30#63642C2211yescharacteristic
31C4×C3⋊(C3⋊C8)C2211yescharacteristic
32C3×#815C2212yes—
33C3×Dic3⋊C8C2212yes—
34C3×Dic3⋊C8C2212yes—
35C3×C8×Dic3C2212yes—
36C3×C4⋊(C3⋊C8)C2211yescharacteristic
37C3×C4⋊(C3⋊C8)C2211yescharacteristic
38#63572C2212yes—
39Dic3⋊(C3⋊C8)C2212yes—
40Dic3⋊(C3⋊C8)C2212yes—
41#63554C2212yes—
42Dic3×C3⋊C8C2212yes—
43C4×D3×Q8—31no—
44C4×D3×Q8—31no—
45C3×#446—31no—
46C3×C8×Q8—31no—
47#5024—31no—
48Q8×C3⋊C8—31no—
49#4837—32no—
50#4837—32no—
51#4837—31no—
52#4837—31no—
53#4831—31no—
54#4831—32no—
55#4830—31no—
56#4828—32no—
57#4707—32no—
58#4701—32no—
59C4×ΓL(1,𝔽25)—31no—
60C4×ΓL(1,𝔽25)—31no—
61C4×ΓL(1,𝔽25)—32no—
62C4×ΓL(1,𝔽25)—61no—
63#4679—32no—
64#4679D311yescharacteristic, complemented
65C8×Dic6D311yescharacteristic, complemented
66C2×C4×C3⋊D3C2×C411yescharacteristic
67C2×C4×C3⋊D3C2×C411yescharacteristic
68C2×C4×C3⋊D3C2311yescharacteristic
69C2×C3×Dic6C2×C411yescharacteristic
70C2×C3×Dic6C2×C411yescharacteristic
71C2×#627C2×C412yes—
72C2×#624C2×C412yes—
73C2×#624C2×C412yes—
74#3797—24no—
75C2×C32×C8C2311yescharacteristic
76C2×C32×C8C2311yescharacteristic
77C32×C42C2311yescharacteristic
78C4×C3⋊Dic3C2×C411yescharacteristic
79C4×C3⋊Dic3C2×C411yescharacteristic
80C4×C3⋊Dic3C2311yescharacteristic
81C2×C3⋊(C3⋊C8)C2311yescharacteristic
82C2×C3⋊(C3⋊C8)C2311yescharacteristic
83#3743—22no—
84#3743Q814yescomplemented
85C3×C4⋊Dic3C2×C411yescharacteristic
86C3×C4⋊Dic3C2×C411yescharacteristic
87C3×Dic3⋊C4C2×C412yes—
88C3×Dic3⋊C4C2×C412yes—
89C3×C4×Dic3C2312yes—
90C3×C4×Dic3C2312yes—
91C2×C3×C3⋊C8C2312yes—
92C2×C3×C3⋊C8C2312yes—
93#3724C2×C412yes—
94Dic3⋊Dic3C2×C412yes—
95Dic3⋊Dic3C2×C412yes—
96Dic32C2×C412yes—
97Dic32C2×C412yes—
98#3712—24no—
99Q8×M16—91noSylow
100C2×D3×Q8—31no—
101C2×D3×Q8—31no—
102C3×C4×Q8—31no—
103C3×C4×Q8—31no—
104Q8×Dic3—31no—
105Q8×Dic3—31no—
106C2×ΓL(1,𝔽25)—31no—
107C2×ΓL(1,𝔽25)—31no—
108C2×ΓL(1,𝔽25)—31no—
109C2×ΓL(1,𝔽25)—31no—
110C2×ΓL(1,𝔽25)—32no—
111C2×ΓL(1,𝔽25)—32no—
112C2×ΓL(1,𝔽25)—61no—
113C2×ΓL(1,𝔽25)—61no—
114D3×C4⋊C4—31no—
115D3×C4⋊C4—31no—
116D3×C4⋊C4—32no—
117D3×C4⋊C4—32no—
118#887—31no—
119#887—31no—
120#887—32no—
121#887—32no—
122C42×D3—31no—
123C42×D3—31no—
124C42×D3—32no—
125C42×D3—32no—
126C42×D3—61no—
127C4×Dic6—32no—
128C4×Dic6—32no—
129C4×Dic6C2×D311yescharacteristic
130C4×Dic6C2×D311yescharacteristic
131C3×C4⋊C8—32no—
132C3×C4⋊C8—32no—
133C3×C4⋊C8—31no—
134C3×C4⋊C8—31no—
135C3×#97—32no—
136C3×C4×C8—21no—
137C3×C4×C8—32no—
138C3×C4×C8C2×D311yescharacteristic
139C3×C4×C8C2×D311yescharacteristic
140#815—31no—
141#815—31no—
142#815—31no—
143#815—31no—
144#815—32no—
145#815—61no—
146#815—61no—
147#815C2×D312yes—
148Dic3⋊C8—32no—
149Dic3⋊C8—32no—
150Dic3⋊C8—32no—
151Dic3⋊C8—32no—
152Dic3⋊C8C2×D312yes—
153Dic3⋊C8C2×D312yes—
154C8×Dic3—32no—
155C8×Dic3C2×D312yes—
156C4⋊(C3⋊C8)—32no—
157C4⋊(C3⋊C8)—32no—
158C4⋊(C3⋊C8)C2×D311yescharacteristic
159C4⋊(C3⋊C8)C2×D311yescharacteristic
160#803—32no—
161C4×C3⋊C8—31no—
162C4×C3⋊C8—31no—
163C4×C3⋊C8—32no—
164C4×C3⋊C8—61no—
165C22×C3⋊D3C22×C411yescharacteristic
166C2×C32×C4C22×C411yescharacteristic
167C2×C32×C4C22×C411yescharacteristic
168C2×C32×C4C2411yescharacteristic
169C2×C3⋊Dic3C22×C411yescharacteristic
170C2×C3⋊Dic3C22×C411yescharacteristic
171C2×C3⋊Dic3C22×C411yescharacteristic
172C4×C3⋊D3—22no—
173C4×C3⋊D3—41no—
174C4×C3⋊D3C2×Q812yes—
175C4×C3⋊D3#4112yes—
176C2×C3×Dic3C22×C412yes—
177C2×C3×Dic3C22×C412yes—
178C2×C3×Dic3C22×C412yes—
179C2×C3×Dic3C22×C412yes—
180C3×Dic6—22no—
181C3×Dic6M1614yescomplemented
182#627—24no—
183#624—24no—
184#624—42no—
185C32×C8—21no—
186C32×C8C2×Q812yescomplemented
187C3⋊(C3⋊C8)—21no—
188C3⋊(C3⋊C8)C2×Q812yescomplemented
189C3×C3⋊C8—22no—
190C3×C3⋊C8—22no—
191C2×C4×Q8—91no—
192#446—31no—
193#446—92no—
194#446—92no—
195#446—91no—
196C8×Q8—31no—
197C8×Q8—91no—
198#423—92no—
199#423—91no—
200C4×M16—92no—
201C4×M16—91no—
202C2×C3×Q8—31no—
203C2×C3×Q8—31no—
204D3×Q8—64no—
205D3×Q8—64no—
206C2×C4×D3—31no—
207C2×C4×D3—31no—
208C2×C4×D3—31no—
209C2×C4×D3—31no—
210C2×C4×D3—32no—
211C2×C4×D3—32no—
212C2×C4×D3—31no—
213C2×C4×D3—31no—
214C2×C4×D3—32no—
215C2×C4×D3—32no—
216C2×C4×D3—61no—
217C2×C4×D3—61no—
218C2×C4×D3—61no—
219C2×Dic6—32no—
220C2×Dic6—32no—
221C2×Dic6C4×D311yescharacteristic
222C2×Dic6C4×D311yescharacteristic
223C2×C3×C8—21no—
224C2×C3×C8—21no—
225C2×C3×C8—32no—
226C2×C3×C8—32no—
227C2×C3×C8C22×D311yescharacteristic
228C2×C3×C8C22×D311yescharacteristic
229C2×C3×C8C22×D311yescharacteristic
230C2×C3×C8C22×D311yescharacteristic
231C3×C4⋊C4—32no—
232C3×C4⋊C4—32no—
233C3×C4⋊C4—31no—
234C3×C4⋊C4—31no—
235C3×C42—21no—
236C3×C42—32no—
237C3×C42—32no—
238C3×C42C22×D311yescharacteristic
239C3×C42C22×D311yescharacteristic
240C4⋊Dic3—32no—
241C4⋊Dic3—32no—
242C4⋊Dic3C4×D311yescharacteristic
243C4⋊Dic3C4×D311yescharacteristic
244Dic3⋊C4—32no—
245Dic3⋊C4—32no—
246Dic3⋊C4—32no—
247Dic3⋊C4—32no—
248Dic3⋊C4C4×D312yes—
249Dic3⋊C4C4×D312yes—
250C4×Dic3—31no—
251C4×Dic3—31no—
252C4×Dic3—31no—
253C4×Dic3—31no—
254C4×Dic3—31no—
255C4×Dic3—31no—
256C4×Dic3—32no—
257C4×Dic3—32no—
258C4×Dic3—32no—
259C4×Dic3—32no—
260C4×Dic3—61no—
261C4×Dic3—61no—
262C4×Dic3—61no—
263C4×Dic3C22×D312yes—
264C4×Dic3C22×D312yes—
265C2×C3⋊C8—31no—
266C2×C3⋊C8—31no—
267C2×C3⋊C8—31no—
268C2×C3⋊C8—31no—
269C2×C3⋊C8—61no—
270C2×C3⋊C8—61no—
271C2×C3⋊C8C22×D312yes—
272C2×C3⋊C8C22×D312yes—
273ΓL(1,𝔽25)—34no—
274ΓL(1,𝔽25)—34no—
275ΓL(1,𝔽25)—64no—
276ΓL(1,𝔽25)—64no—
277ΓL(1,𝔽25)—64no—
278ΓL(1,𝔽25)—62no—
279ΓL(1,𝔽25)—62no—
280ΓL(1,𝔽25)—64no—
281C22×C32C23×C411yescharacteristic
282C2×C3⋊D3—21no—
283C2×C3⋊D3—21no—
284C2×C3⋊D3C4×Q812yes—
285C32×C4—21no—
286C32×C4C2×M1612yes—
287C32×C4C22×Q811yescharacteristic
288C32×C4C2×#4111yescharacteristic
289C3⋊Dic3—21no—
290C3⋊Dic3—21no—
291C3⋊Dic3C4×Q812yes—
292C3×Dic3—22no—
293C3×Dic3—22no—
294C3×Dic3—22no—
295C3×Dic3C2×M1614yes—
296C22×Q8—91no—
297C2×M16—92no—
298C2×M16—91no—
299C2×M16—91no—
300C4×Q8—31no—
301C4×Q8—31no—
302C4×Q8—92no—
303C4×Q8—92no—
304C4×Q8—91no—
305C4×Q8—91no—
306C2×C4⋊C4—92no—
307C2×C4⋊C4—91no—
308C2×C42—92no—
309C2×C42—91no—
310C4⋊C8—32no—
311C4⋊C8—32no—
312C4⋊C8—31no—
313C4⋊C8—31no—
314C4⋊C8—92no—
315C4⋊C8—92no—
316C4⋊C8—92no—
317#97—32no—
318#97—92no—
319#97—91no—
320#97—91no—
321C4×C8—32no—
322C4×C8—92no—
323C4×C8—91no—
324C4×C8D3211yescharacteristic
325C22×D3—31no—
326C22×D3—31no—
327C22×D3—61no—
328C3×Q8—34no—
329C3×Q8—62no—
330C2×C3×C4—21no—
331C2×C3×C4—21no—
332C2×C3×C4—21no—
333C2×C3×C4—32no—
334C2×C3×C4—32no—
335C2×C3×C4—32no—
336C2×C3×C4—32no—
337C2×C3×C4C2×C4×D311yescharacteristic
338C2×C3×C4C2×C4×D311yescharacteristic
339C2×C3×C4C2×C4×D311yescharacteristic
340C2×C3×C4C2×C4×D311yescharacteristic
341C2×C3×C4C23×D311yescharacteristic
342C2×C3×C4C23×D311yescharacteristic
343C2×Dic3—31no—
344C2×Dic3—31no—
345C2×Dic3—31no—
346C2×Dic3—31no—
347C2×Dic3—31no—
348C2×Dic3—31no—
349C2×Dic3—61no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3C2×C4×D312yes—
353C2×Dic3C2×C4×D312yes—
354C2×Dic3C2×C4×D312yes—
355C2×Dic3C2×C4×D312yes—
356C4×D3—32no—
357C4×D3—32no—
358C4×D3—32no—
359C4×D3—32no—
360C4×D3—62no—
361C4×D3—62no—
362C4×D3—64no—
363C4×D3—62no—
364C4×D3—62no—
365C4×D3—64no—
366C4×D3—122no—
367C4×D3—122no—
368C4×D3—122no—
369C4×D3—121no—
370C4×D3—121no—
371C4×D3—122no—
372Dic6—22no—
373Dic6—64no—
374Dic6—64no—
375Dic6ΓL(1,𝔽25)14yescomplemented
376C3×C8—22no—
377C3×C8—21no—
378C3×C8—21no—
379C3×C8—22no—
380C3×C8—62no—
381C3×C8—62no—
382C3×C8D3×Q812yescomplemented
383C3×C8D3×Q812yescomplemented
384C3⋊C8—22no—
385C3⋊C8—22no—
386C3⋊C8—32no—
387C3⋊C8—32no—
388C3⋊C8—62no—
389C3⋊C8—62no—
390C3⋊C8—61no—
391C3⋊C8—61no—
392C2×C32C2×C4×Q811yescharacteristic
393C2×C32C22×M1611yescharacteristic
394C2×C32#56811yescharacteristic
395C3⋊D3—22no—
396C2×Q8—31no—
397C2×Q8—31no—
398C2×Q8—92no—
399C2×Q8—184no—
400C22×C4—92no—
401C22×C4—92no—
402C22×C4—91no—
403C22×C4—91no—
404C22×C4—91no—
405M16—94no—
406M16—182no—
407M16—184no—
408C2×C8—32no—
409C2×C8—32no—
410C2×C8—91no—
411C2×C8—91no—
412C2×C8C2×D3211yescharacteristic
413C2×C8C2×D3211yescharacteristic
414C4⋊C4—32no—
415C4⋊C4—32no—
416C4⋊C4—31no—
417C4⋊C4—31no—
418C4⋊C4—92no—
419C4⋊C4—92no—
420C4⋊C4—92no—
421C42—32no—
422C42—32no—
423C42—92no—
424C42—92no—
425C42—91no—
426C42—91no—
427C42—91no—
428C42C2×D3211yescharacteristic
429C22×C3—21no—
430C22×C3C22×C4×D311yescharacteristic
431C22×C3C22×C4×D311yescharacteristic
432C2×D3—32no—
433C2×D3—32no—
434C2×D3—62no—
435C2×D3—61no—
436C2×D3—61no—
437C2×D3—61no—
438C2×D3—61no—
439C2×D3—121no—
440C2×D3—121no—
441C3×C4—22no—
442C3×C4—21no—
443C3×C4—21no—
444C3×C4—21no—
445C3×C4—22no—
446C3×C4—21no—
447C3×C4—34no—
448C3×C4—62no—
449C3×C4—62no—
450C3×C4—62no—
451C3×C4C2×ΓL(1,𝔽25)12yes—
452C3×C4D3×M1612yes—
453C3×C4C2×D3×Q811yescharacteristic
454C3×C4C2×D3×Q811yescharacteristic
455C3×C4D3×#4111yescharacteristic
456C3×C4D3×#4111yescharacteristic
457Dic3—22no—
458Dic3—22no—
459Dic3—22no—
460Dic3—32no—
461Dic3—32no—
462Dic3—62no—
463Dic3—61no—
464Dic3—61no—
465Dic3—61no—
466Dic3—61no—
467Dic3—121no—
468Dic3—121no—
469Dic3C2×ΓL(1,𝔽25)14yes—
470C32Q8×M1611yescharacteristic, complemented, Sylow
471C23—91no—
472Q8—34no—
473Q8—62no—
474Q8—184no—
475C2×C4—32no—
476C2×C4—32no—
477C2×C4—32no—
478C2×C4—32no—
479C2×C4—91no—
480C2×C4—92no—
481C2×C4—92no—
482C2×C4—91no—
483C2×C4—91no—
484C2×C4—184no—
485C2×C4—182no—
486C2×C4—361no—
487C2×C4—362no—
488C2×C4C4×D3211yescharacteristic
489C2×C4C4×D3211yescharacteristic
490C2×C4C22×D3211yescharacteristic
491C8—21no—
492C8—62no—
493C8—62no—
494C8—92no—
495C8—181no—
496C8#379712yescomplemented
497C2×C3—21no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3C4×D3×Q811yescharacteristic
501C2×C3C4×D3×Q811yescharacteristic
502C2×C3C22×ΓL(1,𝔽25)11yescharacteristic
503C2×C3C2×D3×M1611yescharacteristic
504C2×C3#575011yescharacteristic
505C2×C3#575011yescharacteristic
506D3—62no—
507D3—62no—
508D3—122no—
509C22—92no—
510C22—181no—
511C22—181no—
512C22C2×C4×D3211yescharacteristic
513C4—21no—
514C4—34no—
515C4—62no—
516C4—62no—
517C4—62no—
518C4—92no—
519C4—181no—
520C4—181no—
521C4D3×ΓL(1,𝔽25)12yes—
522C4C2×#379711yescharacteristic
523C4#6430611yescharacteristic
524C3—21no—
525C3Q8×ΓL(1,𝔽25)11yescharacteristic, complemented
526C3#7996211yescharacteristic, complemented
527C2—182no—
528C2C4×#379711yescharacteristic
529C2C2×D3×ΓL(1,𝔽25)11yescharacteristic
530C2#1059467011yescharacteristic
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
3. C3×#4679C2semidirect
4. C3×C8×Dic6C2semidirect
5. #10590784C2semidirect
8. #10590646C2semidirect
10. #10590259C2semidirect
11. #10590250C2semidirect
64. #4679D3semidirect
65. C8×Dic6D3semidirect
84. #3743Q8semidirect
181. C3×Dic6M16semidirect
186. C32×C8C2×Q8semidirect
188. C3⋊(C3⋊C8)C2×Q8semidirect
375. Dic6ΓL(1,𝔽25)semidirect
382. C3×C8D3×Q8semidirect
383. C3×C8D3×Q8semidirect
470. C32Q8×M16semidirect
496. C8#3797semidirect
525. C3Q8×ΓL(1,𝔽25)semidirect
526. C3#79962semidirect

Maximal subgroups

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

subgroupindex
C4×#37972
C4×#37432
C3×#46792
C3×C8×Dic62
#105907842
#105906662
#105906642
#105906462
#105903402
#105902592
#105902502
#799623
Q8×ΓL(1,𝔽25)3