Finite Group Databasebeta

← #11787030

Subgroups — #11787030

Order 1152.

593 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63814C211yesmaximal, characteristic
2C2×#63649C211yesmaximal, characteristic, complemented
3C2×C3×#825C211yesmaximal, characteristic
4C3×#5119C211yesmaximal, characteristic, complemented
5#10591083C211yesmaximal, characteristic
6#10589161C211yesmaximal, characteristic, complemented
7#10589155C211yesmaximal, characteristic, complemented
8#70545—31nomaximal
9#70376—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×#3748C2211yescharacteristic
13C22×C3⋊(C3⋊C8)C2211yescharacteristic
14C3×#926C2211yescharacteristic
15C2×C3×C4×Dic3C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C2×#3711—21no—
18#63814—22no—
19#63814—22no—
20C2×D3×C3⋊C8—21no—
21#63649—22no—
22C3×#825—22no—
23C3×Dic3⋊C8—21no—
24C3×Dic3⋊C8—21no—
25#63565C411yescharacteristic, complemented
26#63565C411yescharacteristic, complemented
27#63562C411yescharacteristic, complemented
28#63562C411yescharacteristic, complemented
29C2×#948—31no—
30C2×#901—31no—
31C3×#424—31no—
32C2×C3×#104—31no—
33#5233—31no—
34C2×#825D311yescharacteristic
35C2×#822—31no—
36C2×#822—31no—
37C2×#822—61no—
38#5119D311yescharacteristic, complemented
39#5002—31no—
40#4550—31no—
41#4532—31no—
42#4524—31no—
43#4523—31no—
44C22×C32×C4C2311yescharacteristic
45C2×C3×#67—21no—
46C22×C3×Dic3—21no—
47C3×#235—21no—
48C3×#235—21no—
49C3×#235D412yes—
50C3×#235Q812yes—
51C2×C3×D12C2×C411yescharacteristic
52C2×C3×C4×D3—21no—
53C2×C3×Dic6C2×C411yescharacteristic
54C32×M16—21no—
55C32×M16—21no—
56C2×C32×C8—21no—
57#3748—21no—
58#3748D412yes—
59C2×C3⋊(C3⋊C8)—21no—
60C2×C3⋊(C3⋊C8)—21no—
61C2×C3⋊(C3⋊C8)—21no—
62C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
63C2×C3⋊(C3⋊C8)C2×C411yescharacteristic
64C3×#217—21no—
65C3×C4⋊Dic3C2×C411yescharacteristic
66C3×C4⋊Dic3C2×C411yescharacteristic
67C3×C4×Dic3—21no—
68C3×C4×Dic3—21no—
69C3×C4×Dic3—21no—
70C2×C3×C3⋊C8—21no—
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8D411yescharacteristic
73C2×C3×C3⋊C8D411yescharacteristic
74#3711—24no—
75D3×C3⋊C8—24no—
76#1866—91noSylow
77C2×C3×#41—31no—
78C2×#235C2×D311yescharacteristic
79C2×C3×M16—21no—
80C2×C3×M16C2×D311yescharacteristic
81C2×C3×M16C2×D311yescharacteristic
82C22×C3×C8—31no—
83C3×#117—31no—
84C2×C3×C42—31no—
85#948—62no—
86#948—62no—
87#926C2×D311yescharacteristic
88C2×C4×Dic3C2×D311yescharacteristic
89C2×#208—31no—
90C2×#208—31no—
91C2×#208—61no—
92C2×#208—61no—
93C22×C3⋊C8—31no—
94C22×C3⋊C8—31no—
95C22×C3⋊C8—61no—
96C22×C3⋊C8—61no—
97C22×C3⋊C8C2×D311yescharacteristic
98#901—62no—
99#901—62no—
100C2×ΓL(1,𝔽25)—61no—
101C2×D3×C8—61no—
102C3×C4⋊C8—61no—
103C3×C4⋊C8—61no—
104C3×#104—62no—
105#835—31no—
106#835—31no—
107#832—31no—
108#832—31no—
109#825—22no—
110#822—62no—
111#822—62no—
112#822—64no—
113Dic3⋊C8—21no—
114Dic3⋊C8—21no—
115#810—31no—
116#810—31no—
117#809—31no—
118#809—31no—
119C23×C32#4111yescharacteristic
120C22×C3×D3—21no—
121C2×C32×C4C22×C411yescharacteristic
122C2×C32×C4C2×D411yescharacteristic
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4C2×Q811yescharacteristic
126C2×C32×C4#4111yescharacteristic
127C3×#67—22no—
128C3×#67—22no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—22no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—41no—
134C2×C3×Dic3—41no—
135C3×D12—21no—
136C3×D12C4⋊C412yes—
137C3×C4×D3—22no—
138C3×C4×D3—22no—
139C3×Dic6—21no—
140C3×Dic6C4⋊C412yes—
141C32×C8—41no—
142C3⋊(C3⋊C8)—21no—
143C3⋊(C3⋊C8)—21no—
144C3⋊(C3⋊C8)—22no—
145C3×C3⋊C8—21no—
146C3×C3⋊C8—21no—
147C2×#131—91no—
148C2×#108—91no—
149#424—31no—
150C2×#104—31no—
151#416—91no—
152#339—91no—
153#336—91no—
154C3×#41—32no—
155C3×#41—32no—
156C3×#41—61no—
157C3×#41—61no—
158C2×C3×Q8—31no—
159C2×C3×D4—31no—
160C2×C3×D4—61no—
161C22×C3×C4—21no—
162C22×C3×C4—61no—
163C22×C3×C4—61no—
164C22×C3×C4C22×D311yescharacteristic
165C22×C3×C4C22×D311yescharacteristic
166C2×#67—21no—
167C22×Dic3—21no—
168#235—21no—
169#235—21no—
170#235Dic612yes—
171#235#6712yes—
172C2×D12C4×D311yescharacteristic
173C2×C4×D3—21no—
174C2×Dic6C4×D311yescharacteristic
175C3×M16—22no—
176C3×M16—21no—
177C3×M16—21no—
178C3×M16—21no—
179C3×M16—21no—
180C3×M16—41no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—31no—
184C2×C3×C8—31no—
185C2×C3×C8—41no—
186C2×C3×C8—61no—
187C2×C3×C8—61no—
188C3×C4⋊C4—31no—
189C3×C4⋊C4—31no—
190C3×C22⋊C4—61no—
191C3×C42—61no—
192C3×C42—61no—
193C3×C42—61no—
194#217—21no—
195C4⋊Dic3C4×D311yescharacteristic
196C4⋊Dic3C4×D311yescharacteristic
197C4×Dic3—21no—
198C4×Dic3—21no—
199C4×Dic3—21no—
200#208—32no—
201#208—32no—
202#208—62no—
203#208—64no—
204#208—61no—
205#208—61no—
206#208—121no—
207C2×C3⋊C8—21no—
208C2×C3⋊C8—21no—
209C2×C3⋊C8—31no—
210C2×C3⋊C8—31no—
211C2×C3⋊C8—31no—
212C2×C3⋊C8—31no—
213C2×C3⋊C8—61no—
214C2×C3⋊C8—61no—
215C2×C3⋊C8—64no—
216C2×C3⋊C8—62no—
217C2×C3⋊C8—62no—
218C2×C3⋊C8—61no—
219C2×C3⋊C8—61no—
220C2×C3⋊C8—61no—
221C2×C3⋊C8—61no—
222C2×C3⋊C8—61no—
223C2×C3⋊C8—61no—
224C2×C3⋊C8—121no—
225C2×C3⋊C8#6711yescharacteristic
226C2×C3⋊C8#6711yescharacteristic
227ΓL(1,𝔽25)—64no—
228D3×C8—64no—
229C22×C32—21no—
230C22×C32—21no—
231C22×C32C4×D411yescharacteristic
232C22×C32C22⋊Q811yescharacteristic
233C22×C32#12311yescharacteristic
234C2×C3×D3—22no—
235C2×C3×D3—41no—
236C2×C3×D3—41no—
237C32×C4C2×C4⋊C411yescharacteristic
238C32×C4C4×D411yescharacteristic
239C32×C4#12011yescharacteristic
240C32×C4C22⋊Q811yescharacteristic
241C3×Dic3—22no—
242C3×Dic3—42no—
243C3×Dic3—41no—
244C3×Dic3—41no—
245C2×#41—31no—
246#131—182no—
247#131—182no—
248C2×M16—91no—
249C2×M16—181no—
250C2×M16D3211yescharacteristic
251C22×C8—31no—
252C22×C8—91no—
253C22×C8—181no—
254#117—31no—
255C2×C42—31no—
256#108—182no—
257C4⋊C8—61no—
258C4⋊C8—61no—
259#104—62no—
260#103—91no—
261#103—91no—
262#102—91no—
263#102—91no—
264C23×C3—21no—
265C23×C3—61no—
266C23×C3#23511yescharacteristic
267C23×C3#23711yescharacteristic
268C22×D3—21no—
269C3×Q8—32no—
270C3×Q8—61no—
271C3×D4—32no—
272C3×D4—62no—
273C3×D4—62no—
274C3×D4—61no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—21no—
281C2×C3×C4—61no—
282C2×C3×C4—62no—
283C2×C3×C4—62no—
284C2×C3×C4—62no—
285C2×C3×C4—61no—
286C2×C3×C4—61no—
287C2×C3×C4—121no—
288C2×C3×C4—121no—
289C2×C3×C4C2×Dic611yescharacteristic
290C2×C3×C4C2×C4×D311yescharacteristic
291C2×C3×C4C2×C4×D311yescharacteristic
292C2×C3×C4D3×D411yescharacteristic
293C2×C3×C4D3×D411yescharacteristic
294C2×C3×C4D3×D411yescharacteristic
295C2×C3×C4D3×D411yescharacteristic
296C2×C3×C4#23711yescharacteristic
297C2×C3×C4D3×Q811yescharacteristic
298C2×C3×C4#23911yescharacteristic
299C2×C3×C4C2×#6711yescharacteristic
300C2×C3×C4C2×#6711yescharacteristic
301#67—22no—
302#67—22no—
303C2×Dic3—21no—
304C2×Dic3—22no—
305C2×Dic3—21no—
306C2×Dic3—21no—
307C2×Dic3—41no—
308C2×Dic3—41no—
309D12—21no—
310D12Dic3⋊C412yes—
311C4×D3—22no—
312C4×D3—22no—
313Dic6—21no—
314Dic6Dic3⋊C412yes—
315C3×C8—42no—
316C3×C8—41no—
317C3×C8—41no—
318C3×C8—61no—
319C3×C8—61no—
320C3⋊C8—21no—
321C3⋊C8—21no—
322C3⋊C8—62no—
323C3⋊C8—62no—
324C3⋊C8—61no—
325C3⋊C8—61no—
326C3⋊C8—61no—
327C3⋊C8—61no—
328C3⋊C8—62no—
329C3⋊C8—62no—
330C3⋊C8—122no—
331C2×C32—21no—
332C2×C32—21no—
333C2×C32#38511yescharacteristic
334C2×C32#44311yescharacteristic
335C2×C32#44411yescharacteristic
336C3×D3—42no—
337#41—32no—
338#41—32no—
339#41—61no—
340#41—61no—
341C2×Q8—31no—
342C2×D4—31no—
343C2×D4—61no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4C2×D3211yescharacteristic
347M16—21no—
348M16—21no—
349M16—92no—
350M16—184no—
351M16—181no—
352C2×C8—21no—
353C2×C8—31no—
354C2×C8—31no—
355C2×C8—61no—
356C2×C8—61no—
357C2×C8—91no—
358C2×C8—91no—
359C2×C8—184no—
360C2×C8—181no—
361C2×C8—181no—
362C2×C8—181no—
363C4⋊C4—31no—
364C4⋊C4—31no—
365C22⋊C4—61no—
366C42—61no—
367C42—61no—
368C42—61no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—21no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—62no—
379C22×C3—121no—
380C22×C3—121no—
381C22×C3#87811yescharacteristic
382C22×C3#67⋊C411yescharacteristic
383C22×C3#89411yescharacteristic
384C22×C3#89611yescharacteristic
385C22×C3C4×#6711yescharacteristic
386C22×C3#93511yescharacteristic
387C2×D3—22no—
388C2×D3—41no—
389C2×D3—41no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—62no—
395C3×C4—122no—
396C3×C4—121no—
397C3×C4—121no—
398C3×C4#87811yescharacteristic
399C3×C4#67⋊C411yescharacteristic
400C3×C4D3×C4⋊C411yescharacteristic
401C3×C4#89311yescharacteristic
402C3×C4C2×Dic3⋊C411yescharacteristic
403C3×C4#93711yescharacteristic
404C3×C4#93711yescharacteristic
405C3×C4#94611yescharacteristic
406Dic3—22no—
407Dic3—42no—
408Dic3—41no—
409Dic3—41no—
410C32#186611yescharacteristic, complemented, Sylow
411C23—61no—
412C23#379511yescharacteristic
413Q8—32no—
414Q8—61no—
415D4—32no—
416D4—62no—
417D4—62no—
418D4—61no—
419C2×C4—61no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4D3×Dic611yescharacteristic
428C2×C4#379611yescharacteristic
429C2×C4C4×D3211yescharacteristic
430C2×C4D3×#6711yescharacteristic
431C2×C4D3×#6711yescharacteristic
432C2×C4#381111yescharacteristic
433C8—41no—
434C8—61no—
435C8—61no—
436C8—181no—
437C8—181no—
438C8—182no—
439C2×C3—21no—
440C2×C3—21no—
441C2×C3—21no—
442C2×C3—21no—
443C2×C3—21no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—41no—
447C2×C3—41no—
448C2×C3—122no—
449C2×C3#482311yescharacteristic
450C2×C3#482411yescharacteristic
451C2×C3#494611yescharacteristic
452C2×C3#498911yescharacteristic
453C2×C3#514411yescharacteristic
454C2×C3#514511yescharacteristic
455D3—42no—
456C22—21no—
457C22—21no—
458C22—62no—
459C22—121no—
460C22—121no—
461C22#6387911yescharacteristic
462C22#6388411yescharacteristic
463C22#6389611yescharacteristic
464C4—62no—
465C4—122no—
466C4—121no—
467C4—121no—
468C4D3×Dic3⋊C411yescharacteristic
469C4#6388111yescharacteristic
470C4#6388711yescharacteristic
471C4#6398311yescharacteristic
472C3—21no—
473C3#7037611yescharacteristic, complemented
474C3#7054511yescharacteristic, complemented
475C2—21no—
476C2—21no—
477C2—122no—
478C2#1059097511yescharacteristic
479C2#1059097611yescharacteristic
480C2#1059137211yescharacteristic
subgroupG/H
centerC2×C4D3×#67
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32C4×D4
prime radicalC2×C3×#235C22
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1866
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. C2×#63649C2semidirect
4. C3×#5119C2semidirect
6. #10589161C2semidirect
7. #10589155C2semidirect
25. #63565C4semidirect
26. #63565C4semidirect
27. #63562C4semidirect
28. #63562C4semidirect
38. #5119D3semidirect
410. C32#1866semidirect
473. C3#70376semidirect
474. C3#70545semidirect

Maximal subgroups

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

subgroupindex
C2×#638142
C2×#636492
C2×C3×#8252
C3×#51192
#105910832
#105891612
#105891552
#705453
#703763