Finite Group Databasebeta

← #11788686

Subgroups — #11788686

Order 1152.

705 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63833C211yesmaximal, characteristic
2C2×#63817C211yesmaximal, characteristic
3#10592665C211yesmaximal, characteristic
4C2×C3×#810C211yesmaximal, characteristic, complemented
5C2×#63560C211yesmaximal, characteristic, complemented
6C3×#4530C211yesmaximal, characteristic, complemented
7#10589166C211yesmaximal, characteristic, complemented
8#70286—31nomaximal
9#69007—31nomaximal
10C2×#3828C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×C4⋊C4C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C2×C3×#208C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C2×#3719—21no—
18#63833—22no—
19#63833C414yescomplemented
20C2×#3717—21no—
21C2×#3712—21no—
22#63817—22no—
23#63817—22no—
24C2×#3710—21no—
25#63637—21no—
26#63636—21no—
27C3×#810—21no—
28C3×#810—21no—
29#63560—22no—
30C2×#942—31no—
31C2×#909—31no—
32C2×#907—31no—
33C2×#901—31no—
34C2×C3×#103—31no—
35C2×#836—31no—
36C2×#824—31no—
37C2×#810D311yescharacteristic, complemented
38#4966—31no—
39#4966—31no—
40#4966—61no—
41C3×#343—31no—
42#4558—31no—
43#4552—31no—
44#4530D311yescharacteristic, complemented
45C22×C32×C4—21no—
46C22×C32×C4C2311yescharacteristic
47C2×#638—21no—
48#3828—21no—
49#3828—21no—
50#3828C2×C412yes—
51#3828D412yes—
52C2×C4⋊(C3⋊D3)D411yescharacteristic
53C2×C4×C3⋊D3—21no—
54C2×#634D411yescharacteristic
55C22×C3×Dic3—21no—
56C2×C3×Dic6—21no—
57C2×C3×Dic6—21no—
58C2×C3×Dic6—21no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6C2×C412yes—
62C32×C4⋊C4—21no—
63C32×C4⋊C4—21no—
64#3748—21no—
65#3748—21no—
66C2×C3⋊(C3⋊C8)—21no—
67C3×#208—21no—
68C3×#208—21no—
69C3×#208C2×C412yes—
70C3×#208D412yes—
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8—21no—
73C2×C3×C3⋊C8—21no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8D411yescharacteristic
76C2×C3×C3⋊C8D411yescharacteristic
77#3719—24no—
78#3717—24no—
79#3712—24no—
80#3710—24no—
81#1908—91noSylow
82C22×C3×Q8—31no—
83C2×#235—31no—
84C2×#235—31no—
85C2×#235—61no—
86C22×Dic6C2×D311yescharacteristic
87C2×C3×M16—31no—
88C2×C3×M16—31no—
89C22×C3×C8—31no—
90C2×C3×C4⋊C4—21no—
91C2×C3×C4⋊C4C2×D311yescharacteristic
92C2×C3×C4⋊C4C2×D311yescharacteristic
93C2×#216—61no—
94#942—34no—
95#942—62no—
96C2×#215—61no—
97C2×#208—31no—
98C2×#208—31no—
99C2×#208—61no—
100C2×#208C2×D311yescharacteristic
101C2×#208C2×D311yescharacteristic
102C22×C3⋊C8C2×D311yescharacteristic
103#909—34no—
104#909—62no—
105#907—62no—
106#907—62no—
107C2×Dic12—61no—
108C2×#204—61no—
109#901—62no—
110#901—62no—
111C2×ΓL(1,𝔽25)—61no—
112C2×ΓL(1,𝔽25)—61no—
113C2×D3×C8—61no—
114C2×D3×C8—61no—
115C3×#103—61no—
116C3×#103—61no—
117#836—62no—
118#824—62no—
119#810—21no—
120#810—21no—
121#810—61no—
122#810—61no—
123#810—62no—
124#809—61no—
125#809—61no—
126#809—62no—
127C23×C32#4111yescharacteristic
128C22×C3⋊D3—21no—
129C2×C32×C4—41no—
130C2×C32×C4—81no—
131C2×C32×C4C22×C411yescharacteristic
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4C2×D411yescharacteristic
134C2×C32×C4C2×D411yescharacteristic
135C2×C32×C4C2×D411yescharacteristic
136C2×C32×C4#4111yescharacteristic
137#638—22no—
138#638—22no—
139C2×C3⋊Dic3—21no—
140C4⋊(C3⋊D3)—21no—
141C4⋊(C3⋊D3)C22⋊C412yes—
142C4×C3⋊D3—22no—
143C4×C3⋊D3—22no—
144#634—21no—
145#634C22⋊C412yes—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—41no—
148C2×C3×Dic3—41no—
149C3×Dic6—22no—
150C3×Dic6—22no—
151C3×Dic6—41no—
152C3×Dic6—41no—
153C3⋊(C3⋊C8)—41no—
154C3×C3⋊C8—22no—
155C3×C3⋊C8—21no—
156C3×C3⋊C8—22no—
157C3×C3⋊C8—21no—
158C2×#137—91no—
159C2×#131—91no—
160#417—91no—
161C2×#103—31no—
162C2×#101—91no—
163#343—31no—
164#341—91no—
165C2×C3×Q8—32no—
166C2×C3×Q8—61no—
167C2×C3×Q8—61no—
168C2×C3×Q8—61no—
169C2×C3×Q8—61no—
170C2×C3×Q8—61no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—21no—
174C22×C3×C4—41no—
175C22×C3×C4—61no—
176C22×C3×C4C22×D311yescharacteristic
177C22×C3×C4C22×D311yescharacteristic
178C2×#67—61no—
179C2×#67—61no—
180C2×#67—121no—
181C22×Dic3—21no—
182#235—32no—
183#235—32no—
184#235—32no—
185#235—32no—
186#235—62no—
187#235—62no—
188#235—61no—
189#235—61no—
190#235—61no—
191#235—61no—
192#235—121no—
193#235—121no—
194C2×D12—31no—
195C2×D12—31no—
196C2×D12—61no—
197C2×C4×D3—61no—
198C2×C4×D3—61no—
199C2×C4×D3—121no—
200C2×Dic6—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6—31no—
206C2×Dic6—31no—
207C2×Dic6—61no—
208C2×Dic6C4×D312yes—
209C3×M16—32no—
210C3×M16—32no—
211C3×M16—61no—
212C3×M16—61no—
213C2×C3×C8—31no—
214C2×C3×C8—31no—
215C2×C3×C8—61no—
216C2×C3×C8—61no—
217C2×C3×C8—61no—
218C2×C3×C8—61no—
219C3×C4⋊C4—22no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—21no—
224C3×C4⋊C4—41no—
225#216—64no—
226#215—64no—
227#208—21no—
228#208—21no—
229#208—61no—
230#208—61no—
231#208—61no—
232#208—61no—
233#208—121no—
234#208—121no—
235#208C4×D312yes—
236#208D1212yes—
237C2×C3⋊C8—21no—
238C2×C3⋊C8—21no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—62no—
242C2×C3⋊C8—61no—
243C2×C3⋊C8—61no—
244C2×C3⋊C8#6711yescharacteristic
245C2×C3⋊C8#6711yescharacteristic
246Dic12—64no—
247#204—64no—
248ΓL(1,𝔽25)—64no—
249ΓL(1,𝔽25)—64no—
250D3×C8—64no—
251D3×C8—64no—
252C22×C32—21no—
253C22×C32—21no—
254C22×C32C4×D411yescharacteristic
255C22×C32#12111yescharacteristic
256C22×C32#12311yescharacteristic
257C2×C3⋊D3—22no—
258C2×C3⋊D3—41no—
259C2×C3⋊D3—41no—
260C32×C4—81no—
261C32×C4C2×C22⋊C411yescharacteristic
262C32×C4C4×D411yescharacteristic
263C32×C4#12011yescharacteristic
264C32×C4#12111yescharacteristic
265C3⋊Dic3—22no—
266C3×Dic3—42no—
267C2×#41—91no—
268C22×Q8—31no—
269#137—94no—
270#137—182no—
271C2×Q16—181no—
272C2×SD16—181no—
273#131—182no—
274#131—182no—
275C2×M16—31no—
276C2×M16—31no—
277C2×M16—91no—
278C2×M16—181no—
279C22×C8—31no—
280C22×C8—181no—
281C2×C4⋊C4D3211yescharacteristic
282#103—61no—
283#103—61no—
284#103—181no—
285#102—181no—
286#101—182no—
287C23×C3—21no—
288C23×C3#23511yescharacteristic
289C23×C3#23711yescharacteristic
290C22×D3—61no—
291C22×D3—61no—
292C22×D3—121no—
293C3×Q8—62no—
294C3×Q8—62no—
295C3×Q8—121no—
296C3×Q8—121no—
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—42no—
304C2×C3×C4—41no—
305C2×C3×C4—41no—
306C2×C3×C4—62no—
307C2×C3×C4—82no—
308C2×C3×C4—81no—
309C2×C3×C4—81no—
310C2×C3×C4—121no—
311C2×C3×C4—121no—
312C2×C3×C4C2×C4×D311yescharacteristic
313C2×C3×C4C2×C4×D311yescharacteristic
314C2×C3×C4C2×D1211yescharacteristic
315C2×C3×C4#23511yescharacteristic
316C2×C3×C4D3×D411yescharacteristic
317C2×C3×C4D3×D411yescharacteristic
318C2×C3×C4D3×D411yescharacteristic
319C2×C3×C4D3×D411yescharacteristic
320C2×C3×C4D3×D411yescharacteristic
321C2×C3×C4D3×D411yescharacteristic
322C2×C3×C4#23911yescharacteristic
323C2×C3×C4C2×#6711yescharacteristic
324#67—62no—
325#67—62no—
326#67—62no—
327#67—62no—
328#67—122no—
329#67—122no—
330C2×Dic3—22no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—61no—
334C2×Dic3—61no—
335C2×Dic3—121no—
336D12—32no—
337D12—32no—
338D12—62no—
339D12—61no—
340D12—61no—
341D12—121no—
342C4×D3—62no—
343C4×D3—62no—
344C4×D3—62no—
345C4×D3—62no—
346C4×D3—122no—
347C4×D3—122no—
348Dic6—22no—
349Dic6—22no—
350Dic6—32no—
351Dic6—32no—
352Dic6—41no—
353Dic6—41no—
354Dic6—62no—
355Dic6—61no—
356Dic6—61no—
357Dic6—121no—
358C3×C8—62no—
359C3×C8—61no—
360C3×C8—61no—
361C3×C8—62no—
362C3⋊C8—22no—
363C3⋊C8—22no—
364C3⋊C8—21no—
365C3⋊C8—21no—
366C3⋊C8—122no—
367C3⋊C8—121no—
368C3⋊C8—121no—
369C2×C32—21no—
370C2×C32—21no—
371C2×C32#38611yescharacteristic
372C2×C32#47111yescharacteristic
373C2×C32#47311yescharacteristic
374C3⋊D3—42no—
375#41—92no—
376#41—92no—
377#41—181no—
378#41—181no—
379C2×Q8—32no—
380C2×Q8—61no—
381C2×Q8—61no—
382C2×Q8—61no—
383C2×Q8—61no—
384C2×Q8—61no—
385C2×Q8—91no—
386C2×D4—91no—
387C2×D4—181no—
388C22×C4—21no—
389C22×C4—61no—
390C22×C4—181no—
391C22×C4C2×D3211yescharacteristic
392Q16—184no—
393SD16—184no—
394M16—32no—
395M16—32no—
396M16—61no—
397M16—61no—
398M16—184no—
399M16—181no—
400M16—181no—
401C2×C8—31no—
402C2×C8—31no—
403C2×C8—61no—
404C2×C8—61no—
405C2×C8—61no—
406C2×C8—61no—
407C2×C8—184no—
408C2×C8—181no—
409C4⋊C4—21no—
410C4⋊C4—21no—
411C22×C3—21no—
412C22×C3—21no—
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—21no—
417C22×C3—21no—
418C22×C3—41no—
419C22×C3—41no—
420C22×C3C4×D1211yescharacteristic
421C22×C3#67⋊C411yescharacteristic
422C22×C3#88311yescharacteristic
423C22×C3#88411yescharacteristic
424C22×C3#89411yescharacteristic
425C22×C3#93811yescharacteristic
426C2×D3—62no—
427C2×D3—62no—
428C2×D3—122no—
429C2×D3—121no—
430C2×D3—121no—
431C2×D3—121no—
432C2×D3—121no—
433C2×D3—241no—
434C2×D3—241no—
435C3×C4—21no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—82no—
440C3×C4—81no—
441C3×C4—81no—
442C3×C4—122no—
443C3×C4D3×C22⋊C411yescharacteristic
444C3×C4D3×C22⋊C411yescharacteristic
445C3×C4#88211yescharacteristic
446C3×C4#88411yescharacteristic
447C3×C4#89311yescharacteristic
448C3×C4#89511yescharacteristic
449C3×C4C4×#6711yescharacteristic
450C3×C4#93711yescharacteristic
451Dic3—42no—
452Dic3—62no—
453Dic3—62no—
454Dic3—122no—
455C32#190811yescharacteristic, complemented, Sylow
456C23—181no—
457C23#380411yescharacteristic
458Q8—62no—
459Q8—62no—
460Q8—92no—
461Q8—121no—
462Q8—121no—
463Q8—181no—
464D4—92no—
465D4—182no—
466D4—182no—
467D4—181no—
468C2×C4—41no—
469C2×C4—62no—
470C2×C4—81no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4—181no—
474C2×C4—182no—
475C2×C4—182no—
476C2×C4#379911yescharacteristic
477C2×C4C4×D3211yescharacteristic
478C2×C4D3×D1211yescharacteristic
479C2×C4#380211yescharacteristic
480C2×C4D3×#6711yescharacteristic
481C2×C4#381111yescharacteristic
482C8—62no—
483C8—61no—
484C8—61no—
485C8—62no—
486C8—361no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—21no—
494C2×C3—41no—
495C2×C3—41no—
496C2×C3#466511yescharacteristic
497C2×C3#474811yescharacteristic
498C2×C3#475511yescharacteristic
499C2×C3#495611yescharacteristic
500C2×C3#517711yescharacteristic
501C2×C3#519211yescharacteristic
502D3—122no—
503D3—122no—
504D3—242no—
505C22—21no—
506C22—21no—
507C22—182no—
508C22—361no—
509C22—361no—
510C22#6385411yescharacteristic
511C22#6388011yescharacteristic
512C22#6395811yescharacteristic
513C4—81no—
514C4—122no—
515C4—182no—
516C4#6390411yescharacteristic
517C4#6391011yescharacteristic
518C4#6392111yescharacteristic
519C4#6392211yescharacteristic
520C3—21no—
521C3#6900711yescharacteristic, complemented
522C3#7028611yescharacteristic, complemented
523C2—21no—
524C2—21no—
525C2—362no—
526C2#1059127511yescharacteristic
527C2#1059140711yescharacteristic
528C2#1059154311yescharacteristic
subgroupG/H
centerC22#63958
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×C4⋊C4C22
socleC22×C32#121
prime radicalC2×#3828C22
cosocleC22×C32×C4C23
hypercenterC2×C4⋊C4D32
hypocenterC32#1908
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
4. C2×C3×#810C2semidirect
5. C2×#63560C2semidirect
6. C3×#4530C2semidirect
7. #10589166C2semidirect
19. #63833C4semidirect
37. C2×#810D3semidirect
44. #4530D3semidirect
455. C32#1908semidirect
521. C3#69007semidirect
522. C3#70286semidirect

Maximal subgroups

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

subgroupindex
C2×#638332
C2×#638172
#105926652
C2×C3×#8102
C2×#635602
C3×#45302
#105891662
#702863
#690073