Finite Group Databasebeta

← #11787906

Subgroups — #11787906

Order 1152.

743 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63863C211yesmaximal, characteristic
2#10594818C211yesmaximal, characteristic, complemented
3C2×C3×#824C211yesmaximal, characteristic, complemented
4C3×#5131C211yesmaximal, characteristic
5#10591086C211yesmaximal, characteristic
6#10589421C211yesmaximal, characteristic, complemented
7#10589160C211yesmaximal, characteristic, complemented
8#70325—31nomaximal
9#70104—31nomaximal
10C22×C3×D12C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×M16C2211yescharacteristic
13#64135C2211yescharacteristic
14C2×C4×C3⋊Dic3C2211yescharacteristic
15C3×#926C2211yescharacteristic
16C2×C3×#208C2211yescharacteristic
17C2×(C2×D3)⋊Dic3—22no—
18#63955—22no—
19#63863—22no—
20#63863C414yescomplemented
21#63843—22no—
22Dic3×Dic6—22no—
23C3×#824—22no—
24C3×#821—22no—
25#63563—22no—
26C2×#947—31no—
27#5812—31no—
28C2×#936—31no—
29#5516—31no—
30C3×#418—31no—
31C2×C3×#101—31no—
32#5215—31no—
33C2×#824D311yescharacteristic, complemented
34#5131—31no—
35#5131D311yescharacteristic
36#4555—31no—
37#4555—31no—
38#4555—31no—
39#4555—61no—
40#4532—31no—
41C23×C3×D3—21no—
42C22×C32×C4C2311yescharacteristic
43C22×C3⋊Dic3—21no—
44C22×C3×Dic3D412yes—
45C2×C3×D12—21no—
46C2×C3×D12—21no—
47C2×C3×D12—21no—
48C2×C3×D12—21no—
49C2×C3×D12—21no—
50C2×C3×D12C2×C412yes—
51C2×C3×Dic6—21no—
52C2×C3×Dic6—21no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6C2×C412yes—
56C2×C3×Dic6D412yes—
57C32×M16—21no—
58C32×M16—21no—
59C2×C32×C8—21no—
60#3757—21no—
61C4⋊(C3⋊Dic3)D412yes—
62C4×C3⋊Dic3—21no—
63C4×C3⋊Dic3—21no—
64C4×C3⋊Dic3C2×C412yes—
65C3×#217—21no—
66C3×C4⋊Dic3D412yes—
67C3×C4×Dic3—21no—
68C3×#208—21no—
69C3×#208—21no—
70C2×C3×C3⋊C8—21no—
71Dic3⋊Dic3—24no—
72(C2×D3)⋊Dic3—28no—
73Dic32—24no—
74#1888—91noSylow
75C22×C3×Q8—31no—
76C22×C3×D4—31no—
77C22×D12C2×D311yescharacteristic
78C22×Dic6C2×D311yescharacteristic
79C2×C3×M16—21no—
80C2×C3×M16—31no—
81C2×C3×M16C2×D311yescharacteristic
82C2×C3×M16C2×D311yescharacteristic
83C3×#117—31no—
84C2×#217—62no—
85#947—34no—
86#947—62no—
87Q8×Dic3—62no—
88Q8×Dic3—62no—
89#936—34no—
90#936—62no—
91C2×(C2×D3)⋊C4—62no—
92#926—31no—
93#926—31no—
94#926—61no—
95#926—62no—
96#926C2×D311yescharacteristic
97C2×C4×Dic3—31no—
98C2×C4×Dic3—31no—
99C2×C4×Dic3—61no—
100C2×#208C2×D311yescharacteristic
101#887—62no—
102#877—62no—
103C4×Dic6—62no—
104C3×#102—62no—
105C3×#101—62no—
106#832—62no—
107#824—22no—
108#821—22no—
109#821—62no—
110C23×C32C2×D411yescharacteristic
111C22×C3×D3—22no—
112C22×C3×D3—44no—
113C22×C3×D3—41no—
114C22×C3×D3—41no—
115C2×C32×C4C22×C411yescharacteristic
116C2×C32×C4C2×D411yescharacteristic
117C2×C32×C4C2×D411yescharacteristic
118C2×C32×C4C2×D411yescharacteristic
119C2×C32×C4#4111yescharacteristic
120C2×C32×C4#4111yescharacteristic
121C2×C3⋊Dic3—22no—
122C2×C3⋊Dic3—22no—
123C2×C3⋊Dic3—42no—
124C2×C3×Dic3—22no—
125C2×C3×Dic3—22no—
126C2×C3×Dic3—22no—
127C2×C3×Dic3C22⋊C414yes—
128C3×D12—22no—
129C3×D12—22no—
130C3×D12—41no—
131C3×D12—41no—
132C3×Dic6—22no—
133C3×Dic6—22no—
134C3×Dic6—22no—
135C3×Dic6—22no—
136C32×C8—41no—
137C3×C3⋊C8—41no—
138C2×#124—91no—
139#519—91no—
140#418—31no—
141#418—91no—
142C2×#101—31no—
143#339—91no—
144#339—91no—
145C24×C3—61no—
146C23×D3—21no—
147C2×C3×Q8—32no—
148C2×C3×Q8—32no—
149C2×C3×Q8—61no—
150C2×C3×Q8—61no—
151C2×C3×Q8—61no—
152C2×C3×Q8—61no—
153C2×C3×D4—32no—
154C2×C3×D4—61no—
155C2×C3×D4—61no—
156C2×C3×D4—61no—
157C2×C3×D4—61no—
158C2×C3×D4—61no—
159C22×C3×C4—21no—
160C22×C3×C4—32no—
161C22×C3×C4C22×D311yescharacteristic
162C22×C3×C4C22×D311yescharacteristic
163C22×Dic3—61no—
164C22×Dic3—61no—
165C22×Dic3—121no—
166C22×Dic3D1212yes—
167C2×D12—21no—
168C2×D12—21no—
169C2×D12—21no—
170C2×D12—21no—
171C2×D12—21no—
172C2×D12C4×D312yes—
173C2×Dic6—21no—
174C2×Dic6—21no—
175C2×Dic6—21no—
176C2×Dic6—21no—
177C2×Dic6C4×D312yes—
178C2×Dic6#6712yes—
179C3×M16—22no—
180C3×M16—21no—
181C3×M16—21no—
182C3×M16—21no—
183C3×M16—21no—
184C3×M16—41no—
185C3×M16—61no—
186C3×M16—61no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C2×C3×C8—41no—
190C2×C3×C8—61no—
191C3×C4⋊C4—32no—
192C3×C22⋊C4—61no—
193C3×C42—61no—
194#217—21no—
195#217—61no—
196#217—61no—
197#217—68no—
198#217—121no—
199(C2×D3)⋊C4—68no—
200C4⋊Dic3—32no—
201C4⋊Dic3—32no—
202C4⋊Dic3—62no—
203C4⋊Dic3—64no—
204C4⋊Dic3#6712yes—
205Dic3⋊C4—64no—
206C4×Dic3—21no—
207C4×Dic3—32no—
208C4×Dic3—32no—
209C4×Dic3—62no—
210C4×Dic3—64no—
211C4×Dic3—64no—
212C4×Dic3—61no—
213C4×Dic3—61no—
214C4×Dic3—61no—
215C4×Dic3—61no—
216C4×Dic3—121no—
217C4×Dic3—121no—
218#208—21no—
219#208—21no—
220C2×C3⋊C8—21no—
221C22×C32—22no—
222C22×C32C2×C22⋊C411yescharacteristic
223C22×C32#12011yescharacteristic
224C22×C32#12311yescharacteristic
225C2×C3×D3—44no—
226C2×C3×D3—42no—
227C2×C3×D3—82no—
228C2×C3×D3—82no—
229C32×C4C4×D411yescharacteristic
230C32×C4C4×D411yescharacteristic
231C32×C4#12111yescharacteristic
232C32×C4#12111yescharacteristic
233C3⋊Dic3—42no—
234C3⋊Dic3—42no—
235C3×Dic3—24no—
236C3×Dic3—42no—
237C22×Q8—31no—
238C22×D4—31no—
239C2×M16—31no—
240C2×M16D3211yescharacteristic
241#124—94no—
242#124—182no—
243C4×Q8—182no—
244C4×Q8—182no—
245#117—31no—
246#117—91no—
247#117—182no—
248C2×C22⋊C4—182no—
249C2×C42—91no—
250#102—62no—
251#102—182no—
252#101—62no—
253C23×C3—21no—
254C23×C3—62no—
255C23×C3—124no—
256C23×C3—121no—
257C23×C3—121no—
258C23×C3C2×D1211yescharacteristic
259C23×C3D3×D411yescharacteristic
260C22×D3—22no—
261C22×D3—44no—
262C22×D3—41no—
263C22×D3—41no—
264C3×Q8—62no—
265C3×Q8—62no—
266C3×Q8—62no—
267C3×Q8—62no—
268C3×D4—62no—
269C3×D4—62no—
270C3×D4—121no—
271C3×D4—121no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—34no—
279C2×C3×C4—62no—
280C2×C3×C4—62no—
281C2×C3×C4—62no—
282C2×C3×C4C2×C4×D311yescharacteristic
283C2×C3×C4C2×C4×D311yescharacteristic
284C2×C3×C4#23511yescharacteristic
285C2×C3×C4D3×D411yescharacteristic
286C2×C3×C4D3×D411yescharacteristic
287C2×C3×C4D3×D411yescharacteristic
288C2×C3×C4D3×D411yescharacteristic
289C2×C3×C4#23711yescharacteristic
290C2×C3×C4#23711yescharacteristic
291C2×C3×C4#23711yescharacteristic
292C2×C3×C4C2×#6711yescharacteristic
293C2×C3×C4C2×#6711yescharacteristic
294C2×Dic3—22no—
295C2×Dic3—22no—
296C2×Dic3—22no—
297C2×Dic3—62no—
298C2×Dic3—62no—
299C2×Dic3—62no—
300C2×Dic3—62no—
301C2×Dic3—122no—
302C2×Dic3—122no—
303C2×Dic3—122no—
304C2×Dic3—122no—
305C2×Dic3—124no—
306C2×Dic3(C2×D3)⋊C414yes—
307D12—22no—
308D12—22no—
309D12—41no—
310D12—41no—
311Dic6—22no—
312Dic6—22no—
313Dic6—22no—
314Dic6—22no—
315C3×C8—42no—
316C3×C8—41no—
317C3×C8—41no—
318C3×C8—121no—
319C3⋊C8—41no—
320C2×C32—21no—
321C2×C32—21no—
322C2×C32#38611yescharacteristic
323C2×C32#45412yes—
324C3×D3—82no—
325C24—61no—
326C2×Q8—32no—
327C2×Q8—32no—
328C2×Q8—61no—
329C2×Q8—61no—
330C2×Q8—61no—
331C2×Q8—61no—
332C2×D4—32no—
333C2×D4—61no—
334C2×D4—61no—
335C2×D4—61no—
336C2×D4—61no—
337C2×D4—61no—
338C22×C4—32no—
339C22×C4—181no—
340C22×C4C2×D3211yescharacteristic
341M16—21no—
342M16—21no—
343M16—61no—
344M16—61no—
345C2×C8—21no—
346C2×C8—61no—
347C4⋊C4—32no—
348C4⋊C4—92no—
349C4⋊C4—184no—
350C22⋊C4—61no—
351C22⋊C4—188no—
352C22⋊C4—181no—
353C42—61no—
354C42—92no—
355C42—184no—
356C42—181no—
357C42—181no—
358C22×C3—22no—
359C22×C3—22no—
360C22×C3—21no—
361C22×C3—21no—
362C22×C3—21no—
363C22×C3—42no—
364C22×C3—124no—
365C22×C3—122no—
366C22×C3—242no—
367C22×C3—242no—
368C22×C3D3×C22⋊C411yescharacteristic
369C22×C3#88211yescharacteristic
370C22×C3#88311yescharacteristic
371C22×C3#88611yescharacteristic
372C22×C3C2×(C2×D3)⋊C411yescharacteristic
373C22×C3#93711yescharacteristic
374C2×D3—44no—
375C2×D3—42no—
376C2×D3—82no—
377C2×D3—82no—
378C3×C4—21no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—64no—
383C3×C4—122no—
384C3×C4#67⋊C411yescharacteristic
385C3×C4#67⋊C411yescharacteristic
386C3×C4#67⋊C411yescharacteristic
387C3×C4#88411yescharacteristic
388C3×C4C4×#6711yescharacteristic
389C3×C4#93811yescharacteristic
390C3×C4#93911yescharacteristic
391C3×C4#93911yescharacteristic
392Dic3—24no—
393Dic3—42no—
394Dic3—122no—
395Dic3—122no—
396Dic3—124no—
397Dic3—122no—
398Dic3—122no—
399Dic3—242no—
400C32#188811yescharacteristic, complemented, Sylow
401C23—62no—
402C23—124no—
403C23—121no—
404C23—121no—
405C23D3×D1211yescharacteristic
406Q8—62no—
407Q8—62no—
408Q8—62no—
409Q8—62no—
410D4—62no—
411D4—62no—
412D4—121no—
413D4—121no—
414C2×C4—34no—
415C2×C4—62no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—182no—
419C2×C4—182no—
420C2×C4—362no—
421C2×C4#379511yescharacteristic
422C2×C4C4×D3211yescharacteristic
423C2×C4#380211yescharacteristic
424C2×C4#380511yescharacteristic
425C2×C4D3×#6711yescharacteristic
426C2×C4D3×#6711yescharacteristic
427C8—41no—
428C8—121no—
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—22no—
434C2×C3—21no—
435C2×C3—41no—
436C2×C3—41no—
437C2×C3—242no—
438C2×C3#475312yes—
439C2×C3#495611yescharacteristic
440C2×C3#495811yescharacteristic
441C2×C3#520112yes—
442D3—82no—
443C22—22no—
444C22—124no—
445C22—122no—
446C22—242no—
447C22—242no—
448C22#6389111yescharacteristic
449C22D3×(C2×D3)⋊C411yescharacteristic
450C22#6392211yescharacteristic
451C4—64no—
452C4—122no—
453C4—362no—
454C4—362no—
455C4#6387911yescharacteristic
456C4#6388511yescharacteristic
457C4#6390211yescharacteristic
458C4#6397011yescharacteristic
459C3—21no—
460C3#7010411yescharacteristic, complemented
461C3#7032511yescharacteristic, complemented
462C2—21no—
463C2—21no—
464C2—242no—
465C2#1059102012yes—
466C2#1059136311yescharacteristic
subgroupG/H
centerC22#63922
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×M16C22
socleC22×C32#120
prime radicalC22×C3×D12C22
cosocleC22×C32×C4C23
hypercenterC2×M16D32
hypocenterC32#1888
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
2. #10594818C2semidirect
3. C2×C3×#824C2semidirect
6. #10589421C2semidirect
7. #10589160C2semidirect
20. #63863C4semidirect
33. C2×#824D3semidirect
400. C32#1888semidirect
460. C3#70104semidirect
461. C3#70325semidirect

Maximal subgroups

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

subgroupindex
C2×#638632
#105948182
C2×C3×#8242
C3×#51312
#105910862
#105894212
#105891602
#703253
#701043