Finite Group Databasebeta

← #11789729

Subgroups — #11789729

Order 1152.

503 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1M16×C3⋊Dic3C211yesmaximal, characteristic, complemented
2C3×#5139C211yesmaximal, characteristic
3C3×#5124C211yesmaximal, characteristic, complemented
4C2×#63568C211yesmaximal, characteristic
5#10591076C211yesmaximal, characteristic
6#10589169C211yesmaximal, characteristic, complemented
7#10589160C211yesmaximal, characteristic, complemented
8#70727—31nomaximal
9#70401—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C2×C32×M16C2211yescharacteristic
12C2×C4×C3⋊Dic3C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C3×#926C2211yescharacteristic
15C2×C3×#208C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17#63642—21no—
18C8×C3⋊Dic3—21no—
19C3×#825—22no—
20C3×#822—22no—
21#63568—21no—
22#63568—21no—
23#63560C412yescomplemented
24#63558C412yescomplemented
25C3×#430—31no—
26C3×#421—31no—
27#5237—31no—
28C2×#837—31no—
29#5139D311yescharacteristic
30C2×#825—31no—
31#5132—31no—
32#5124D311yescharacteristic, complemented
33Dic3×M16—31no—
34Dic3×M16—31no—
35Dic3×M16—61no—
36#4556—31no—
37#4555—31no—
38#4533—31no—
39#4532—31no—
40C22×C32×C4C2311yescharacteristic
41C22×C3⋊Dic3—21no—
42C2×C3×#67—21no—
43C3×#235—21no—
44C3×#235—21no—
45C3×#235—21no—
46C3×#235—21no—
47C2×C3×D12C2×C411yescharacteristic
48C2×C3×C4×D3—21no—
49C2×C3×Dic6C2×C411yescharacteristic
50C32×M16—21no—
51C32×M16D412yes—
52C2×C32×C8—21no—
53C4×C3⋊Dic3—21no—
54C4×C3⋊Dic3D411yescharacteristic
55C4×C3⋊Dic3D411yescharacteristic
56#3748—21no—
57#3748C2×C412yes—
58C2×C3⋊(C3⋊C8)—21no—
59C3×#217—21no—
60C3×C4⋊Dic3—21no—
61C3×C4×Dic3—21no—
62C3×#208—21no—
63C3×#208—21no—
64C3×#208—21no—
65C3×#208C2×C412yes—
66C2×C3×C3⋊C8—21no—
67C2×C3×C3⋊C8—21no—
68#1932—91noSylow
69C2×C3×#41—31no—
70C2×#235C2×D311yescharacteristic
71C2×C3×M16—21no—
72C2×C3×M16—31no—
73C2×C3×M16—31no—
74C2×C3×M16C2×D311yescharacteristic
75C2×C3×M16C2×D311yescharacteristic
76C3×#117—31no—
77#926C2×D311yescharacteristic
78C2×C4×Dic3—31no—
79C2×C4×Dic3—31no—
80C2×C4×Dic3—61no—
81C2×#208—31no—
82C2×#208—31no—
83C2×#208—61no—
84C2×#208C2×D311yescharacteristic
85C2×#208C2×D311yescharacteristic
86C3×#108—62no—
87C3×#104—62no—
88#837—61no—
89#837—61no—
90#836—32no—
91#833—32no—
92#825—22no—
93#825—61no—
94#825—61no—
95#824—32no—
96#823—32no—
97#822—22no—
98#815—61no—
99#815—61no—
100#815—62no—
101C8×Dic3—61no—
102C8×Dic3—61no—
103C8×Dic3—62no—
104C23×C32#4111yescharacteristic
105C22×C3×D3—21no—
106C2×C32×C4C22×C411yescharacteristic
107C2×C32×C4C2×D411yescharacteristic
108C2×C32×C4C2×D411yescharacteristic
109C2×C32×C4#4111yescharacteristic
110C2×C32×C4#4111yescharacteristic
111C2×C32×C4#4111yescharacteristic
112C2×C3⋊Dic3—21no—
113C2×C3⋊Dic3—21no—
114C2×C3⋊Dic3—42no—
115C3×#67—41no—
116C3×#67—41no—
117C2×C3×Dic3—21no—
118C2×C3×Dic3—41no—
119C3×D12—21no—
120C3×D12—21no—
121C3×C4×D3—41no—
122C3×C4×D3—41no—
123C3×Dic6—21no—
124C3×Dic6—21no—
125C32×C8—22no—
126C3⋊(C3⋊C8)—22no—
127C3×C3⋊C8—22no—
128C3×C3⋊C8—41no—
129#430—31no—
130#421—31no—
131C2×#104—91no—
132#413—91no—
133C4×M16—91no—
134#344—91no—
135#339—91no—
136C3×#41—61no—
137C3×#41—61no—
138C3×#41—61no—
139C3×#41—61no—
140C2×C3×Q8—31no—
141C2×C3×D4—31no—
142C2×C3×D4—61no—
143C22×C3×C4—21no—
144C22×C3×C4—61no—
145C22×C3×C4C22×D311yescharacteristic
146C22×C3×C4C22×D311yescharacteristic
147C2×#67—21no—
148C22×Dic3—62no—
149C22×Dic3—61no—
150C22×Dic3—61no—
151#235—21no—
152#235—21no—
153#235—21no—
154#235—21no—
155C2×D12C4×D311yescharacteristic
156C2×C4×D3—21no—
157C2×Dic6C4×D311yescharacteristic
158C3×M16—22no—
159C3×M16—21no—
160C3×M16—21no—
161C3×M16—22no—
162C3×M16—32no—
163C3×M16—61no—
164C3×M16—61no—
165C3×M16—61no—
166C3×M16#6712yes—
167C3×M16#6712yes—
168C2×C3×C8—22no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—61no—
172C2×C3×C8—61no—
173C3×C4⋊C4—61no—
174C3×C22⋊C4—61no—
175C3×C42—61no—
176#217—21no—
177C4⋊Dic3—21no—
178C4×Dic3—21no—
179C4×Dic3—31no—
180C4×Dic3—31no—
181C4×Dic3—31no—
182C4×Dic3—31no—
183C4×Dic3—61no—
184C4×Dic3—61no—
185C4×Dic3—62no—
186C4×Dic3—61no—
187C4×Dic3—61no—
188#208—21no—
189#208—21no—
190#208—21no—
191#208—32no—
192#208—32no—
193#208—62no—
194#208—62no—
195#208—61no—
196#208—61no—
197#208C4×D312yes—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—21no—
200C2×C3⋊C8—62no—
201C2×C3⋊C8—61no—
202C2×C3⋊C8—61no—
203C22×C32—41no—
204C22×C32#11711yescharacteristic
205C22×C32#12311yescharacteristic
206C22×C32#12411yescharacteristic
207C2×C3×D3—41no—
208C2×C3×D3—81no—
209C32×C4C4×D411yescharacteristic
210C32×C4C4×D411yescharacteristic
211C32×C4#12111yescharacteristic
212C32×C4#12611yescharacteristic
213C3⋊Dic3—22no—
214C3⋊Dic3—41no—
215C3×Dic3—41no—
216C3×Dic3—81no—
217C2×#41—31no—
218C2×M16—31no—
219C2×M16—31no—
220C2×M16—91no—
221C2×M16D3211yescharacteristic
222#117—31no—
223C2×C42—91no—
224#108—62no—
225#104—62no—
226#104—181no—
227#104—181no—
228#101—92no—
229#100—92no—
230#97—181no—
231C4×C8—181no—
232C23×C3—21no—
233C23×C3—61no—
234C23×C3#23711yescharacteristic
235C23×C3#23911yescharacteristic
236C22×D3—21no—
237C3×Q8—61no—
238C3×Q8—61no—
239C3×D4—61no—
240C3×D4—61no—
241C3×D4—121no—
242C3×D4—121no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—21no—
249C2×C3×C4—61no—
250C2×C3×C4—121no—
251C2×C3×C4—121no—
252C2×C3×C4—121no—
253C2×C3×C4C2×C4×D311yescharacteristic
254C2×C3×C4C2×C4×D311yescharacteristic
255C2×C3×C4#23511yescharacteristic
256C2×C3×C4#23511yescharacteristic
257C2×C3×C4D3×D411yescharacteristic
258C2×C3×C4D3×D411yescharacteristic
259C2×C3×C4#23711yescharacteristic
260C2×C3×C4#23711yescharacteristic
261C2×C3×C4#23711yescharacteristic
262C2×C3×C4#23911yescharacteristic
263C2×C3×C4C2×#6711yescharacteristic
264C2×C3×C4C2×#6711yescharacteristic
265#67—41no—
266#67—41no—
267C2×Dic3—21no—
268C2×Dic3—41no—
269C2×Dic3—62no—
270C2×Dic3—61no—
271C2×Dic3—61no—
272C2×Dic3—62no—
273C2×Dic3—61no—
274C2×Dic3—61no—
275C2×Dic3—122no—
276C2×Dic3—122no—
277C2×Dic3—124no—
278D12—21no—
279D12—21no—
280C4×D3—41no—
281C4×D3—41no—
282Dic6—21no—
283Dic6—21no—
284C3×C8—24no—
285C3×C8—22no—
286C3×C8—22no—
287C3×C8—62no—
288C3×C8—121no—
289C3⋊C8—22no—
290C3⋊C8—41no—
291C3⋊C8—64no—
292C3⋊C8—62no—
293C3⋊C8—62no—
294C2×C32—21no—
295C2×C32—21no—
296C2×C32—21no—
297C2×C32#38811yescharacteristic
298C3×D3—81no—
299#41—61no—
300#41—61no—
301#41—61no—
302#41—61no—
303C2×Q8—31no—
304C2×D4—31no—
305C2×D4—61no—
306C22×C4—61no—
307C22×C4—181no—
308C22×C4C2×D3211yescharacteristic
309M16—21no—
310M16—32no—
311M16—61no—
312M16—61no—
313M16—61no—
314M16—92no—
315M16—181no—
316M16#62512yes—
317C2×C8—21no—
318C2×C8—61no—
319C2×C8—61no—
320C2×C8—181no—
321C4⋊C4—61no—
322C22⋊C4—61no—
323C42—61no—
324C42—91no—
325C42—91no—
326C42—181no—
327C22×C3—21no—
328C22×C3—21no—
329C22×C3—21no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—42no—
333C22×C3—121no—
334C22×C3—241no—
335C22×C3#87711yescharacteristic
336C22×C3#88311yescharacteristic
337C22×C3#89211yescharacteristic
338C22×C3#89411yescharacteristic
339C22×C3#93611yescharacteristic
340C22×C3#94711yescharacteristic
341C2×D3—41no—
342C2×D3—81no—
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—21no—
347C3×C4—121no—
348C3×C4—241no—
349C3×C4#87911yescharacteristic
350C3×C4#67⋊C411yescharacteristic
351C3×C4#89311yescharacteristic
352C3×C4#89811yescharacteristic
353C3×C4C4×#6711yescharacteristic
354C3×C4C4×#6711yescharacteristic
355C3×C4#93911yescharacteristic
356C3×C4#93911yescharacteristic
357Dic3—41no—
358Dic3—64no—
359Dic3—62no—
360Dic3—62no—
361Dic3—81no—
362Dic3—122no—
363Dic3—121no—
364Dic3—121no—
365C32#193211yescharacteristic, complemented, Sylow
366C23—61no—
367C23#379611yescharacteristic
368Q8—61no—
369Q8—61no—
370D4—61no—
371D4—61no—
372D4—121no—
373D4—121no—
374C2×C4—61no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—181no—
379C2×C4—181no—
380C2×C4—362no—
381C2×C4#379611yescharacteristic
382C2×C4#379811yescharacteristic
383C2×C4C4×D3211yescharacteristic
384C2×C4#380511yescharacteristic
385C2×C4C2×#62511yescharacteristic
386C2×C4#381111yescharacteristic
387C8—22no—
388C8—62no—
389C8—121no—
390C8—182no—
391C2×C3—22no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—21no—
397C2×C3—21no—
398C2×C3—21no—
399C2×C3—41no—
400C2×C3—41no—
401C2×C3—241no—
402C2×C3#494511yescharacteristic
403C2×C3#499011yescharacteristic
404D3—81no—
405C22—41no—
406C22—121no—
407C22—241no—
408C22#6385311yescharacteristic
409C22#6386311yescharacteristic
410C22#6387811yescharacteristic
411C4—121no—
412C4—182no—
413C4—241no—
414C4—361no—
415C4#6386211yescharacteristic
416C4#6388111yescharacteristic
417C4C4×#62511yescharacteristic
418C4#6398011yescharacteristic
419C3—21no—
420C3#7040111yescharacteristic, complemented
421C3#7072711yescharacteristic, complemented
422C2—21no—
423C2—21no—
424C2—21no—
425C2—241no—
426C2#1059128311yescharacteristic
subgroupG/H
centerC4#63980
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC2×C32#388
prime radicalC2×C3×#235C22
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1932
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. M16×C3⋊Dic3C2semidirect
3. C3×#5124C2semidirect
6. #10589169C2semidirect
7. #10589160C2semidirect
23. #63560C4semidirect
24. #63558C4semidirect
32. #5124D3semidirect
365. C32#1932semidirect
420. C3#70401semidirect
421. C3#70727semidirect

Maximal subgroups

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

subgroupindex
M16×C3⋊Dic32
C3×#51392
C3×#51242
C2×#635682
#105910762
#105891692
#105891602
#707273
#704013