Finite Group Databasebeta

← #11896065

Subgroups — #11896065

Order 1152.

737 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5542C211yesmaximal, characteristic
2#10593856C211yesmaximal, characteristic, complemented
3C4×#3754C211yesmaximal, characteristic
4C3×#5008C211yesmaximal, characteristic
5#10591488C212yesmaximal
6#10591441C212yesmaximal
7#10590556C212yesmaximal, complemented
8#10590480C212yesmaximal, complemented
9#10590388C211yesmaximal, characteristic, complemented
10#10590290C211yesmaximal, characteristic, complemented
11#10590259C211yesmaximal, characteristic, complemented
12#83677—31nomaximal
13#81091—31nomaximal
14C32×C4×D4C2211yescharacteristic
15C2×#3754C2211yescharacteristic
16C4×#634C2211yescharacteristic
17C3×#936C2212yes—
18C3×#926—21no—
19C3×#924—21no—
20C3×#887C2212yes—
21C3×#879—22no—
22C3×#869C2211yescharacteristic
23C3×#869C2211yescharacteristic
24C3×C4×Dic6C2211yescharacteristic
25#63919—21no—
26#63918C2212yes—
27#63844C2212yes—
28#63843—21no—
29#63839—22no—
30C2×#3719C2212yes—
31#63659C2211yescharacteristic
32#63637C2211yescharacteristic
33#63635C2211yescharacteristic
34C4×C3⋊(C3⋊C8)C2211yescharacteristic
35C3×#832C2212yes—
36C3×#808C2212yes—
37C3×C4⋊(C3⋊C8)C2211yescharacteristic
38Dic3⋊(C3⋊C8)C2212yes—
39#63566C2212yes—
40#63565C2212yes—
41#63565C2212yes—
42#63554C2212yes—
43C3×#556—31no—
44#5722—31no—
45Q8×Dic6—31no—
46#5542D311yescharacteristic
47C3×#476—31no—
48#5093—31no—
49#5090—31no—
50#5079—31no—
51#5055—32no—
52#5054—32no—
53#5030—32no—
54#5024—31no—
55#5020—32no—
56C4×#214—31no—
57C4×#214—31no—
58C4×#214—61no—
59#5008D311yescharacteristic
60#4854—32no—
61#4853—32no—
62#4849—31no—
63#4831—31no—
64#4767—32no—
65#4765—32no—
66C2×C32×D4C2311yescharacteristic
67C22×C32×C4—21no—
68C2×#634C2311yescharacteristic
69C2×C3×Dic6—21no—
70C2×C3×Dic6D411yescharacteristic
71C2×C3×Dic6C2312yes—
72C2×#627—22no—
73C32×C4⋊C4C2311yescharacteristic
74C32×C22⋊C4—21no—
75C32×C42C2311yescharacteristic
76#3754—22no—
77C4⋊(C3⋊Dic3)C2311yescharacteristic
78#3750—21no—
79C4×C3⋊Dic3—21no—
80C2×C3⋊(C3⋊C8)C2311yescharacteristic
81C2×C3⋊(C3⋊C8)C2311yescharacteristic
82C3×#217—22no—
83C3×#217—22no—
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—21no—
87C3×C4⋊Dic3D411yescharacteristic
88C3×C4⋊Dic3C2312yes—
89C3×Dic3⋊C4—22no—
90C3×Dic3⋊C4D412yes—
91C3×C4×Dic3—22no—
92C3×C4×Dic3C2312yes—
93C2×C3×C3⋊C8C2312yes—
94#3724—22no—
95Dic3⋊Dic3—22no—
96Dic32—22no—
97#3719—24no—
98#3369—91noSylow
99C3×C4⋊Q8—31no—
100C3×C4⋊Q8—31no—
101C3×#126—62no—
102C3×#124—32no—
103C3×C22⋊Q8—61no—
104C3×C4×Q8—31no—
105C3×C4×Q8—32no—
106C3×C4×D4—21no—
107C3×C4×D4C2×D311yescharacteristic
108C3×C4×D4C2×D311yescharacteristic
109C3×#117—61no—
110Q8×Dic3—61no—
111#944—62no—
112C2×#216—32no—
113#936C2×D312yes—
114C2×#214—31no—
115C2×#214—31no—
116C2×#214—61no—
117#926—21no—
118#924—21no—
119C2×Dic12—32no—
120#888—32no—
121#888—61no—
122#888—62no—
123#888—61no—
124#887—32no—
125#887—61no—
126#887C2×D312yes—
127#879—22no—
128#869—32no—
129#869C2×D311yescharacteristic
130#869C2×D311yescharacteristic
131C4×Dic6—31no—
132C4×Dic6—31no—
133C4×Dic6—32no—
134C4×Dic6—61no—
135C4×Dic6C2×D311yescharacteristic
136C3×#106—32no—
137C3×C4⋊C8—31no—
138C3×#102—32no—
139#835—32no—
140#835—32no—
141#835—32no—
142#832—31no—
143#832—31no—
144#832—61no—
145#832C2×D312yes—
146#816—32no—
147#815—32no—
148Dic3⋊C8—32no—
149#810—32no—
150#810—32no—
151#810—31no—
152#810—31no—
153#810—61no—
154#808—31no—
155#808—31no—
156#808—61no—
157#808C2×D312yes—
158C4⋊(C3⋊C8)—32no—
159C4⋊(C3⋊C8)C2×D311yescharacteristic
160#803—32no—
161C4×C3⋊C8—31no—
162C4×C3⋊C8—31no—
163C4×C3⋊C8—61no—
164C23×C32—21no—
165C32×D4—21no—
166C32×D4—21no—
167C2×C32×C4—21no—
168C2×C32×C4—42no—
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2411yescharacteristic
172C2×C3⋊Dic3—21no—
173C2×C3⋊Dic3—21no—
174#634—21no—
175#634#4112yes—
176C2×C3×Dic3—22no—
177C2×C3×Dic3—22no—
178C2×C3×Dic3C2×D412yes—
179C2×C3×Dic3C2×D412yes—
180C3×Dic6—22no—
181C3×Dic6—22no—
182C3×Dic6—22no—
183C3×Dic6—81no—
184#627—82no—
185C3⋊(C3⋊C8)—21no—
186C3⋊(C3⋊C8)—21no—
187C3×C3⋊C8—22no—
188Q82—91no—
189#556—31no—
190#489—92no—
191#487—91no—
192#476—31no—
193#475—92no—
194#464—92no—
195#462—91no—
196#446—91no—
197#441—92no—
198C4×SD16—91no—
199C2×C3×Q8—31no—
200C2×C3×Q8—32no—
201C2×C3×Q8—61no—
202C2×C3×D4—21no—
203C2×C3×D4C22×D311yescharacteristic
204C2×C3×D4C22×D311yescharacteristic
205C22×C3×C4—21no—
206C22×C3×C4—21no—
207C22×C3×C4—41no—
208C2×Dic6—21no—
209C2×Dic6—31no—
210C2×Dic6—31no—
211C2×Dic6—61no—
212C2×Dic6—62no—
213C2×Dic6—62no—
214C2×Dic6#6711yescharacteristic
215C2×Dic6C22×D312yes—
216C2×C3×C8—32no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—32no—
219C3×C4⋊C4—31no—
220C3×C4⋊C4—32no—
221C3×C4⋊C4—62no—
222C3×C4⋊C4—61no—
223C3×C4⋊C4—61no—
224C3×C4⋊C4—61no—
225C3×C4⋊C4C22×D311yescharacteristic
226C3×C4⋊C4C22×D311yescharacteristic
227C3×C22⋊C4—21no—
228C3×C22⋊C4—21no—
229C3×C22⋊C4—41no—
230C3×C22⋊C4—62no—
231C3×C22⋊C4—62no—
232C3×C42—21no—
233C3×C42—32no—
234C3×C42—62no—
235C3×C42C22×D311yescharacteristic
236C3×C42C22×D311yescharacteristic
237#217—22no—
238#217—22no—
239#216—64no—
240#214—62no—
241#214—62no—
242#214—64no—
243C4⋊Dic3—21no—
244C4⋊Dic3—21no—
245C4⋊Dic3—21no—
246C4⋊Dic3—31no—
247C4⋊Dic3—31no—
248C4⋊Dic3—61no—
249C4⋊Dic3—62no—
250C4⋊Dic3#6711yescharacteristic
251C4⋊Dic3C22×D312yes—
252Dic3⋊C4—22no—
253Dic3⋊C4—62no—
254Dic3⋊C4—61no—
255Dic3⋊C4—61no—
256Dic3⋊C4—62no—
257Dic3⋊C4—62no—
258Dic3⋊C4—121no—
259Dic3⋊C4#6712yes—
260C4×Dic3—22no—
261C4×Dic3—62no—
262C4×Dic3—61no—
263C4×Dic3—61no—
264C4×Dic3—62no—
265C4×Dic3—121no—
266C4×Dic3C22×D312yes—
267C2×C3⋊C8—31no—
268C2×C3⋊C8—31no—
269C2×C3⋊C8—31no—
270C2×C3⋊C8—31no—
271C2×C3⋊C8—61no—
272C2×C3⋊C8—61no—
273C2×C3⋊C8C22×D312yes—
274Dic12—64no—
275C22×C32—41no—
276C22×C32—41no—
277C22×C32—41no—
278C22×C32C22×D411yescharacteristic
279C32×C4—21no—
280C32×C4—41no—
281C32×C4#13712yes—
282C32×C4C2×#4111yescharacteristic
283C32×C42−1+411yescharacteristic
284C3⋊Dic3—22no—
285C3⋊Dic3—41no—
286C3×Dic3—22no—
287C3×Dic3—22no—
288C3×Dic3—42no—
289C3×Dic3—42no—
290C2×Q16—92no—
291C2×SD16—91no—
292C4⋊Q8—31no—
293C4⋊Q8—31no—
294C4⋊Q8—92no—
295C4⋊Q8—181no—
296C4⋊Q8—182no—
297#126—62no—
298#124—32no—
299C22⋊Q8—61no—
300C4×Q8—31no—
301C4×Q8—32no—
302C4×Q8—92no—
303C4×Q8—91no—
304C4×Q8—181no—
305C4×D4D3211yescharacteristic
306#117—61no—
307#106—32no—
308#106—91no—
309C4⋊C8—31no—
310C4⋊C8—92no—
311#103—92no—
312#103—92no—
313#103—92no—
314#103—91no—
315#102—32no—
316#102—91no—
317#97—92no—
318C4×C8—91no—
319C23×C3—21no—
320C23×C3—21no—
321C23×C3—41no—
322C3×Q8—62no—
323C3×Q8—62no—
324C3×Q8—62no—
325C3×Q8—241no—
326C3×D4—22no—
327C3×D4—21no—
328C3×D4—21no—
329C3×D4—21no—
330C3×D4—21no—
331C3×D4—41no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—32no—
338C2×C3×C4—32no—
339C2×C3×C4—41no—
340C2×C3×C4—42no—
341C2×C3×C4—42no—
342C2×C3×C4—62no—
343C2×C3×C4—62no—
344C2×C3×C4—82no—
345C2×C3×C4D3×D411yescharacteristic
346C2×C3×C4D3×D411yescharacteristic
347C2×C3×C4C2×#6711yescharacteristic
348C2×C3×C4C2×#6711yescharacteristic
349C2×C3×C4C23×D311yescharacteristic
350C2×C3×C4C23×D311yescharacteristic
351C2×Dic3—22no—
352C2×Dic3—22no—
353C2×Dic3—61no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—61no—
357C2×Dic3—121no—
358C2×Dic3—121no—
359C2×Dic3C2×#6712yes—
360C2×Dic3C2×#6712yes—
361Dic6—22no—
362Dic6—22no—
363Dic6—22no—
364Dic6—32no—
365Dic6—32no—
366Dic6—62no—
367Dic6—61no—
368Dic6—61no—
369Dic6—81no—
370Dic6—121no—
371Dic6—242no—
372Dic6—242no—
373C3×C8—62no—
374C3⋊C8—22no—
375C3⋊C8—62no—
376C3⋊C8—61no—
377C3⋊C8—61no—
378C3⋊C8—61no—
379C3⋊C8—61no—
380C3⋊C8—121no—
381C2×C32—81no—
382C2×C32#54811yescharacteristic
383C2×C32C2×#13711yescharacteristic
384C2×C32#57711yescharacteristic
385C2×Q8—31no—
386C2×Q8—32no—
387C2×Q8—61no—
388C2×Q8—91no—
389C2×Q8—182no—
390C2×D4C2×D3211yescharacteristic
391C22×C4—21no—
392Q16—184no—
393SD16—182no—
394C2×C8—32no—
395C2×C8—91no—
396C2×C8—91no—
397C4⋊C4—31no—
398C4⋊C4—32no—
399C4⋊C4—32no—
400C4⋊C4—62no—
401C4⋊C4—61no—
402C4⋊C4—61no—
403C4⋊C4—61no—
404C4⋊C4—91no—
405C4⋊C4—182no—
406C4⋊C4—182no—
407C4⋊C4—181no—
408C4⋊C4C2×D3211yescharacteristic
409C22⋊C4—21no—
410C22⋊C4—62no—
411C22⋊C4—62no—
412C42—32no—
413C42—62no—
414C42—182no—
415C42—181no—
416C42C2×D3211yescharacteristic
417C22×C3—21no—
418C22×C3—42no—
419C22×C3—41no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—41no—
423C22×C3—41no—
424C22×C3—41no—
425C22×C3—81no—
426C22×C3—81no—
427C22×C3C2×D3×D411yescharacteristic
428C22×C3C22×#6711yescharacteristic
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—22no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—41no—
437C3×C4—62no—
438C3×C4—62no—
439C3×C4—81no—
440C3×C4—122no—
441C3×C4—122no—
442C3×C4#91512yes—
443C3×C4#95112yes—
444C3×C4#100711yescharacteristic
445C3×C4D3×#4111yescharacteristic
446C3×C4D3×#4111yescharacteristic
447C3×C4#101011yescharacteristic
448Dic3—22no—
449Dic3—22no—
450Dic3—42no—
451Dic3—42no—
452Dic3—62no—
453Dic3—62no—
454Dic3—122no—
455Dic3—121no—
456Dic3—121no—
457Dic3—241no—
458C32#336911yescharacteristic, complemented, Sylow
459C23—21no—
460Q8—62no—
461Q8—62no—
462Q8—62no—
463Q8—92no—
464Q8—181no—
465Q8—241no—
466Q8—722no—
467D4—21no—
468D4—21no—
469C2×C4—21no—
470C2×C4—32no—
471C2×C4—32no—
472C2×C4—42no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—181no—
476C2×C4—181no—
477C2×C4D3×#6711yescharacteristic
478C2×C4D3×#6711yescharacteristic
479C2×C4C22×D3211yescharacteristic
480C8—62no—
481C8—181no—
482C8—181no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—82no—
487C2×C3—81no—
488C2×C3—81no—
489C2×C3#563311yescharacteristic
490C2×C3C2×#91511yescharacteristic
491C2×C3#577511yescharacteristic
492C2×C3C2×#95111yescharacteristic
493C2×C3#583111yescharacteristic
494C2×C3#583711yescharacteristic
495C22—41no—
496C22—41no—
497C22—41no—
498C22C2×D3×#6711yescharacteristic
499C4—21no—
500C4—41no—
501C4—62no—
502C4—62no—
503C4—122no—
504C4—122no—
505C4—182no—
506C4—361no—
507C4#6392912yes—
508C4#6429711yescharacteristic
509C4#6430611yescharacteristic
510C3—21no—
511C3#8109111yescharacteristic, complemented
512C3#8367711yescharacteristic, complemented
513C2—81no—
514C2#1059483011yescharacteristic
515C2C2×#6392911yescharacteristic
516C2#1059545511yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×D4C22
socleC22×C32C22×D4
prime radicalC2×C32×D4C23
hypercenterC4×D4D32
hypocenterC32#3369
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
2. #10593856C2semidirect
7. #10590556C2semidirect
8. #10590480C2semidirect
9. #10590388C2semidirect
10. #10590290C2semidirect
11. #10590259C2semidirect
458. C32#3369semidirect
511. C3#81091semidirect
512. C3#83677semidirect

Maximal subgroups

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

subgroupindex
C3×#55422
#105938562
C4×#37542
C3×#50082
#105914882
#105914412
#105905562
#105904802
#105903882
#105902902
#105902592
#836773
#810913