Finite Group Databasebeta

← #11788676

Subgroups — #11788676

Order 1152.

697 conjugacy classes of subgroups.

The subgroup lattice has 519 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#907C211yesmaximal, characteristic
2C2×#63831C211yesmaximal, characteristic
3C2×#63651C211yesmaximal, characteristic, complemented
4C2×#63567C211yesmaximal, characteristic, complemented
5#10591087C211yesmaximal, characteristic
6C3×#4554C211yesmaximal, characteristic, complemented
7#10589166C211yesmaximal, characteristic, complemented
8#72019—31nomaximal
9#69007—31nomaximal
10C22×#634C2211yescharacteristic
11C2×C3×#235C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C2×C3×C4⋊Dic3C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C3×#907—22no—
18C3×#907—22no—
19C2×C3×ΓL(1,𝔽25)—21no—
20C2×C3×D3×C8—21no—
21C2×#3719—21no—
22#63831—22no—
23#63831C414yescomplemented
24C2×#3716—21no—
25#63651—22no—
26#63567—21no—
27#63567—21no—
28#63565—21no—
29#63562—21no—
30C2×C3×#131—31no—
31C2×#951—31no—
32C2×#909—31no—
33C2×#907D311yescharacteristic
34#5232—31no—
35C2×#824—31no—
36C2×#824—31no—
37C2×#824—61no—
38C2×#816—31no—
39C2×#810—31no—
40#4966—31no—
41C3×#343—31no—
42#4558—31no—
43#4554D311yescharacteristic, complemented
44#4552—31no—
45C22×C32×C4C2311yescharacteristic
46C22×C3⋊Dic3—21no—
47C2×#634—21no—
48C2×#634—21no—
49C2×#634—21no—
50C2×#634—21no—
51C2×#634—21no—
52C2×#634C2×C412yes—
53C2×C3×#67—21no—
54C22×C3×Dic3—21no—
55C3×#235—21no—
56C3×#235—21no—
57C3×#235C2×C412yes—
58C3×#235D412yes—
59C2×C3×D12D411yescharacteristic
60C2×C3×C4×D3—21no—
61C2×C3×Dic6D411yescharacteristic
62C32×M16—21no—
63C32×M16D412yes—
64C2×C32×C8—21no—
65#3748—21no—
66#3748—21no—
67C2×C3⋊(C3⋊C8)—21no—
68C3×C4⋊Dic3—21no—
69C3×C4⋊Dic3—21no—
70C3×#208—21no—
71C3×#208C2×C412yes—
72C2×C3×C3⋊C8—21no—
73C2×C3×C3⋊C8—21no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8D411yescharacteristic
76C2×C3×C3⋊C8D411yescharacteristic
77C3×ΓL(1,𝔽25)—24no—
78C3×D3×C8—24no—
79#3719—24no—
80#3716—24no—
81#1908—91noSylow
82C2×C3×#41—31no—
83C2×#235C2×D311yescharacteristic
84C22×Dic6—31no—
85C22×Dic6—31no—
86C22×Dic6—61no—
87C3×#131—62no—
88C3×#131—62no—
89C2×C3×M16—21no—
90C2×C3×M16—31no—
91C2×C3×M16—61no—
92C2×C3×M16C2×D311yescharacteristic
93C2×C3×M16C2×D311yescharacteristic
94C22×C3×C8—31no—
95C22×C3×C8—61no—
96C2×C3×C4⋊C4—31no—
97#951—34no—
98#951—62no—
99C2×#216—61no—
100C2×#214—61no—
101C2×C4⋊Dic3C2×D311yescharacteristic
102C2×#208—31no—
103C2×#208—31no—
104C2×#208—61no—
105C2×#208C2×D311yescharacteristic
106C22×C3⋊C8C2×D311yescharacteristic
107#909—34no—
108#909—62no—
109#907—22no—
110#907—22no—
111C2×Dic12—61no—
112C2×#204—61no—
113C2×ΓL(1,𝔽25)—21no—
114C2×D3×C8—21no—
115#835—61no—
116#832—61no—
117#824—64no—
118#824—62no—
119#824—62no—
120#816—61no—
121#816—61no—
122#810—61no—
123#810—61no—
124#810—61no—
125#809—61no—
126C23×C32#4111yescharacteristic
127C22×C3×D3—21no—
128C2×C32×C4C22×C411yescharacteristic
129C2×C32×C4C2×D411yescharacteristic
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4#4111yescharacteristic
134C2×C3⋊Dic3—22no—
135C2×C3⋊Dic3—41no—
136C2×C3⋊Dic3—41no—
137#634—22no—
138#634—22no—
139#634—41no—
140#634—41no—
141C3×#67—22no—
142C3×#67—22no—
143C2×C3×Dic3—21no—
144C2×C3×Dic3—41no—
145C2×C3×Dic3—81no—
146C3×D12—21no—
147C3×D12C22⋊C412yes—
148C3×C4×D3—22no—
149C3×C4×D3—22no—
150C3×Dic6—21no—
151C3×Dic6C22⋊C412yes—
152C32×C8—22no—
153C3⋊(C3⋊C8)—41no—
154C3×C3⋊C8—22no—
155C3×C3⋊C8—21no—
156C3×C3⋊C8—21no—
157C2×#137—91no—
158C2×#131—31no—
159#417—91no—
160C2×#103—91no—
161C2×#101—91no—
162#343—31no—
163#341—91no—
164C3×#41—32no—
165C3×#41—32no—
166C3×#41—61no—
167C3×#41—61no—
168C2×C3×Q8—31no—
169C2×C3×D4—31no—
170C2×C3×D4—61no—
171C22×C3×C4—21no—
172C22×C3×C4—61no—
173C22×C3×C4—61no—
174C22×C3×C4C22×D311yescharacteristic
175C22×C3×C4C22×D311yescharacteristic
176C2×#67—21no—
177C22×Dic3—21no—
178C22×Dic3—62no—
179C22×Dic3—61no—
180C22×Dic3—61no—
181#235—21no—
182#235—21no—
183#235C4×D312yes—
184#235#6712yes—
185C2×D12D1211yescharacteristic
186C2×C4×D3—21no—
187C2×Dic6—32no—
188C2×Dic6—32no—
189C2×Dic6—62no—
190C2×Dic6—61no—
191C2×Dic6—61no—
192C2×Dic6—61no—
193C2×Dic6—61no—
194C2×Dic6—61no—
195C2×Dic6—61no—
196C2×Dic6—61no—
197C2×Dic6—61no—
198C2×Dic6—62no—
199C2×Dic6—61no—
200C2×Dic6—61no—
201C2×Dic6—121no—
202C2×Dic6—121no—
203C2×Dic6—121no—
204C2×Dic6—121no—
205C2×Dic6D1211yescharacteristic
206C3×M16—21no—
207C3×M16—21no—
208C3×M16—22no—
209C3×M16—32no—
210C3×M16—41no—
211C3×M16—64no—
212C3×M16—61no—
213C3×M16D1212yes—
214C3×M16#6712yes—
215C2×C3×C8—21no—
216C2×C3×C8—21no—
217C2×C3×C8—31no—
218C2×C3×C8—31no—
219C2×C3×C8—41no—
220C2×C3×C8—64no—
221C2×C3×C8—61no—
222C2×C3×C8—61no—
223C2×C3×C8—61no—
224C3×C4⋊C4—61no—
225C3×C4⋊C4—61no—
226#216—64no—
227#214—64no—
228C4⋊Dic3—21no—
229C4⋊Dic3—21no—
230#208—21no—
231#208—62no—
232#208—61no—
233#208—61no—
234#208—61no—
235#208—61no—
236#208—121no—
237#208C4×D312yes—
238C2×C3⋊C8—21no—
239C2×C3⋊C8—21no—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—61no—
242C2×C3⋊C8—61no—
243C2×C3⋊C8—121no—
244C2×C3⋊C8#6711yescharacteristic
245C2×C3⋊C8#6711yescharacteristic
246Dic12—64no—
247#204—64no—
248ΓL(1,𝔽25)—24no—
249D3×C8—24no—
250C22×C32—21no—
251C22×C32—21no—
252C22×C32C4×D411yescharacteristic
253C22×C32#12111yescharacteristic
254C22×C32#12311yescharacteristic
255C2×C3×D3—22no—
256C2×C3×D3—41no—
257C2×C3×D3—41no—
258C32×C4C2×C22⋊C411yescharacteristic
259C32×C4C4×D411yescharacteristic
260C32×C4#12011yescharacteristic
261C32×C4#12111yescharacteristic
262C3⋊Dic3—42no—
263C3×Dic3—22no—
264C3×Dic3—81no—
265C2×#41—31no—
266C22×Q8—91no—
267#137—94no—
268#137—182no—
269C2×Q16—181no—
270C2×SD16—181no—
271#131—62no—
272#131—62no—
273C2×M16—31no—
274C2×M16—61no—
275C2×M16—91no—
276C2×M16D3211yescharacteristic
277C22×C8—31no—
278C22×C8—61no—
279C2×C4⋊C4—31no—
280#103—181no—
281#103—181no—
282#103—181no—
283#102—181no—
284#101—182no—
285C23×C3—21no—
286C23×C3—61no—
287C23×C3#23511yescharacteristic
288C23×C3#23511yescharacteristic
289C22×D3—21no—
290C3×Q8—32no—
291C3×Q8—61no—
292C3×D4—32no—
293C3×D4—62no—
294C3×D4—62no—
295C3×D4—61no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—61no—
303C2×C3×C4—62no—
304C2×C3×C4—62no—
305C2×C3×C4—121no—
306C2×C3×C4—241no—
307C2×C3×C4C2×C4×D311yescharacteristic
308C2×C3×C4C2×C4×D311yescharacteristic
309C2×C3×C4C2×D1211yescharacteristic
310C2×C3×C4C2×D1211yescharacteristic
311C2×C3×C4#23511yescharacteristic
312C2×C3×C4D3×D411yescharacteristic
313C2×C3×C4D3×D411yescharacteristic
314C2×C3×C4D3×D411yescharacteristic
315C2×C3×C4#23911yescharacteristic
316C2×C3×C4C2×#6711yescharacteristic
317C2×C3×C4C2×#6711yescharacteristic
318C2×C3×C4C2×#6711yescharacteristic
319#67—22no—
320#67—22no—
321C2×Dic3—21no—
322C2×Dic3—41no—
323C2×Dic3—64no—
324C2×Dic3—62no—
325C2×Dic3—62no—
326C2×Dic3—81no—
327C2×Dic3—122no—
328C2×Dic3—122no—
329C2×Dic3—121no—
330C2×Dic3—121no—
331C2×Dic3—121no—
332C2×Dic3—121no—
333D12—21no—
334D12(C2×D3)⋊C412yes—
335C4×D3—22no—
336C4×D3—22no—
337Dic6—21no—
338Dic6—62no—
339Dic6—62no—
340Dic6—62no—
341Dic6—62no—
342Dic6—122no—
343Dic6—122no—
344Dic6—122no—
345Dic6—122no—
346Dic6—121no—
347Dic6—121no—
348Dic6—121no—
349Dic6—121no—
350Dic6(C2×D3)⋊C412yes—
351C3×C8—22no—
352C3×C8—22no—
353C3×C8—42no—
354C3×C8—61no—
355C3×C8—61no—
356C3×C8—62no—
357C3⋊C8—22no—
358C3⋊C8—21no—
359C3⋊C8—21no—
360C3⋊C8—122no—
361C3⋊C8—121no—
362C3⋊C8—121no—
363C2×C32—21no—
364C2×C32—21no—
365C2×C32#38611yescharacteristic
366C2×C32#47111yescharacteristic
367C2×C32#47311yescharacteristic
368C3×D3—42no—
369#41—32no—
370#41—32no—
371#41—61no—
372#41—61no—
373C2×Q8—31no—
374C2×Q8—92no—
375C2×Q8—181no—
376C2×Q8—181no—
377C2×Q8—181no—
378C2×Q8—181no—
379C2×Q8—181no—
380C2×D4—31no—
381C2×D4—61no—
382C22×C4—61no—
383C22×C4—61no—
384C22×C4—181no—
385C22×C4C2×D3211yescharacteristic
386Q16—184no—
387SD16—184no—
388M16—21no—
389M16—32no—
390M16—64no—
391M16—61no—
392M16—181no—
393M16—181no—
394M16#62612yes—
395C2×C8—21no—
396C2×C8—31no—
397C2×C8—31no—
398C2×C8—64no—
399C2×C8—61no—
400C2×C8—61no—
401C2×C8—61no—
402C2×C8—181no—
403C4⋊C4—61no—
404C4⋊C4—61no—
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—21no—
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—21no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3—62no—
415C22×C3—121no—
416C22×C3—121no—
417C22×C3C4×D1211yescharacteristic
418C22×C3#88411yescharacteristic
419C22×C3#89411yescharacteristic
420C22×C3C4×#6711yescharacteristic
421C22×C3#92911yescharacteristic
422C22×C3#93011yescharacteristic
423C2×D3—22no—
424C2×D3—41no—
425C2×D3—41no—
426C3×C4—21no—
427C3×C4—21no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—62no—
431C3×C4—241no—
432C3×C4D3×C22⋊C411yescharacteristic
433C3×C4#88211yescharacteristic
434C3×C4#89311yescharacteristic
435C3×C4#89511yescharacteristic
436C3×C4C2×(C2×D3)⋊C411yescharacteristic
437C3×C4C4×#6711yescharacteristic
438C3×C4#93011yescharacteristic
439C3×C4#95311yescharacteristic
440Dic3—22no—
441Dic3—81no—
442Dic3—124no—
443Dic3—122no—
444Dic3—122no—
445C32#190811yescharacteristic, complemented, Sylow
446C23—61no—
447C23#379811yescharacteristic
448Q8—32no—
449Q8—61no—
450Q8—182no—
451Q8—182no—
452Q8—361no—
453Q8—361no—
454D4—32no—
455D4—62no—
456D4—62no—
457D4—61no—
458C2×C4—61no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—121no—
462C2×C4—182no—
463C2×C4—241no—
464C2×C4—361no—
465C2×C4—361no—
466C2×C4#379911yescharacteristic
467C2×C4C4×D3211yescharacteristic
468C2×C4D3×D1211yescharacteristic
469C2×C4C2×#62611yescharacteristic
470C2×C4D3×#6711yescharacteristic
471C2×C4D3×#6711yescharacteristic
472C8—22no—
473C8—62no—
474C8—61no—
475C8—61no—
476C8—361no—
477C2×C3—21no—
478C2×C3—21no—
479C2×C3—21no—
480C2×C3—21no—
481C2×C3—21no—
482C2×C3—21no—
483C2×C3—21no—
484C2×C3—41no—
485C2×C3—41no—
486C2×C3—122no—
487C2×C3#466511yescharacteristic
488C2×C3#474811yescharacteristic
489C2×C3#475511yescharacteristic
490C2×C3#514111yescharacteristic
491C2×C3#514311yescharacteristic
492C2×C3#521311yescharacteristic
493D3—42no—
494C22—21no—
495C22—21no—
496C22—62no—
497C22—121no—
498C22—121no—
499C22#6385011yescharacteristic
500C22C4×#62611yescharacteristic
501C22#6390611yescharacteristic
502C4—62no—
503C4—241no—
504C4—362no—
505C4#6390411yescharacteristic
506C4#6390911yescharacteristic
507C4D3×(C2×D3)⋊C411yescharacteristic
508C4#6397711yescharacteristic
509C3—21no—
510C3#6900711yescharacteristic, complemented
511C3#7201911yescharacteristic, complemented
512C2—21no—
513C2—21no—
514C2—122no—
515C2#1059099611yescharacteristic
516C2#1059099911yescharacteristic
517C2#1059138211yescharacteristic
subgroupG/H
centerC22#63906
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32#121
prime radicalC2×C3×#235C22
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1908
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
3. C2×#63651C2semidirect
4. C2×#63567C2semidirect
6. C3×#4554C2semidirect
7. #10589166C2semidirect
23. #63831C4semidirect
43. #4554D3semidirect
445. C32#1908semidirect
510. C3#69007semidirect
511. C3#72019semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#9072
C2×#638312
C2×#636512
C2×#635672
#105910872
C3×#45542
#105891662
#720193
#690073