Finite Group Databasebeta

← #11785491

Subgroups — #11785491

Order 1152.

599 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63817C211yesmaximal, characteristic
2#10592705C211yesmaximal, characteristic
3C3×#4998C211yesmaximal, characteristic, complemented
4#10591232C211yesmaximal, characteristic
5#10591122C211yesmaximal, characteristic, complemented
6C3×#4531C211yesmaximal, characteristic, complemented
7#10589158C211yesmaximal, characteristic, complemented
8#70664—31nomaximal
9#68710—31nomaximal
10C2×#3828C2211yescharacteristic
11C32×#117C2211yescharacteristic
12#64135C2211yescharacteristic
13C3×#926C2211yescharacteristic
14C3×#926C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C2×#3712C412yescomplemented
18#63817—21no—
19#63817—21no—
20#63817—21no—
21#63817—21no—
22C2×#3710C412yescomplemented
23#63633—22no—
24C3×#808—21no—
25C3×#807—21no—
26#63581—22no—
27#63575—21no—
28#63573—21no—
29C2×#907—31no—
30C2×#901—31no—
31C3×#427—31no—
32#5106—31no—
33#5006—31no—
34#5006—31no—
35#5006—31no—
36#5006—31no—
37#5006—61no—
38#4998—31no—
39#4998D311yescharacteristic, complemented
40C3×H(3,ℤ/4)—31no—
41#4557—31no—
42#4531—31no—
43#4531D311yescharacteristic, complemented
44C22×C32×C4C2311yescharacteristic
45C2×#638D412yes—
46#3828—21no—
47#3828—21no—
48#3828—21no—
49#3828—21no—
50C2×C4⋊(C3⋊D3)Q811yescharacteristic
51C2×C4×C3⋊D3C2×C412yes—
52C2×#634Q811yescharacteristic
53C32×C4⋊C4—21no—
54C32×C22⋊C4—21no—
55C32×C42—21no—
56#3757—21no—
57C4⋊(C3⋊Dic3)—21no—
58C4×C3⋊Dic3—21no—
59C3×#217—21no—
60C3×#217—21no—
61C3×C4⋊Dic3—21no—
62C3×C4⋊Dic3—21no—
63C3×C4×Dic3—21no—
64C3×C4×Dic3—21no—
65C3×#208—21no—
66C3×#208—21no—
67C2×C3×C3⋊C8C2×C412yes—
68C2×C3×C3⋊C8C2×C411yescharacteristic
69C2×C3×C3⋊C8C2×C411yescharacteristic
70C2×C3×C3⋊C8D411yescharacteristic
71C2×C3×C3⋊C8D411yescharacteristic
72C2×C3×C3⋊C8D411yescharacteristic
73C2×C3×C3⋊C8D411yescharacteristic
74#3712—22no—
75#3712—22no—
76#3710—22no—
77#3710—22no—
78#1812—91noSylow
79C2×#235—31no—
80C2×#235—31no—
81C2×#235—61no—
82C2×C3×M16—31no—
83C22×C3×C8—31no—
84C3×#117—21no—
85C3×#117—31no—
86C3×#117—31no—
87C3×#117C2×D311yescharacteristic
88C3×#117C2×D311yescharacteristic
89#926—31no—
90#926—31no—
91#926—61no—
92#926C2×D311yescharacteristic
93#926C2×D311yescharacteristic
94C2×#208C2×D311yescharacteristic
95C22×C3⋊C8C2×D311yescharacteristic
96#907—61no—
97#907—61no—
98#907—61no—
99#907—61no—
100#901—61no—
101#901—61no—
102#901—61no—
103#901—61no—
104C2×ΓL(1,𝔽25)—32no—
105C2×ΓL(1,𝔽25)—32no—
106C2×D3×C8—32no—
107C2×D3×C8—32no—
108C3×#107—61no—
109C3×#106—61no—
110#818—61no—
111#817—61no—
112#808—21no—
113#808—61no—
114#807—21no—
115#807—61no—
116#806—62no—
117#806—62no—
118#806—62no—
119#806—62no—
120#806—64no—
121C23×C32C2×D411yescharacteristic
122C22×C3⋊D3C4⋊C412yes—
123C2×C32×C4—41no—
124C2×C32×C4C22×C411yescharacteristic
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4C2×Q811yescharacteristic
128C2×C32×C4#4111yescharacteristic
129C2×C32×C4#4111yescharacteristic
130#638—22no—
131#638—22no—
132C2×C3⋊Dic3—41no—
133C2×C3⋊Dic3C4⋊C412yes—
134C4⋊(C3⋊D3)—21no—
135C4⋊(C3⋊D3)—21no—
136C4×C3⋊D3—22no—
137C4×C3⋊D3—22no—
138#634—21no—
139#634—21no—
140C2×C3×Dic3—41no—
141C2×C3×Dic3—41no—
142C3×C3⋊C8—22no—
143C3×C3⋊C8C22⋊C411yescharacteristic
144C3×C3⋊C8C22⋊C411yescharacteristic
145C3×C3⋊C8C22⋊C411yescharacteristic
146C3×C3⋊C8C22⋊C411yescharacteristic
147C2×#131—91no—
148#427—31no—
149#427—91no—
150#410—91no—
151#410—91no—
152H(3,ℤ/4)—31no—
153H(3,ℤ/4)—91no—
154C22×C3×C4—21no—
155C22×C3×C4C22×D311yescharacteristic
156C22×C3×C4C22×D311yescharacteristic
157C2×#67—32no—
158C2×#67—32no—
159C2×#67—62no—
160#235—61no—
161#235—61no—
162#235—61no—
163#235—61no—
164#235—61no—
165#235—61no—
166#235—61no—
167#235—61no—
168#235—121no—
169#235—121no—
170#235—121no—
171#235—121no—
172C2×D12—31no—
173C2×D12—31no—
174C2×D12—61no—
175C2×C4×D3—32no—
176C2×C4×D3—32no—
177C2×C4×D3—62no—
178C2×Dic6—31no—
179C2×Dic6—31no—
180C2×Dic6—61no—
181C3×M16—61no—
182C3×M16—61no—
183C2×C3×C8—31no—
184C2×C3×C8—31no—
185C2×C3×C8—31no—
186C2×C3×C8—31no—
187C2×C3×C8—31no—
188C2×C3×C8—31no—
189C2×C3×C8—32no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—41no—
193C3×C4⋊C4—61no—
194C3×C4⋊C4—61no—
195C3×C22⋊C4—22no—
196C3×C22⋊C4—21no—
197C3×C22⋊C4—21no—
198C3×C22⋊C4—61no—
199C3×C22⋊C4—61no—
200C3×C42—21no—
201C3×C42—21no—
202C3×C42—41no—
203C3×C42—61no—
204C3×C42—61no—
205#217—21no—
206#217—21no—
207#217—62no—
208#217—61no—
209#217—61no—
210C4⋊Dic3—21no—
211C4⋊Dic3—21no—
212C4⋊Dic3—61no—
213C4⋊Dic3—61no—
214C4⋊Dic3—121no—
215C4×Dic3—21no—
216C4×Dic3—21no—
217C4×Dic3—61no—
218C4×Dic3—61no—
219C4×Dic3—121no—
220#208—21no—
221#208—21no—
222C2×C3⋊C8C4×D312yes—
223C2×C3⋊C8C4×D311yescharacteristic
224C2×C3⋊C8C4×D311yescharacteristic
225C2×C3⋊C8D1211yescharacteristic
226C2×C3⋊C8D1211yescharacteristic
227C2×C3⋊C8#6711yescharacteristic
228C2×C3⋊C8#6711yescharacteristic
229ΓL(1,𝔽25)—62no—
230ΓL(1,𝔽25)—62no—
231ΓL(1,𝔽25)—62no—
232ΓL(1,𝔽25)—62no—
233D3×C8—62no—
234D3×C8—62no—
235D3×C8—62no—
236D3×C8—62no—
237C22×C32—41no—
238C22×C32C2×C4⋊C411yescharacteristic
239C22×C32#12111yescharacteristic
240C22×C32#12111yescharacteristic
241C2×C3⋊D3—22no—
242C2×C3⋊D3—42no—
243C32×C4—81no—
244C32×C4C2×C22⋊C411yescharacteristic
245C32×C4#11711yescharacteristic
246C32×C4C22⋊Q811yescharacteristic
247C32×C4C22⋊Q811yescharacteristic
248C3⋊Dic3—22no—
249C3⋊Dic3—81no—
250C3×Dic3—81no—
251C3×Dic3—81no—
252C2×#41—91no—
253#131—181no—
254#131—181no—
255#131—181no—
256#131—181no—
257C2×M16—31no—
258C2×M16—92no—
259C22×C8—31no—
260C22×C8—92no—
261#117—31no—
262#117—31no—
263#117—91no—
264#117D3211yescharacteristic
265#107—61no—
266#107—181no—
267#106—61no—
268#106—181no—
269#99—182no—
270#99—182no—
271C23×C3—21no—
272C23×C3D3×D411yescharacteristic
273C23×C3D3×D411yescharacteristic
274C22×D3—32no—
275C22×D3—32no—
276C22×D3—62no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—42no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—121no—
287C2×C3×C4—121no—
288C2×C3×C4C2×C4×D311yescharacteristic
289C2×C3×C4C2×C4×D311yescharacteristic
290C2×C3×C4C2×D1211yescharacteristic
291C2×C3×C4#23511yescharacteristic
292C2×C3×C4#23511yescharacteristic
293C2×C3×C4D3×D411yescharacteristic
294C2×C3×C4D3×D411yescharacteristic
295C2×C3×C4#23711yescharacteristic
296C2×C3×C4D3×Q811yescharacteristic
297C2×C3×C4D3×Q811yescharacteristic
298C2×C3×C4#23911yescharacteristic
299C2×C3×C4C2×#6711yescharacteristic
300#67—62no—
301#67—62no—
302#67—62no—
303#67—62no—
304#67—122no—
305#67—122no—
306C2×Dic3—32no—
307C2×Dic3—32no—
308C2×Dic3—41no—
309C2×Dic3—41no—
310C2×Dic3—62no—
311C2×Dic3—122no—
312C2×Dic3—121no—
313C2×Dic3—121no—
314D12—61no—
315D12—61no—
316D12—61no—
317D12—61no—
318D12—121no—
319D12—121no—
320C4×D3—62no—
321C4×D3—62no—
322C4×D3—62no—
323C4×D3—62no—
324C4×D3—122no—
325C4×D3—122no—
326Dic6—61no—
327Dic6—61no—
328Dic6—61no—
329Dic6—61no—
330Dic6—121no—
331Dic6—121no—
332C3×C8—31no—
333C3×C8—31no—
334C3×C8—31no—
335C3×C8—31no—
336C3×C8—62no—
337C3⋊C8—22no—
338C3⋊C8(C2×D3)⋊C411yescharacteristic
339C3⋊C8(C2×D3)⋊C411yescharacteristic
340C3⋊C8(C2×D3)⋊C411yescharacteristic
341C3⋊C8(C2×D3)⋊C411yescharacteristic
342C2×C32—21no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32#38011yescharacteristic
346C3⋊D3—42no—
347#41—181no—
348#41—181no—
349#41—181no—
350#41—181no—
351C2×Q8—91no—
352C2×D4—92no—
353C2×D4—91no—
354C22×C4—92no—
355C22×C4C2×D3211yescharacteristic
356M16—61no—
357M16—61no—
358M16—182no—
359M16—182no—
360C2×C8—31no—
361C2×C8—31no—
362C2×C8—31no—
363C2×C8—31no—
364C2×C8—31no—
365C2×C8—31no—
366C2×C8—32no—
367C2×C8—182no—
368C2×C8—182no—
369C4⋊C4—21no—
370C4⋊C4—61no—
371C4⋊C4—61no—
372C4⋊C4—181no—
373C22⋊C4—21no—
374C22⋊C4—61no—
375C22⋊C4—61no—
376C22⋊C4—181no—
377C42—21no—
378C42—61no—
379C42—61no—
380C42—181no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—41no—
385C22×C3—41no—
386C22×C3—81no—
387C22×C3#88411yescharacteristic
388C22×C3#88411yescharacteristic
389C22×C3D3×C4⋊C411yescharacteristic
390C22×C3D3×C4⋊C411yescharacteristic
391C22×C3#89511yescharacteristic
392C22×C3#93811yescharacteristic
393C2×D3—62no—
394C2×D3—62no—
395C2×D3—122no—
396C2×D3—122no—
397C2×D3—122no—
398C2×D3—242no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—21no—
403C3×C4—81no—
404C3×C4—81no—
405C3×C4—82no—
406C3×C4—241no—
407C3×C4—241no—
408C3×C4#87211yescharacteristic
409C3×C4D3×C22⋊C411yescharacteristic
410C3×C4#89211yescharacteristic
411C3×C4#89611yescharacteristic
412C3×C4#89611yescharacteristic
413C3×C4#89711yescharacteristic
414C3×C4C2×(C2×D3)⋊C411yescharacteristic
415C3×C4#94611yescharacteristic
416Dic3—62no—
417Dic3—62no—
418Dic3—81no—
419Dic3—81no—
420Dic3—122no—
421Dic3—241no—
422Dic3—241no—
423Dic3—242no—
424C32#181211yescharacteristic, complemented, Sylow
425C23—92no—
426C23#381111yescharacteristic
427Q8—181no—
428Q8—181no—
429D4—182no—
430D4—182no—
431D4—181no—
432D4—181no—
433C2×C4—41no—
434C2×C4—92no—
435C2×C4—121no—
436C2×C4—121no—
437C2×C4—182no—
438C2×C4—182no—
439C2×C4—361no—
440C2×C4#379711yescharacteristic
441C2×C4#379911yescharacteristic
442C2×C4C4×D3211yescharacteristic
443C2×C4D3×D1211yescharacteristic
444C2×C4#380411yescharacteristic
445C2×C4D3×#6711yescharacteristic
446C8—31no—
447C8—31no—
448C8—31no—
449C8—31no—
450C8—62no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—21no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—41no—
459C2×C3—41no—
460C2×C3—41no—
461C2×C3#466711yescharacteristic
462C2×C3#498711yescharacteristic
463D3—122no—
464D3—122no—
465D3—242no—
466C22—41no—
467C22—182no—
468C22—362no—
469C22#6388311yescharacteristic
470C22#6391011yescharacteristic
471C22#6395811yescharacteristic
472C4—81no—
473C4—182no—
474C4—241no—
475C4—241no—
476C4—721no—
477C4#6383611yescharacteristic
478C4#6388811yescharacteristic
479C4#6389511yescharacteristic
480C4D3×(C2×D3)⋊C411yescharacteristic
481C3—21no—
482C3#6871011yescharacteristic, complemented
483C3#7066411yescharacteristic, complemented
484C2—21no—
485C2—21no—
486C2—21no—
487C2—362no—
488C2#1059136711yescharacteristic
subgroupG/H
centerC4#63836
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC32×#117C22
socleC2×C32#380
prime radicalC2×#3828C22
cosocleC22×C32×C4C23
hypercenter#117D32
hypocenterC32#1812
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. C3×#4998C2semidirect
5. #10591122C2semidirect
6. C3×#4531C2semidirect
7. #10589158C2semidirect
17. C2×#3712C4semidirect
22. C2×#3710C4semidirect
39. #4998D3semidirect
43. #4531D3semidirect
424. C32#1812semidirect
482. C3#68710semidirect
483. C3#70664semidirect

Maximal subgroups

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

subgroupindex
C2×#638172
#105927052
C3×#49982
#105912322
#105911222
C3×#45312
#105891582
#706643
#687103