Finite Group Databasebeta

← #11788919

Subgroups — #11788919

Order 1152.

745 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596216C211yesmaximal, characteristic
2#10594866C211yesmaximal, characteristic
3C2×C3×#824C211yesmaximal, characteristic, complemented
4C2×#63560C211yesmaximal, characteristic, complemented
5#10591078C211yesmaximal, characteristic
6C3×#4559C211yesmaximal, characteristic, complemented
7#10589169C211yesmaximal, characteristic, complemented
8#70124—31nomaximal
9#69041—31nomaximal
10C2×#3828C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14#64116—22no—
15M16×C3⋊D3—22no—
16#64110—22no—
17C3×#926C2211yescharacteristic
18C2×C3×#208C2211yescharacteristic
19C2×C3×#208C2211yescharacteristic
20#63966C414yes—
21#63871—22no—
22#63866—21no—
23#63864—21no—
24C3×#824—22no—
25#63561—21no—
26#63560—22no—
27#63559—21no—
28#5816—31no—
29#5745—31no—
30#5745—31no—
31#5745—61no—
32#5526—31no—
33C2×C3×#101—31no—
34C2×#836—31no—
35C2×#824—31no—
36C2×#824D311yescharacteristic, complemented
37#5132—31no—
38#5132—31no—
39C3×#344—31no—
40#4559D311yescharacteristic, complemented
41#4556—31no—
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×M16D412yes—
61C2×C32×C8—21no—
62#3748—21no—
63#3748D412yes—
64C2×C3⋊(C3⋊C8)—21no—
65#3743—24no—
66C8×C3⋊D3—24no—
67C3×#217C2×C412yes—
68C3×C4⋊Dic3—21no—
69C3×C4×Dic3—21no—
70C3×#208—21no—
71C3×#208—21no—
72C3×#208—21no—
73C3×#208—21no—
74C2×C3×C3⋊C8—21no—
75C2×C3×C3⋊C8—21no—
76Dic3⋊Dic3—24no—
77#3722—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—64no—
101#907—62no—
102#907—62no—
103D3×M16—62no—
104D3×M16—62no—
105D3×M16—64no—
106#901—62no—
107#901—62no—
108#901—64no—
109#897—62no—
110#886—34no—
111#875—61no—
112#869—61no—
113C3×#101—62no—
114#836—62no—
115#824—22no—
116#824—61no—
117#824—61no—
118#824—62no—
119#823—61no—
120#823—61no—
121C23×C32C22×C411yescharacteristic
122C22×C3⋊D3C22⋊C412yes—
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4C2×D411yescharacteristic
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4#4111yescharacteristic
128C2×C32×C4#4111yescharacteristic
129#638—22no—
130#638—22no—
131C2×C3⋊Dic3C22⋊C412yes—
132C4⋊(C3⋊D3)—21no—
133C4⋊(C3⋊D3)—21no—
134C4×C3⋊D3—22no—
135C4×C3⋊D3—22no—
136#634—21no—
137#634—21no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—22no—
141C2×C3×Dic3—22no—
142C3×Dic6—41no—
143C3×Dic6—41no—
144C3×Dic6—41no—
145C3×Dic6—41no—
146C32×C8—22no—
147C3⋊(C3⋊C8)—22no—
148C3×C3⋊C8—41no—
149C3×C3⋊C8—41no—
150#568—91no—
151#536—91no—
152#413—91no—
153C2×#101—31no—
154C2×#101—91no—
155#344—31no—
156#344—91no—
157C2×C3×Q8—61no—
158C2×C3×Q8—61no—
159C2×C3×Q8—61no—
160C2×C3×Q8—61no—
161C2×C3×Q8—61no—
162C2×C3×Q8—61no—
163C22×C3×C4—21no—
164C22×C3×C4—32no—
165C22×C3×C4C22×D311yescharacteristic
166C22×C3×C4C22×D311yescharacteristic
167C2×#67—32no—
168C2×#67—32no—
169C2×#67—62no—
170C22×Dic3C4×D312yes—
171#235—62no—
172#235—62no—
173#235—62no—
174#235—62no—
175#235—61no—
176#235—61no—
177#235—61no—
178#235—61no—
179#235—61no—
180#235—61no—
181#235—61no—
182#235—61no—
183C2×D12—31no—
184C2×D12—31no—
185C2×D12—61no—
186C2×C4×D3—32no—
187C2×C4×D3—32no—
188C2×C4×D3—62no—
189C2×Dic6—21no—
190C2×Dic6—21no—
191C2×Dic6—21no—
192C2×Dic6—21no—
193C2×Dic6—21no—
194C2×Dic6—21no—
195C2×Dic6—31no—
196C2×Dic6—31no—
197C2×Dic6—61no—
198C3×M16—22no—
199C3×M16—21no—
200C3×M16—21no—
201C3×M16—22no—
202C3×M16—61no—
203C3×M16—61no—
204C3×M16—61no—
205C3×M16—61no—
206C3×M16D1212yes—
207C3×M16#6712yes—
208C2×C3×C8—22no—
209C2×C3×C8—21no—
210C2×C3×C8—21no—
211C2×C3×C8—61no—
212C2×C3×C8—61no—
213C3×C4⋊C4—61no—
214C3×C22⋊C4—32no—
215C3×C42—61no—
216#217C4×D312yes—
217(C2×D3)⋊C4—64no—
218(C2×D3)⋊C4—64no—
219C4⋊Dic3—21no—
220C4⋊Dic3—64no—
221Dic3⋊C4—64no—
222C4×Dic3—21no—
223#208—21no—
224#208—21no—
225#208—21no—
226#208—21no—
227#208—32no—
228#208—32no—
229#208—62no—
230#208—62no—
231#208—61no—
232#208—61no—
233C2×C3⋊C8—21no—
234C2×C3⋊C8—21no—
235C2×C3⋊C8—62no—
236C2×C3⋊C8—61no—
237C2×C3⋊C8—61no—
238ΓL(1,𝔽25)—68no—
239ΓL(1,𝔽25)—64no—
240ΓL(1,𝔽25)—64no—
241D3×C8—68no—
242D3×C8—64no—
243D3×C8—64no—
244C22×C32—41no—
245C22×C32C2×C22⋊C411yescharacteristic
246C22×C32C4×D411yescharacteristic
247C22×C32C4×D411yescharacteristic
248C2×C3⋊D3—22no—
249C2×C3⋊D3—42no—
250C32×C4#12011yescharacteristic
251C32×C4#12111yescharacteristic
252C32×C4#12111yescharacteristic
253C32×C4#12311yescharacteristic
254C3⋊Dic3—22no—
255C3×Dic3—42no—
256C3×Dic3—42no—
257C2×#41—91no—
258C22×Q8—31no—
259#131—182no—
260#131—182no—
261C2×M16—31no—
262C2×M16—31no—
263C2×M16—91no—
264C2×M16—182no—
265C2×M16D3211yescharacteristic
266C4⋊Q8—181no—
267#124—181no—
268#123—94no—
269C22⋊Q8—182no—
270#117—31no—
271#101—62no—
272#101—181no—
273#101—182no—
274#100—181no—
275C23×C3—21no—
276C23×C3C2×C4×D311yescharacteristic
277C23×C3C2×C4×D311yescharacteristic
278C22×D3—32no—
279C22×D3—32no—
280C22×D3—62no—
281C3×Q8—121no—
282C3×Q8—121no—
283C3×Q8—121no—
284C3×Q8—121no—
285C2×C3×C4—21no—
286C2×C3×C4—21no—
287C2×C3×C4—21no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—21no—
291C2×C3×C4—62no—
292C2×C3×C4—62no—
293C2×C3×C4—62no—
294C2×C3×C4—62no—
295C2×C3×C4C2×D1211yescharacteristic
296C2×C3×C4#23511yescharacteristic
297C2×C3×C4#23511yescharacteristic
298C2×C3×C4D3×D411yescharacteristic
299C2×C3×C4D3×D411yescharacteristic
300C2×C3×C4D3×D411yescharacteristic
301C2×C3×C4D3×D411yescharacteristic
302C2×C3×C4D3×D411yescharacteristic
303C2×C3×C4D3×D411yescharacteristic
304C2×C3×C4#23711yescharacteristic
305C2×C3×C4#23911yescharacteristic
306C2×C3×C4C2×#6711yescharacteristic
307#67—64no—
308#67—64no—
309#67—62no—
310#67—62no—
311#67—62no—
312#67—62no—
313C2×Dic3—22no—
314C2×Dic3—22no—
315C2×Dic3—22no—
316C2×Dic3—22no—
317C2×Dic3—32no—
318C2×Dic3—32no—
319C2×Dic3—62no—
320D12—62no—
321D12—62no—
322D12—61no—
323D12—61no—
324D12—61no—
325D12—61no—
326C4×D3—64no—
327C4×D3—64no—
328C4×D3—62no—
329C4×D3—62no—
330C4×D3—62no—
331C4×D3—62no—
332Dic6—41no—
333Dic6—41no—
334Dic6—41no—
335Dic6—41no—
336Dic6—62no—
337Dic6—62no—
338Dic6—61no—
339Dic6—61no—
340Dic6—61no—
341Dic6—61no—
342C3×C8—24no—
343C3×C8—22no—
344C3×C8—22no—
345C3×C8—121no—
346C3×C8—121no—
347C3⋊C8—41no—
348C3⋊C8—41no—
349C3⋊C8—64no—
350C3⋊C8—62no—
351C3⋊C8—62no—
352C2×C32—21no—
353C2×C32—21no—
354C2×C32—21no—
355C2×C32#38611yescharacteristic
356C3⋊D3—42no—
357#41—181no—
358#41—181no—
359#41—181no—
360#41—181no—
361C2×Q8—61no—
362C2×Q8—61no—
363C2×Q8—61no—
364C2×Q8—61no—
365C2×Q8—61no—
366C2×Q8—61no—
367C2×Q8—91no—
368C2×D4—92no—
369C2×D4—91no—
370C22×C4—32no—
371C22×C4—92no—
372C22×C4C2×D3211yescharacteristic
373M16—21no—
374M16—61no—
375M16—61no—
376M16—61no—
377M16—61no—
378M16—92no—
379M16—184no—
380M16—181no—
381M16#62612yes—
382C2×C8—21no—
383C2×C8—61no—
384C2×C8—61no—
385C2×C8—184no—
386C2×C8—181no—
387C4⋊C4—61no—
388C4⋊C4—184no—
389C22⋊C4—32no—
390C22⋊C4—184no—
391C42—61no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—81no—
398C22×C3C4×D1211yescharacteristic
399C22×C3D3×C22⋊C411yescharacteristic
400C22×C3D3×C22⋊C411yescharacteristic
401C22×C3#67⋊C411yescharacteristic
402C22×C3#89311yescharacteristic
403C22×C3C4×#6711yescharacteristic
404C2×D3—64no—
405C2×D3—62no—
406C2×D3—62no—
407C2×D3—122no—
408C2×D3—122no—
409C2×D3—242no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—122no—
415C3×C4—122no—
416C3×C4#88211yescharacteristic
417C3×C4#88311yescharacteristic
418C3×C4#88411yescharacteristic
419C3×C4#88411yescharacteristic
420C3×C4#89411yescharacteristic
421C3×C4#89511yescharacteristic
422C3×C4#93711yescharacteristic
423C3×C4#93811yescharacteristic
424Dic3—42no—
425Dic3—42no—
426Dic3—64no—
427Dic3—62no—
428Dic3—62no—
429C32#191511yescharacteristic, complemented, Sylow
430C23—92no—
431C23C4×D3211yescharacteristic
432Q8—121no—
433Q8—121no—
434Q8—121no—
435Q8—121no—
436Q8—181no—
437Q8—181no—
438D4—182no—
439D4—182no—
440D4—181no—
441D4—181no—
442C2×C4—62no—
443C2×C4—62no—
444C2×C4—62no—
445C2×C4—62no—
446C2×C4—92no—
447C2×C4—182no—
448C2×C4—182no—
449C2×C4#379611yescharacteristic
450C2×C4#379811yescharacteristic
451C2×C4#380211yescharacteristic
452C2×C4C2×#62611yescharacteristic
453C2×C4#381111yescharacteristic
454C2×C4#381111yescharacteristic
455C8—22no—
456C8—121no—
457C8—121no—
458C8—182no—
459C2×C3—21no—
460C2×C3—21no—
461C2×C3—21no—
462C2×C3—21no—
463C2×C3—21no—
464C2×C3—21no—
465C2×C3—21no—
466C2×C3—41no—
467C2×C3—41no—
468C2×C3—41no—
469C2×C3#466511yescharacteristic
470C2×C3#495611yescharacteristic
471D3—124no—
472D3—122no—
473D3—122no—
474C22—41no—
475C22—182no—
476C22—362no—
477C22#6388111yescharacteristic
478C22C4×#62611yescharacteristic
479C22#6392111yescharacteristic
480C4—122no—
481C4—122no—
482C4—182no—
483C4#6385311yescharacteristic
484C4#6391211yescharacteristic
485C4#6391611yescharacteristic
486C4#6398111yescharacteristic
487C3—21no—
488C3#6904111yescharacteristic, complemented
489C3#7012411yescharacteristic, complemented
490C2—21no—
491C2—21no—
492C2—21no—
493C2—362no—
494C2#1059138711yescharacteristic
subgroupG/H
centerC2#10591387
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
3. C2×C3×#824C2semidirect
4. C2×#63560C2semidirect
6. C3×#4559C2semidirect
7. #10589169C2semidirect
36. C2×#824D3semidirect
40. #4559D3semidirect
429. C32#1915semidirect
488. C3#69041semidirect
489. C3#70124semidirect

Maximal subgroups

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

subgroupindex
#105962162
#105948662
C2×C3×#8242
C2×#635602
#105910782
C3×#45592
#105891692
#701243
#690413