Finite Group Databasebeta

← #11861512

Subgroups — #11861512

Order 1152.

839 conjugacy classes of subgroups.

The subgroup lattice has 525 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#3797C211yesmaximal, characteristic
2#10592536C211yesmaximal, characteristic
3C3×#4831C211yesmaximal, characteristic, complemented
4C3×#4830C211yesmaximal, characteristic, complemented
5#10590786C212yesmaximal, complemented
6#10590785C212yesmaximal, complemented
7#10590666C212yesmaximal
8#10590664C212yesmaximal
9C4×#3712C211yesmaximal, characteristic
10#10590258C211yesmaximal, characteristic, complemented
11C3⋊C8×Dic6C211yesmaximal, characteristic, complemented
12#79962—31nomaximal
13Q8×ΓL(1,𝔽25)—31nomaximal
14C2×#3797C411yescharacteristic
15C2×#3743C2212yes—
16C42×C3⋊D3C2211yescharacteristic
17C3×C4×Dic6C2211yescharacteristic
18C3×C4×Dic6C2211yescharacteristic
19C4×#627C2212yes—
20#63900C411yescharacteristic
21#63883C412yes—
22C4×#624C2212yes—
23#63843C411yescharacteristic
24#63843C411yescharacteristic
25#63838C412yes—
26C2×#3712C2211yescharacteristic
27C2×#3712C2211yescharacteristic
28C32×C4⋊C8C2211yescharacteristic
29#63641C2212yes—
30C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
31C3×#815C2212yes—
32C3×Dic3⋊C8C2212yes—
33C3×Dic3⋊C8C2212yes—
34C3×C8×Dic3C2212yes—
35C3×C4×C3⋊C8C2211yescharacteristic
36C3×C4×C3⋊C8C2211yescharacteristic
37Dic3⋊(C3⋊C8)C2212yes—
38Dic3⋊(C3⋊C8)C2212yes—
39#63555C2211yescharacteristic
40#63555C2211yescharacteristic
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—31no—
50#4837—31no—
51#4837—32no—
52#4837—32no—
53#4837—61no—
54#4831—32no—
55#4831D311yescharacteristic, complemented
56#4830D311yescharacteristic, complemented
57#4828—32no—
58#4707—32no—
59#4701—32no—
60C4×ΓL(1,𝔽25)—32no—
61C4×ΓL(1,𝔽25)—31no—
62C4×ΓL(1,𝔽25)—31no—
63#4679—31no—
64#4679—32no—
65C8×Dic6—31no—
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×C8C2312yes—
76C32×C42C2311yescharacteristic
77C4×C3⋊Dic3C2×C411yescharacteristic
78C4×C3⋊Dic3C2×C411yescharacteristic
79C4×C3⋊Dic3C2311yescharacteristic
80C2×C3⋊(C3⋊C8)C2312yes—
81#3743—24no—
82C3×C4⋊Dic3C2×C411yescharacteristic
83C3×C4⋊Dic3C2×C411yescharacteristic
84C3×Dic3⋊C4C2×C412yes—
85C3×Dic3⋊C4C2×C412yes—
86C3×C4×Dic3C2312yes—
87C3×C4×Dic3C2312yes—
88C2×C3×C3⋊C8C2311yescharacteristic
89C2×C3×C3⋊C8C2311yescharacteristic
90C2×C3×C3⋊C8C2311yescharacteristic
91C2×C3×C3⋊C8C2311yescharacteristic
92#3724C2×C412yes—
93Dic3⋊Dic3C2×C412yes—
94Dic3⋊Dic3C2×C412yes—
95Dic32C2×C412yes—
96Dic32C2×C412yes—
97#3712—22no—
98#3712Q814yescomplemented
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)—32no—
109C2×ΓL(1,𝔽25)—32no—
110C2×ΓL(1,𝔽25)—31no—
111C2×ΓL(1,𝔽25)—31no—
112C2×ΓL(1,𝔽25)—62no—
113D3×C4⋊C4—31no—
114D3×C4⋊C4—31no—
115D3×C4⋊C4—32no—
116D3×C4⋊C4—32no—
117#887—31no—
118#887—31no—
119#887—32no—
120#887—32no—
121C42×D3—31no—
122C42×D3—31no—
123C42×D3—32no—
124C42×D3—32no—
125C42×D3—61no—
126C4×Dic6—32no—
127C4×Dic6—32no—
128C4×Dic6C2×D311yescharacteristic
129C4×Dic6C2×D311yescharacteristic
130C3×C4⋊C8—21no—
131C3×C4⋊C8—32no—
132C3×C4⋊C8—32no—
133C3×C4⋊C8C2×D311yescharacteristic
134C3×C4⋊C8C2×D311yescharacteristic
135C3×#97—32no—
136C3×C4×C8—32no—
137C3×C4×C8—31no—
138C3×C4×C8—31no—
139#815—32no—
140#815—31no—
141#815—31no—
142#815—31no—
143#815—31no—
144#815C2×D312yes—
145Dic3⋊C8—32no—
146Dic3⋊C8—32no—
147Dic3⋊C8—32no—
148Dic3⋊C8—32no—
149Dic3⋊C8—62no—
150Dic3⋊C8C2×D312yes—
151Dic3⋊C8C2×D312yes—
152C8×Dic3—32no—
153C8×Dic3C2×D312yes—
154C4⋊(C3⋊C8)—31no—
155C4⋊(C3⋊C8)—31no—
156C4⋊(C3⋊C8)—32no—
157C4⋊(C3⋊C8)—32no—
158C4⋊(C3⋊C8)—61no—
159#803—32no—
160C4×C3⋊C8—32no—
161C4×C3⋊C8C2×D311yescharacteristic
162C4×C3⋊C8C2×D311yescharacteristic
163C22×C3⋊D3C22×C411yescharacteristic
164C2×C32×C4C22×C411yescharacteristic
165C2×C32×C4C22×C411yescharacteristic
166C2×C32×C4C2411yescharacteristic
167C2×C3⋊Dic3C22×C411yescharacteristic
168C2×C3⋊Dic3C22×C411yescharacteristic
169C2×C3⋊Dic3C22×C411yescharacteristic
170C4×C3⋊D3—22no—
171C4×C3⋊D3—41no—
172C4×C3⋊D3C2×Q812yes—
173C4×C3⋊D3#4112yes—
174C2×C3×Dic3C22×C412yes—
175C2×C3×Dic3C22×C412yes—
176C2×C3×Dic3C22×C412yes—
177C2×C3×Dic3C22×C412yes—
178C3×Dic6—22no—
179C3×Dic6M1614yescomplemented
180#627—24no—
181#624—24no—
182#624—42no—
183C32×C8—22no—
184C3⋊(C3⋊C8)—22no—
185C3×C3⋊C8—21no—
186C3×C3⋊C8—21no—
187C3×C3⋊C8C2×Q812yescomplemented
188C3×C3⋊C8C2×Q812yescomplemented
189C2×C4×Q8—91no—
190#446—31no—
191#446—92no—
192#446—91no—
193#446—92no—
194C8×Q8—31no—
195C8×Q8—91no—
196#423—92no—
197#423—91no—
198C4×M16—91no—
199C4×M16—92no—
200C2×C3×Q8—31no—
201C2×C3×Q8—31no—
202D3×Q8—64no—
203D3×Q8—64no—
204C2×C4×D3—31no—
205C2×C4×D3—31no—
206C2×C4×D3—31no—
207C2×C4×D3—31no—
208C2×C4×D3—32no—
209C2×C4×D3—32no—
210C2×C4×D3—31no—
211C2×C4×D3—31no—
212C2×C4×D3—32no—
213C2×C4×D3—32no—
214C2×C4×D3—61no—
215C2×C4×D3—61no—
216C2×C4×D3—61no—
217C2×Dic6—32no—
218C2×Dic6—32no—
219C2×Dic6C4×D311yescharacteristic
220C2×Dic6C4×D311yescharacteristic
221C2×C3×C8—22no—
222C2×C3×C8—31no—
223C2×C3×C8—31no—
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C2×C3×C8C22×D312yes—
227C2×C3×C8C22×D312yes—
228C3×C4⋊C4—32no—
229C3×C4⋊C4—32no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—31no—
232C3×C42—21no—
233C3×C42—32no—
234C3×C42—32no—
235C3×C42C22×D311yescharacteristic
236C3×C42C22×D311yescharacteristic
237C4⋊Dic3—32no—
238C4⋊Dic3—32no—
239C4⋊Dic3C4×D311yescharacteristic
240C4⋊Dic3C4×D311yescharacteristic
241Dic3⋊C4—32no—
242Dic3⋊C4—32no—
243Dic3⋊C4—32no—
244Dic3⋊C4—32no—
245Dic3⋊C4C4×D312yes—
246Dic3⋊C4C4×D312yes—
247C4×Dic3—31no—
248C4×Dic3—31no—
249C4×Dic3—31no—
250C4×Dic3—31no—
251C4×Dic3—31no—
252C4×Dic3—31no—
253C4×Dic3—32no—
254C4×Dic3—32no—
255C4×Dic3—32no—
256C4×Dic3—32no—
257C4×Dic3—61no—
258C4×Dic3—61no—
259C4×Dic3—61no—
260C4×Dic3C22×D312yes—
261C4×Dic3C22×D312yes—
262C2×C3⋊C8—32no—
263C2×C3⋊C8—32no—
264C2×C3⋊C8—62no—
265C2×C3⋊C8C22×D311yescharacteristic
266C2×C3⋊C8C22×D311yescharacteristic
267C2×C3⋊C8C22×D311yescharacteristic
268C2×C3⋊C8C22×D311yescharacteristic
269ΓL(1,𝔽25)—34no—
270ΓL(1,𝔽25)—34no—
271ΓL(1,𝔽25)—62no—
272ΓL(1,𝔽25)—62no—
273ΓL(1,𝔽25)—64no—
274ΓL(1,𝔽25)—64no—
275ΓL(1,𝔽25)—124no—
276C22×C32C23×C411yescharacteristic
277C2×C3⋊D3—21no—
278C2×C3⋊D3—21no—
279C2×C3⋊D3C4×Q812yes—
280C32×C4—21no—
281C32×C4C2×M1612yes—
282C32×C4C22×Q811yescharacteristic
283C32×C4C2×#4111yescharacteristic
284C3⋊Dic3—21no—
285C3⋊Dic3—21no—
286C3⋊Dic3C4×Q812yes—
287C3×Dic3—22no—
288C3×Dic3—22no—
289C3×Dic3—22no—
290C3×Dic3C2×M1614yes—
291C22×Q8—91no—
292C2×M16—91no—
293C2×M16—91no—
294C2×M16—92no—
295C4×Q8—31no—
296C4×Q8—31no—
297C4×Q8—92no—
298C4×Q8—92no—
299C4×Q8—91no—
300C4×Q8—91no—
301C2×C4⋊C4—92no—
302C2×C4⋊C4—91no—
303C2×C42—92no—
304C2×C42—91no—
305C4⋊C8—32no—
306C4⋊C8—32no—
307C4⋊C8—92no—
308C4⋊C8—92no—
309C4⋊C8—92no—
310C4⋊C8—91no—
311C4⋊C8D3211yescharacteristic
312#97—32no—
313#97—92no—
314#97—91no—
315#97—91no—
316C4×C8—32no—
317C4×C8—31no—
318C4×C8—31no—
319C4×C8—92no—
320C22×D3—31no—
321C22×D3—31no—
322C22×D3—61no—
323C3×Q8—34no—
324C3×Q8—62no—
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—32no—
329C2×C3×C4—32no—
330C2×C3×C4—32no—
331C2×C3×C4—32no—
332C2×C3×C4C2×C4×D311yescharacteristic
333C2×C3×C4C2×C4×D311yescharacteristic
334C2×C3×C4C2×C4×D311yescharacteristic
335C2×C3×C4C2×C4×D311yescharacteristic
336C2×C3×C4C23×D311yescharacteristic
337C2×C3×C4C23×D311yescharacteristic
338C2×Dic3—31no—
339C2×Dic3—31no—
340C2×Dic3—31no—
341C2×Dic3—31no—
342C2×Dic3—31no—
343C2×Dic3—31no—
344C2×Dic3—61no—
345C2×Dic3—61no—
346C2×Dic3—61no—
347C2×Dic3C2×C4×D312yes—
348C2×Dic3C2×C4×D312yes—
349C2×Dic3C2×C4×D312yes—
350C2×Dic3C2×C4×D312yes—
351C4×D3—32no—
352C4×D3—32no—
353C4×D3—32no—
354C4×D3—32no—
355C4×D3—62no—
356C4×D3—62no—
357C4×D3—64no—
358C4×D3—62no—
359C4×D3—62no—
360C4×D3—64no—
361C4×D3—122no—
362C4×D3—122no—
363C4×D3—122no—
364C4×D3—121no—
365C4×D3—121no—
366C4×D3—122no—
367Dic6—22no—
368Dic6—64no—
369Dic6—64no—
370Dic6ΓL(1,𝔽25)14yescomplemented
371C3×C8—22no—
372C3×C8—22no—
373C3×C8—32no—
374C3×C8—32no—
375C3×C8—42no—
376C3×C8—61no—
377C3×C8—61no—
378C3⋊C8—21no—
379C3⋊C8—21no—
380C3⋊C8—62no—
381C3⋊C8—62no—
382C3⋊C8—122no—
383C3⋊C8D3×Q812yescomplemented
384C3⋊C8D3×Q812yescomplemented
385C2×C32C2×C4×Q811yescharacteristic
386C2×C32C22×M1611yescharacteristic
387C2×C32#56811yescharacteristic
388C3⋊D3—22no—
389C2×Q8—31no—
390C2×Q8—31no—
391C2×Q8—92no—
392C2×Q8—184no—
393C22×C4—92no—
394C22×C4—92no—
395C22×C4—91no—
396C22×C4—91no—
397C22×C4—91no—
398M16—94no—
399M16—184no—
400M16—182no—
401C2×C8—31no—
402C2×C8—31no—
403C2×C8—31no—
404C2×C8—31no—
405C2×C8—92no—
406C2×C8C2×D3212yes—
407C4⋊C4—32no—
408C4⋊C4—32no—
409C4⋊C4—31no—
410C4⋊C4—31no—
411C4⋊C4—92no—
412C4⋊C4—92no—
413C4⋊C4—92no—
414C42—32no—
415C42—32no—
416C42—92no—
417C42—92no—
418C42—91no—
419C42—91no—
420C42—91no—
421C42C2×D3211yescharacteristic
422C22×C3—21no—
423C22×C3C22×C4×D311yescharacteristic
424C22×C3C22×C4×D311yescharacteristic
425C2×D3—32no—
426C2×D3—32no—
427C2×D3—62no—
428C2×D3—61no—
429C2×D3—61no—
430C2×D3—61no—
431C2×D3—61no—
432C2×D3—121no—
433C2×D3—121no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—22no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—34no—
440C3×C4—41no—
441C3×C4—62no—
442C3×C4—62no—
443C3×C4—62no—
444C3×C4C2×ΓL(1,𝔽25)12yes—
445C3×C4D3×M1612yes—
446C3×C4C2×D3×Q811yescharacteristic
447C3×C4C2×D3×Q811yescharacteristic
448C3×C4D3×#4111yescharacteristic
449C3×C4D3×#4111yescharacteristic
450Dic3—22no—
451Dic3—22no—
452Dic3—22no—
453Dic3—32no—
454Dic3—32no—
455Dic3—62no—
456Dic3—62no—
457Dic3—61no—
458Dic3—61no—
459Dic3—61no—
460Dic3—61no—
461Dic3—121no—
462Dic3C2×ΓL(1,𝔽25)14yes—
463C32Q8×M1611yescharacteristic, complemented, Sylow
464C23—91no—
465Q8—34no—
466Q8—62no—
467Q8—184no—
468C2×C4—32no—
469C2×C4—32no—
470C2×C4—32no—
471C2×C4—32no—
472C2×C4—91no—
473C2×C4—92no—
474C2×C4—92no—
475C2×C4—91no—
476C2×C4—91no—
477C2×C4—184no—
478C2×C4—182no—
479C2×C4—361no—
480C2×C4—362no—
481C2×C4C4×D3211yescharacteristic
482C2×C4C4×D3211yescharacteristic
483C2×C4C22×D3211yescharacteristic
484C8—22no—
485C8—32no—
486C8—32no—
487C8—61no—
488C8—61no—
489C8—182no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3C4×D3×Q811yescharacteristic
494C2×C3C4×D3×Q811yescharacteristic
495C2×C3C22×ΓL(1,𝔽25)11yescharacteristic
496C2×C3C2×D3×M1611yescharacteristic
497C2×C3#575011yescharacteristic
498C2×C3#575011yescharacteristic
499D3—62no—
500D3—62no—
501D3—122no—
502C22—92no—
503C22—181no—
504C22—181no—
505C22C2×C4×D3211yescharacteristic
506C4—21no—
507C4—34no—
508C4—62no—
509C4—62no—
510C4—62no—
511C4—92no—
512C4—181no—
513C4—181no—
514C4D3×ΓL(1,𝔽25)12yes—
515C4#6431211yescharacteristic
516C4D32×Q811yescharacteristic
517C3—21no—
518C3Q8×ΓL(1,𝔽25)11yescharacteristic, complemented
519C3#7996211yescharacteristic, complemented
520C2—182no—
521C2#1059436311yescharacteristic
522C2C2×D3×ΓL(1,𝔽25)11yescharacteristic
523C2#1059470811yescharacteristic
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×#4831C2semidirect
4. C3×#4830C2semidirect
5. #10590786C2semidirect
6. #10590785C2semidirect
10. #10590258C2semidirect
11. C3⋊C8×Dic6C2semidirect
55. #4831D3semidirect
56. #4830D3semidirect
98. #3712Q8semidirect
179. C3×Dic6M16semidirect
187. C3×C3⋊C8C2×Q8semidirect
188. C3×C3⋊C8C2×Q8semidirect
370. Dic6ΓL(1,𝔽25)semidirect
383. C3⋊C8D3×Q8semidirect
384. C3⋊C8D3×Q8semidirect
463. C32Q8×M16semidirect
518. C3Q8×ΓL(1,𝔽25)semidirect
519. C3#79962semidirect

Maximal subgroups

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

subgroupindex
C4×#37972
#105925362
C3×#48312
C3×#48302
#105907862
#105907852
#105906662
#105906642
C4×#37122
#105902582
C3⋊C8×Dic62
#799623
Q8×ΓL(1,𝔽25)3