Finite Group Databasebeta

← #11786752

Subgroups — #11786752

Order 1152.

621 conjugacy classes of subgroups.

The subgroup lattice has 489 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5504C211yesmaximal, characteristic
2C3×#4977C211yesmaximal, characteristic, complemented
3#10591139C211yesmaximal, characteristic, complemented
4C2×#63569C211yesmaximal, characteristic
5#10591087C211yesmaximal, characteristic
6#10589396C211yesmaximal, characteristic, complemented
7#10589161C211yesmaximal, characteristic, complemented
8#71665—31nomaximal
9#69562—31nomaximal
10C2×C3×#235C2211yescharacteristic
11C2×C32×C4⋊C4C2211yescharacteristic
12C2×#3748C2211yescharacteristic
13C2×#3748C2211yescharacteristic
14C3×C4×#67—22no—
15C3×#926C2211yescharacteristic
16C2×C3×C4⋊Dic3C2211yescharacteristic
17C2×C3×C4×Dic3C2211yescharacteristic
18C3×#893—21no—
19C3×#892—21no—
20C3×D3×C4⋊C4—21no—
21C3×#887—21no—
22C3×C3⋊#95—22no—
23#63576—21no—
24#63574—21no—
25#63569C414yescomplemented
26#63565—21no—
27#63562—21no—
28C3×#520—31no—
29#5504D311yescharacteristic
30C3×#382—31no—
31C2×#837—31no—
32#5232—31no—
33#4995—31no—
34#4977D311yescharacteristic, complemented
35#4966—31no—
36#4964—31no—
37C2×#805—31no—
38#4532—31no—
39#4530—31no—
40#4530—31no—
41#4530—61no—
42#4523—31no—
43C22×C32×C4D412yes—
44C22×C32×C4C2311yescharacteristic
45C2×C3×#67—21no—
46C22×C3×Dic3—21no—
47C22×C3×Dic3—21no—
48C3×#235—21no—
49C3×#235—21no—
50C3×#235C2×C412yes—
51C3×#235Q812yes—
52C2×C3×D12D411yescharacteristic
53C2×C3×C4×D3—21no—
54C2×C3×C4×D3—22no—
55C2×C3×Dic6D411yescharacteristic
56C32×C4⋊C4—21no—
57C32×C4⋊C4—21no—
58#3748—21no—
59#3748—21no—
60#3748—21no—
61#3748C2×C412yescomplemented
62C2×C3⋊(C3⋊C8)—21no—
63C2×C3⋊(C3⋊C8)—21no—
64C3×#217—21no—
65C3×(C2×D3)⋊C4—22no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3D411yescharacteristic
69C3×C4⋊Dic3D411yescharacteristic
70C3×Dic3⋊C4—22no—
71C3×C4×Dic3—22no—
72C3×C4×Dic3—21no—
73C3×C4×Dic3—21no—
74C3×C4×Dic3C2×C412yescomplemented
75#1860—91noSylow
76C2×C3×#41—31no—
77C2×#235C2×D311yescharacteristic
78C3×C4×Q8—61no—
79C3×C4×D4—61no—
80C3×C4×D4—62no—
81C3×#117—31no—
82C3×#117—61no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—31no—
85C2×C3×C4⋊C4—61no—
86C2×C3×C4⋊C4C2×D311yescharacteristic
87C2×C3×C4⋊C4C2×D311yescharacteristic
88C2×C3×C42—31no—
89C4×#67—22no—
90#926C2×D311yescharacteristic
91C2×C4⋊Dic3C2×D311yescharacteristic
92C2×C4×Dic3C2×D311yescharacteristic
93C2×#208—31no—
94C2×#208—31no—
95C2×#208—31no—
96C2×#208—31no—
97C2×#208—61no—
98C2×#208—61no—
99#893—21no—
100#892—21no—
101D3×C4⋊C4—21no—
102#887—21no—
103C3×#95—62no—
104#837—34no—
105#835—61no—
106#832—61no—
107C3⋊#95—22no—
108#810—61no—
109#809—61no—
110#808—61no—
111#808—61no—
112#807—61no—
113#807—61no—
114#805—34no—
115C23×C32C2×D411yescharacteristic
116C22×C3×D3—21no—
117C2×C32×C4—22no—
118C2×C32×C4—42no—
119C2×C32×C4C22×C411yescharacteristic
120C2×C32×C4C2×D411yescharacteristic
121C2×C32×C4C2×D411yescharacteristic
122C2×C32×C4C2×Q811yescharacteristic
123C2×C32×C4#4111yescharacteristic
124C2×C32×C4#4111yescharacteristic
125C3×#67—22no—
126C3×#67—22no—
127C2×C3×Dic3—21no—
128C2×C3×Dic3—22no—
129C2×C3×Dic3—21no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—41no—
132C2×C3×Dic3—41no—
133C2×C3×Dic3—42no—
134C2×C3×Dic3—41no—
135C3×D12—21no—
136C3×D12C4⋊C412yes—
137C3×C4×D3—22no—
138C3×C4×D3—22no—
139C3×C4×D3—82no—
140C3×Dic6—21no—
141C3×Dic6C4⋊C412yes—
142C3⋊(C3⋊C8)—22no—
143C3⋊(C3⋊C8)—41no—
144#520—31no—
145#428—91no—
146C2×#104—91no—
147#417—91no—
148#382—31no—
149#343—91no—
150#339—91no—
151C3×#41—32no—
152C3×#41—32no—
153C3×#41—61no—
154C3×#41—61no—
155C2×C3×Q8—31no—
156C2×C3×D4—31no—
157C2×C3×D4—61no—
158C22×C3×C4—22no—
159C22×C3×C4—21no—
160C22×C3×C4—62no—
161C22×C3×C4—61no—
162C22×C3×C4—61no—
163C22×C3×C4—61no—
164C22×C3×C4#6712yes—
165C22×C3×C4#6712yes—
166C22×C3×C4C22×D311yescharacteristic
167C22×C3×C4C22×D311yescharacteristic
168C2×#67—21no—
169C22×Dic3—21no—
170C22×Dic3—21no—
171#235—21no—
172#235—21no—
173#235Dic612yes—
174#235C4×D312yes—
175C2×D12#6711yescharacteristic
176C2×C4×D3—21no—
177C2×C4×D3—22no—
178C2×Dic6#6711yescharacteristic
179C3×C4⋊C4—21no—
180C3×C4⋊C4—21no—
181C3×C4⋊C4—21no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—31no—
184C3×C4⋊C4—31no—
185C3×C4⋊C4—41no—
186C3×C4⋊C4—41no—
187C3×C4⋊C4—62no—
188C3×C4⋊C4—61no—
189C3×C4⋊C4—61no—
190C3×C22⋊C4—62no—
191C3×C22⋊C4—61no—
192C3×C42—32no—
193C3×C42—62no—
194C3×C42—61no—
195C3×C42—61no—
196#217—21no—
197(C2×D3)⋊C4—22no—
198C4⋊Dic3—21no—
199C4⋊Dic3—21no—
200C4⋊Dic3#6711yescharacteristic
201C4⋊Dic3#6711yescharacteristic
202Dic3⋊C4—22no—
203C4×Dic3—22no—
204C4×Dic3—21no—
205C4×Dic3—21no—
206C4×Dic3C4×D312yescomplemented
207#208—32no—
208#208—32no—
209#208—62no—
210#208—61no—
211#208—61no—
212#208—62no—
213#208—61no—
214#208—61no—
215#208—61no—
216#208—61no—
217#208—121no—
218#208—121no—
219C2×C3⋊C8—61no—
220C2×C3⋊C8—61no—
221C2×C3⋊C8—61no—
222C2×C3⋊C8—61no—
223C2×C3⋊C8—121no—
224C2×C3⋊C8—121no—
225C22×C32—21no—
226C22×C32—21no—
227C22×C32C4×D411yescharacteristic
228C22×C32#12011yescharacteristic
229C22×C32C22⋊Q811yescharacteristic
230C2×C3×D3—22no—
231C2×C3×D3—41no—
232C2×C3×D3—41no—
233C32×C4—42no—
234C32×C4C2×C4⋊C411yescharacteristic
235C32×C4C4×D411yescharacteristic
236C32×C4C22⋊Q811yescharacteristic
237C32×C4#12311yescharacteristic
238C3×Dic3—22no—
239C3×Dic3—42no—
240C3×Dic3—41no—
241C3×Dic3—41no—
242C3×Dic3—81no—
243C2×#41—31no—
244C2×M16—91no—
245C2×M16—91no—
246C4×Q8—61no—
247C4×D4—61no—
248C4×D4—62no—
249#117—31no—
250#117—61no—
251C2×C4⋊C4—31no—
252C2×C4⋊C4—61no—
253C2×C4⋊C4D3211yescharacteristic
254C2×C42—31no—
255#107—181no—
256#106—181no—
257#104—94no—
258#103—181no—
259#102—181no—
260#95—62no—
261C23×C3—21no—
262C23×C3—61no—
263C23×C3C2×#6711yescharacteristic
264C23×C3C2×#6711yescharacteristic
265C22×D3—21no—
266C3×Q8—32no—
267C3×Q8—61no—
268C3×D4—32no—
269C3×D4—62no—
270C3×D4—62no—
271C3×D4—61no—
272C2×C3×C4—22no—
273C2×C3×C4—22no—
274C2×C3×C4—21no—
275C2×C3×C4—21no—
276C2×C3×C4—21no—
277C2×C3×C4—21no—
278C2×C3×C4—21no—
279C2×C3×C4—21no—
280C2×C3×C4—42no—
281C2×C3×C4—42no—
282C2×C3×C4—42no—
283C2×C3×C4—61no—
284C2×C3×C4—62no—
285C2×C3×C4—62no—
286C2×C3×C4—62no—
287C2×C3×C4—61no—
288C2×C3×C4—61no—
289C2×C3×C4—82no—
290C2×C3×C4—121no—
291C2×C3×C4—121no—
292C2×C3×C4—122no—
293C2×C3×C4—121no—
294C2×C3×C4—242no—
295C2×C3×C4C2×Dic611yescharacteristic
296C2×C3×C4C2×C4×D311yescharacteristic
297C2×C3×C4C2×C4×D311yescharacteristic
298C2×C3×C4#23511yescharacteristic
299C2×C3×C4#23511yescharacteristic
300C2×C3×C4#23511yescharacteristic
301C2×C3×C4D3×D411yescharacteristic
302C2×C3×C4D3×D411yescharacteristic
303C2×C3×C4D3×Q811yescharacteristic
304C2×C3×C4#23911yescharacteristic
305C2×C3×C4C2×#6711yescharacteristic
306C2×C3×C4C2×#6711yescharacteristic
307#67—22no—
308#67—22no—
309C2×Dic3—21no—
310C2×Dic3—22no—
311C2×Dic3—21no—
312C2×Dic3—21no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—42no—
316C2×Dic3—41no—
317D12—21no—
318D12Dic3⋊C412yes—
319C4×D3—22no—
320C4×D3—22no—
321C4×D3—82no—
322Dic6—21no—
323Dic6Dic3⋊C412yes—
324C3⋊C8—62no—
325C3⋊C8—62no—
326C3⋊C8—122no—
327C3⋊C8—122no—
328C3⋊C8—121no—
329C3⋊C8—121no—
330C2×C32—21no—
331C2×C32—21no—
332C2×C32#38511yescharacteristic
333C2×C32#45411yescharacteristic
334C2×C32#45511yescharacteristic
335C3×D3—42no—
336#41—32no—
337#41—32no—
338#41—61no—
339#41—61no—
340C2×Q8—31no—
341C2×D4—31no—
342C2×D4—61no—
343C22×C4—62no—
344C22×C4—61no—
345C22×C4—61no—
346C22×C4—61no—
347C22×C4#62512yes—
348C22×C4C2×D3211yescharacteristic
349M16—92no—
350M16—181no—
351M16—181no—
352M16—181no—
353C2×C8—181no—
354C2×C8—181no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—31no—
358C4⋊C4—31no—
359C4⋊C4—62no—
360C4⋊C4—61no—
361C4⋊C4—61no—
362C22⋊C4—62no—
363C22⋊C4—61no—
364C42—32no—
365C42—62no—
366C42—61no—
367C42—61no—
368C22×C3—21no—
369C22×C3—21no—
370C22×C3—21no—
371C22×C3—21no—
372C22×C3—21no—
373C22×C3—21no—
374C22×C3—21no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—62no—
378C22×C3—121no—
379C22×C3—121no—
380C22×C3#92411yescharacteristic
381C22×C3C4×#6711yescharacteristic
382C22×C3C4×#6711yescharacteristic
383C22×C3#93711yescharacteristic
384C22×C3#94611yescharacteristic
385C22×C3#95311yescharacteristic
386C2×D3—22no—
387C2×D3—41no—
388C2×D3—41no—
389C3×C4—21no—
390C3×C4—21no—
391C3×C4—21no—
392C3×C4—21no—
393C3×C4—42no—
394C3×C4—42no—
395C3×C4—62no—
396C3×C4—82no—
397C3×C4—122no—
398C3×C4—121no—
399C3×C4—121no—
400C3×C4—241no—
401C3×C4D3×C4⋊C411yescharacteristic
402C3×C4#89311yescharacteristic
403C3×C4#89411yescharacteristic
404C3×C4#89611yescharacteristic
405C3×C4C2×Dic3⋊C411yescharacteristic
406C3×C4#92411yescharacteristic
407C3×C4C4×#6711yescharacteristic
408C3×C4#92911yescharacteristic
409Dic3—22no—
410Dic3—42no—
411Dic3—41no—
412Dic3—41no—
413Dic3—81no—
414C32#186011yescharacteristic, complemented, Sylow
415C23—61no—
416C23C2×#62511yescharacteristic
417Q8—32no—
418Q8—61no—
419D4—32no—
420D4—62no—
421D4—62no—
422D4—61no—
423C2×C4—22no—
424C2×C4—42no—
425C2×C4—61no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—61no—
430C2×C4—61no—
431C2×C4—121no—
432C2×C4—121no—
433C2×C4—122no—
434C2×C4—121no—
435C2×C4—242no—
436C2×C4D3×Dic611yescharacteristic
437C2×C4#379811yescharacteristic
438C2×C4#379911yescharacteristic
439C2×C4C4×D3211yescharacteristic
440C2×C4D3×#6711yescharacteristic
441C2×C4D3×#6711yescharacteristic
442C8—182no—
443C8—361no—
444C2×C3—21no—
445C2×C3—21no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—41no—
452C2×C3—41no—
453C2×C3—122no—
454C2×C3#498911yescharacteristic
455C2×C3#506111yescharacteristic
456C2×C3#506211yescharacteristic
457C2×C3#521011yescharacteristic
458C2×C3#524211yescharacteristic
459C2×C3#524511yescharacteristic
460D3—42no—
461C22—21no—
462C22—21no—
463C22—62no—
464C22—121no—
465C22—121no—
466C22#6385611yescharacteristic
467C22C4×#62511yescharacteristic
468C22#6397611yescharacteristic
469C4—42no—
470C4—62no—
471C4—122no—
472C4—121no—
473C4—121no—
474C4—241no—
475C4#6385011yescharacteristic
476C4#6385111yescharacteristic
477C4D3×Dic3⋊C411yescharacteristic
478C4#6390411yescharacteristic
479C3—21no—
480C3#6956211yescharacteristic, complemented
481C3#7166511yescharacteristic, complemented
482C2—21no—
483C2—21no—
484C2—122no—
485C2#1059127711yescharacteristic
486C2#1059151211yescharacteristic
487C2#1059164511yescharacteristic
subgroupG/H
centerC22#63976
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×C4⋊C4C22
socleC22×C32#120
prime radicalC2×C3×#235C22
cosocleC22×C32×C4C23
hypercenterC2×C4⋊C4D32
hypocenterC32#1860
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#4977C2semidirect
3. #10591139C2semidirect
6. #10589396C2semidirect
7. #10589161C2semidirect
25. #63569C4semidirect
34. #4977D3semidirect
61. #3748C2×C4semidirect
74. C3×C4×Dic3C2×C4semidirect
206. C4×Dic3C4×D3semidirect
414. C32#1860semidirect
480. C3#69562semidirect
481. C3#71665semidirect

Maximal subgroups

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

subgroupindex
C3×#55042
C3×#49772
#105911392
C2×#635692
#105910872
#105893962
#105891612
#716653
#695623