Finite Group Databasebeta

← #11788916

Subgroups — #11788916

Order 1152.

691 conjugacy classes of subgroups.

The subgroup lattice has 493 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10594866C211yesmaximal, characteristic
2#10594780C211yesmaximal, characteristic
3#10592831C211yesmaximal, characteristic
4C2×C3×#824C211yesmaximal, characteristic, complemented
5C2×#63560C211yesmaximal, characteristic, complemented
6C3×#4556C211yesmaximal, characteristic, complemented
7#10589170C211yesmaximal, characteristic, complemented
8#70124—31nomaximal
9#69041—31nomaximal
10C2×#3828C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C3×#926C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17#63966C414yes—
18#63871—22no—
19#63866—21no—
20#63864—21no—
21#63818—22no—
22#63817—22no—
23#63817—22no—
24#63651—21no—
25#63650—21no—
26C3×#824—22no—
27#63560—22no—
28#5816—31no—
29#5745—31no—
30#5745—31no—
31#5526—31no—
32C2×C3×#101—31no—
33C2×#836—31no—
34C2×#824—31no—
35C2×#824D311yescharacteristic, complemented
36#5132—31no—
37#5132—31no—
38#5132—61no—
39C3×#344—31no—
40#4559—31no—
41#4556D311yescharacteristic, complemented
42#4533—31no—
43C22×C32×C4C2311yescharacteristic
44C2×#638C2×C412yes—
45#3828—21no—
46#3828—21no—
47#3828—21no—
48#3828—21no—
49C2×C4⋊(C3⋊D3)D411yescharacteristic
50C2×C4×C3⋊D3D412yes—
51C2×#634D411yescharacteristic
52C22×C3×Dic3C2×C412yes—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6—21no—
56C2×C3×Dic6—21no—
57C2×C3×Dic6—21no—
58C2×C3×Dic6—21no—
59C32×M16—21no—
60C32×M16—21no—
61C2×C32×C8—21no—
62#3748—21no—
63#3748—21no—
64C2×C3⋊(C3⋊C8)—21no—
65C3×#217C2×C412yes—
66C3×C4⋊Dic3—21no—
67C3×C4×Dic3—21no—
68C3×#208—21no—
69C3×#208—21no—
70C3×#208D412yes—
71C3×#208D412yes—
72C2×C3×C3⋊C8—21no—
73C2×C3×C3⋊C8—21no—
74Dic3⋊Dic3—24no—
75#3722—24no—
76#3712—24no—
77#3710—24no—
78#1915—91noSylow
79C22×C3×Q8—31no—
80C2×#235—31no—
81C2×#235—31no—
82C2×#235—61no—
83C22×Dic6C2×D311yescharacteristic
84C2×C3×M16—21no—
85C2×C3×M16—31no—
86C2×C3×M16—31no—
87C2×C3×M16C2×D311yescharacteristic
88C2×C3×M16C2×D311yescharacteristic
89C3×#117—31no—
90#947—61no—
91#946—62no—
92#944—61no—
93#929—34no—
94#926C2×D311yescharacteristic
95C2×#208—31no—
96C2×#208—31no—
97C2×#208—61no—
98C2×#208C2×D311yescharacteristic
99C2×#208C2×D311yescharacteristic
100#907—62no—
101#907—62no—
102D3×M16—62no—
103D3×M16—62no—
104#901—62no—
105#901—62no—
106#897—62no—
107#886—34no—
108#875—61no—
109#869—61no—
110C3×#101—62no—
111#836—62no—
112#824—22no—
113#824—61no—
114#824—61no—
115#824—62no—
116#824—62no—
117#823—61no—
118#823—61no—
119#823—62no—
120C23×C32C22×C411yescharacteristic
121C22×C3⋊D3C22⋊C412yes—
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4#4111yescharacteristic
127C2×C32×C4#4111yescharacteristic
128#638—22no—
129#638—22no—
130C2×C3⋊Dic3C22⋊C412yes—
131C4⋊(C3⋊D3)—21no—
132C4⋊(C3⋊D3)—21no—
133C4×C3⋊D3—22no—
134C4×C3⋊D3—22no—
135#634—21no—
136#634—21no—
137C2×C3×Dic3—22no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—22no—
141C3×Dic6—41no—
142C3×Dic6—41no—
143C3×Dic6—41no—
144C3×Dic6—41no—
145C32×C8—41no—
146C3⋊(C3⋊C8)—41no—
147C3×C3⋊C8—22no—
148C3×C3⋊C8—22no—
149#568—91no—
150#536—91no—
151#413—91no—
152C2×#101—31no—
153C2×#101—91no—
154#344—31no—
155#344—91no—
156C2×C3×Q8—61no—
157C2×C3×Q8—61no—
158C2×C3×Q8—61no—
159C2×C3×Q8—61no—
160C2×C3×Q8—61no—
161C2×C3×Q8—61no—
162C22×C3×C4—21no—
163C22×C3×C4—32no—
164C22×C3×C4C22×D311yescharacteristic
165C22×C3×C4C22×D311yescharacteristic
166C2×#67—32no—
167C2×#67—32no—
168C2×#67—62no—
169C22×Dic3C4×D312yes—
170#235—61no—
171#235—61no—
172#235—61no—
173#235—61no—
174#235—61no—
175#235—61no—
176#235—61no—
177#235—61no—
178#235—121no—
179#235—121no—
180#235—121no—
181#235—121no—
182C2×D12—31no—
183C2×D12—31no—
184C2×D12—61no—
185C2×C4×D3—32no—
186C2×C4×D3—32no—
187C2×C4×D3—62no—
188C2×Dic6—21no—
189C2×Dic6—21no—
190C2×Dic6—21no—
191C2×Dic6—21no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6—31no—
195C2×Dic6—31no—
196C2×Dic6—61no—
197C3×M16—22no—
198C3×M16—21no—
199C3×M16—21no—
200C3×M16—21no—
201C3×M16—21no—
202C3×M16—32no—
203C3×M16—32no—
204C3×M16—41no—
205C3×M16—61no—
206C3×M16—61no—
207C2×C3×C8—21no—
208C2×C3×C8—21no—
209C2×C3×C8—41no—
210C2×C3×C8—61no—
211C2×C3×C8—61no—
212C3×C4⋊C4—61no—
213C3×C22⋊C4—32no—
214C3×C42—61no—
215#217C4×D312yes—
216(C2×D3)⋊C4—64no—
217(C2×D3)⋊C4—64no—
218C4⋊Dic3—21no—
219C4⋊Dic3—64no—
220Dic3⋊C4—64no—
221C4×Dic3—21no—
222#208—21no—
223#208—21no—
224#208—62no—
225#208—61no—
226#208—61no—
227#208—61no—
228#208—61no—
229#208—121no—
230#208D1212yes—
231#208#6712yes—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—61no—
235C2×C3⋊C8—61no—
236C2×C3⋊C8—121no—
237ΓL(1,𝔽25)—64no—
238ΓL(1,𝔽25)—64no—
239D3×C8—64no—
240D3×C8—64no—
241C22×C32—41no—
242C22×C32C2×C22⋊C411yescharacteristic
243C22×C32C4×D411yescharacteristic
244C22×C32C4×D411yescharacteristic
245C2×C3⋊D3—22no—
246C2×C3⋊D3—42no—
247C32×C4#12011yescharacteristic
248C32×C4#12111yescharacteristic
249C32×C4#12111yescharacteristic
250C32×C4#12311yescharacteristic
251C3⋊Dic3—22no—
252C3×Dic3—42no—
253C3×Dic3—42no—
254C2×#41—91no—
255C22×Q8—31no—
256#131—182no—
257#131—182no—
258C2×M16—31no—
259C2×M16—31no—
260C2×M16—91no—
261C2×M16—182no—
262C2×M16D3211yescharacteristic
263C4⋊Q8—181no—
264#124—181no—
265#123—94no—
266C22⋊Q8—182no—
267#117—31no—
268#101—62no—
269#101—181no—
270#101—182no—
271#100—181no—
272C23×C3—21no—
273C23×C3C2×C4×D311yescharacteristic
274C23×C3C2×C4×D311yescharacteristic
275C22×D3—32no—
276C22×D3—32no—
277C22×D3—62no—
278C3×Q8—121no—
279C3×Q8—121no—
280C3×Q8—121no—
281C3×Q8—121no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—21no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—62no—
289C2×C3×C4—62no—
290C2×C3×C4—62no—
291C2×C3×C4—62no—
292C2×C3×C4C2×D1211yescharacteristic
293C2×C3×C4#23511yescharacteristic
294C2×C3×C4#23511yescharacteristic
295C2×C3×C4D3×D411yescharacteristic
296C2×C3×C4D3×D411yescharacteristic
297C2×C3×C4D3×D411yescharacteristic
298C2×C3×C4D3×D411yescharacteristic
299C2×C3×C4D3×D411yescharacteristic
300C2×C3×C4D3×D411yescharacteristic
301C2×C3×C4#23711yescharacteristic
302C2×C3×C4#23911yescharacteristic
303C2×C3×C4C2×#6711yescharacteristic
304#67—62no—
305#67—62no—
306#67—62no—
307#67—62no—
308#67—122no—
309#67—122no—
310C2×Dic3—22no—
311C2×Dic3—22no—
312C2×Dic3—22no—
313C2×Dic3—22no—
314C2×Dic3—32no—
315C2×Dic3—32no—
316C2×Dic3—62no—
317D12—61no—
318D12—61no—
319D12—61no—
320D12—61no—
321D12—121no—
322D12—121no—
323C4×D3—62no—
324C4×D3—62no—
325C4×D3—62no—
326C4×D3—62no—
327C4×D3—122no—
328C4×D3—122no—
329Dic6—41no—
330Dic6—41no—
331Dic6—41no—
332Dic6—41no—
333Dic6—61no—
334Dic6—61no—
335Dic6—61no—
336Dic6—61no—
337Dic6—121no—
338Dic6—121no—
339C3×C8—42no—
340C3×C8—41no—
341C3×C8—41no—
342C3×C8—62no—
343C3×C8—62no—
344C3⋊C8—22no—
345C3⋊C8—22no—
346C3⋊C8—122no—
347C3⋊C8—121no—
348C3⋊C8—121no—
349C2×C32—21no—
350C2×C32—21no—
351C2×C32—21no—
352C2×C32#38611yescharacteristic
353C3⋊D3—42no—
354#41—181no—
355#41—181no—
356#41—181no—
357#41—181no—
358C2×Q8—61no—
359C2×Q8—61no—
360C2×Q8—61no—
361C2×Q8—61no—
362C2×Q8—61no—
363C2×Q8—61no—
364C2×Q8—91no—
365C2×D4—92no—
366C2×D4—91no—
367C22×C4—32no—
368C22×C4—92no—
369C22×C4C2×D3211yescharacteristic
370M16—21no—
371M16—21no—
372M16—32no—
373M16—32no—
374M16—61no—
375M16—61no—
376M16—184no—
377M16—181no—
378M16—181no—
379C2×C8—21no—
380C2×C8—61no—
381C2×C8—61no—
382C2×C8—184no—
383C2×C8—181no—
384C4⋊C4—61no—
385C4⋊C4—184no—
386C22⋊C4—32no—
387C22⋊C4—184no—
388C42—61no—
389C22×C3—21no—
390C22×C3—21no—
391C22×C3—21no—
392C22×C3—41no—
393C22×C3—41no—
394C22×C3—81no—
395C22×C3C4×D1211yescharacteristic
396C22×C3D3×C22⋊C411yescharacteristic
397C22×C3D3×C22⋊C411yescharacteristic
398C22×C3#67⋊C411yescharacteristic
399C22×C3#89311yescharacteristic
400C22×C3C4×#6711yescharacteristic
401C2×D3—62no—
402C2×D3—62no—
403C2×D3—122no—
404C2×D3—122no—
405C2×D3—122no—
406C2×D3—242no—
407C3×C4—21no—
408C3×C4—21no—
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—122no—
412C3×C4—122no—
413C3×C4#88211yescharacteristic
414C3×C4#88311yescharacteristic
415C3×C4#88411yescharacteristic
416C3×C4#88411yescharacteristic
417C3×C4#89411yescharacteristic
418C3×C4#89511yescharacteristic
419C3×C4#93711yescharacteristic
420C3×C4#93811yescharacteristic
421Dic3—42no—
422Dic3—42no—
423Dic3—62no—
424Dic3—62no—
425Dic3—122no—
426C32#191511yescharacteristic, complemented, Sylow
427C23—92no—
428C23C4×D3211yescharacteristic
429Q8—121no—
430Q8—121no—
431Q8—121no—
432Q8—121no—
433Q8—181no—
434Q8—181no—
435D4—182no—
436D4—182no—
437D4—181no—
438D4—181no—
439C2×C4—62no—
440C2×C4—62no—
441C2×C4—62no—
442C2×C4—62no—
443C2×C4—92no—
444C2×C4—182no—
445C2×C4—182no—
446C2×C4#379911yescharacteristic
447C2×C4D3×D1211yescharacteristic
448C2×C4#380211yescharacteristic
449C2×C4#380411yescharacteristic
450C2×C4D3×#6711yescharacteristic
451C2×C4#381111yescharacteristic
452C8—41no—
453C8—62no—
454C8—62no—
455C8—361no—
456C2×C3—21no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—41no—
464C2×C3—41no—
465C2×C3—41no—
466C2×C3#466511yescharacteristic
467C2×C3#495611yescharacteristic
468D3—122no—
469D3—122no—
470D3—242no—
471C22—41no—
472C22—182no—
473C22—362no—
474C22#6388011yescharacteristic
475C22#6390411yescharacteristic
476C22#6392111yescharacteristic
477C4—122no—
478C4—122no—
479C4—182no—
480C4#6385411yescharacteristic
481C4#6391011yescharacteristic
482C4#6392211yescharacteristic
483C4#6395811yescharacteristic
484C3—21no—
485C3#6904111yescharacteristic, complemented
486C3#7012411yescharacteristic, complemented
487C2—21no—
488C2—21no—
489C2—21no—
490C2—362no—
491C2#1059127511yescharacteristic
subgroupG/H
centerC2#10591275
commutatorC23×C32C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC2×C32#386
prime radicalC2×#3828C22
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1915
LCSG ▷ C23×C32 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
4. C2×C3×#824C2semidirect
5. C2×#63560C2semidirect
6. C3×#4556C2semidirect
7. #10589170C2semidirect
35. C2×#824D3semidirect
41. #4556D3semidirect
426. C32#1915semidirect
485. C3#69041semidirect
486. C3#70124semidirect

Maximal subgroups

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

subgroupindex
#105948662
#105947802
#105928312
C2×C3×#8242
C2×#635602
C3×#45562
#105891702
#701243
#690413