Finite Group Databasebeta

← #11896070

Subgroups — #11896070

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×Q8×Dic6C211yesmaximal, characteristic, complemented
2#10593865C211yesmaximal, characteristic
3C4×#3755C211yesmaximal, characteristic
4C3×#5031C211yesmaximal, characteristic, complemented
5#10591607C212yesmaximal
6#10591576C212yesmaximal, complemented
7#10590535C212yesmaximal
8#10590486C212yesmaximal, complemented
9#10590300C211yesmaximal, characteristic
10#10590290C211yesmaximal, characteristic, complemented
11#10590259C211yesmaximal, characteristic, complemented
12#83704—31nomaximal
13#80464—31nomaximal
14C32×C4×Q8C2211yescharacteristic
15C2×#3755C2211yescharacteristic
16C4×C4⋊(C3⋊D3)C2211yescharacteristic
17C3×Q8×Dic3C2212yes—
18C2×C3×#216C2212yes—
19C3×#888—22no—
20C3×#888C2212yes—
21C3×#869—21no—
22C3×#869C2211yescharacteristic
23C3×#869C2211yescharacteristic
24C3×C4×Dic6—21no—
25C3×C4×Dic6C2211yescharacteristic
26#63871—21no—
27#63864C2212yes—
28#63849—21no—
29#63848—22no—
30Dic3×Dic6C2212yes—
31#63662C2211yescharacteristic
32#63636C2211yescharacteristic
33#63635C2211yescharacteristic
34C4×C3⋊(C3⋊C8)C2211yescharacteristic
35C3×#810C2212yes—
36C3×C4⋊(C3⋊C8)C2211yescharacteristic
37#63573C2212yes—
38Dic3⋊(C3⋊C8)C2212yes—
39#63565C2212yes—
40#63565C2212yes—
41#63564C2212yes—
42#63554C2212yes—
43C3×Q82—31no—
44#5732—31no—
45#5584—31no—
46Q8×Dic6D311yescharacteristic, complemented
47C3×#462—31no—
48#5185—32no—
49#5184—32no—
50#5094—32no—
51#5093—31no—
52#5089—32no—
53#5084—31no—
54#5079—31no—
55#5031D311yescharacteristic, complemented
56#5030—32no—
57#5028—32no—
58C4×#215—31no—
59C4×#215—31no—
60C4×#215—61no—
61#5024—31no—
62#4842—31no—
63#4831—31no—
64#4794—32no—
65#4791—32no—
66C2×C32×Q8C2311yescharacteristic
67C2×C4⋊(C3⋊D3)C2311yescharacteristic
68C2×C4×C3⋊D3—21no—
69C2×C3×Dic6—21no—
70C2×C3×Dic6—22no—
71C2×C3×Dic6D411yescharacteristic
72C2×C3×Dic6C2312yes—
73C32×C4⋊C4—21no—
74C32×C4⋊C4C2311yescharacteristic
75C32×C42—21no—
76C32×C42C2311yescharacteristic
77#3755—22no—
78#3752—21no—
79C4⋊(C3⋊Dic3)C2311yescharacteristic
80C2×C3⋊(C3⋊C8)C2311yescharacteristic
81C2×C3⋊(C3⋊C8)C2311yescharacteristic
82C3×#216—24no—
83C3×C4⋊Dic3—21no—
84C3×C4⋊Dic3—21no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—22no—
87C3×C4⋊Dic3D411yescharacteristic
88C3×C4⋊Dic3C2312yes—
89C3×Dic3⋊C4—22no—
90C3×Dic3⋊C4D412yes—
91C3×C4×Dic3—22no—
92C3×C4×Dic3C2312yes—
93C2×C3×C3⋊C8C2312yes—
94Dic3⋊Dic3—22no—
95#3722—22no—
96#3722—22no—
97Dic32—22no—
98#3369—91noSylow
99C2×C3×Q16—32no—
100C3×C4⋊Q8—32no—
101C3×C4⋊Q8—31no—
102C3×C4⋊Q8—31no—
103C3×C4⋊Q8—61no—
104C3×C4⋊Q8—62no—
105C3×C4×Q8—21no—
106C3×C4×Q8—31no—
107C3×C4×Q8—32no—
108C3×C4×Q8—61no—
109C3×C4×Q8C2×D311yescharacteristic
110C3×C4×Q8C2×D311yescharacteristic
111#947—32no—
112#946—61no—
113Q8×Dic3—32no—
114Q8×Dic3C2×D312yes—
115C2×#216C2×D312yes—
116C2×#215—31no—
117C2×#215—31no—
118C2×#215—61no—
119#898—62no—
120#898—62no—
121#897—61no—
122#892—61no—
123#892—61no—
124#888—22no—
125#888C2×D312yes—
126#875—32no—
127C4×D12—31no—
128C4×D12—31no—
129C4×D12—61no—
130#869—21no—
131#869C2×D311yescharacteristic
132#869C2×D311yescharacteristic
133C4×Dic6—21no—
134C4×Dic6—32no—
135C4×Dic6C2×D311yescharacteristic
136C3×C4⋊C8—31no—
137C3×#103—32no—
138#835—31no—
139#835—31no—
140#835—32no—
141#835—32no—
142#835—61no—
143#821—32no—
144#817—32no—
145#815—32no—
146Dic3⋊C8—32no—
147#810—32no—
148#810—32no—
149#810C2×D312yes—
150#809—31no—
151#809—31no—
152#809—32no—
153#809—61no—
154#808—31no—
155#808—31no—
156#808—32no—
157#808—61no—
158C4⋊(C3⋊C8)—32no—
159C4⋊(C3⋊C8)C2×D311yescharacteristic
160#803—32no—
161C4×C3⋊C8—31no—
162C4×C3⋊C8—31no—
163C4×C3⋊C8—61no—
164C22×C3⋊D3—21no—
165C32×Q8—21no—
166C32×Q8#4112yes—
167C2×C32×C4—21no—
168C2×C32×C4—21no—
169C2×C32×C4C2×D411yescharacteristic
170C2×C32×C4C2×D411yescharacteristic
171C2×C32×C4C2411yescharacteristic
172C2×C3⋊Dic3—21no—
173C4⋊(C3⋊D3)—21no—
174C4⋊(C3⋊D3)—21no—
175C4×C3⋊D3—42no—
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—82no—
184C3×Dic6—81no—
185C3⋊(C3⋊C8)—21no—
186C3⋊(C3⋊C8)—21no—
187C3×C3⋊C8—22no—
188Q82—31no—
189#556—91no—
190#489—92no—
191#487—91no—
192#476—91no—
193#475—92no—
194#464—92no—
195#462—31no—
196#446—91no—
197#441—92no—
198C4×SD16—91no—
199C2×C3×Q8—21no—
200C2×C3×Q8—31no—
201C2×C3×Q8—32no—
202C2×C3×Q8—62no—
203C2×C3×Q8—61no—
204C2×C3×Q8C22×D311yescharacteristic
205C2×C3×Q8C22×D311yescharacteristic
206C2×D12—31no—
207C2×D12—31no—
208C2×D12—61no—
209C2×C4×D3—61no—
210C2×C4×D3—61no—
211C2×C4×D3—121no—
212C2×Dic6—21no—
213C2×Dic6—22no—
214C2×Dic6#6711yescharacteristic
215C2×Dic6C22×D312yes—
216C3×Q16—64no—
217C2×C3×C8—32no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—21no—
221C3×C4⋊C4—31no—
222C3×C4⋊C4—32no—
223C3×C4⋊C4—32no—
224C3×C4⋊C4—41no—
225C3×C4⋊C4—62no—
226C3×C4⋊C4—62no—
227C3×C4⋊C4—61no—
228C3×C4⋊C4—61no—
229C3×C4⋊C4—61no—
230C3×C4⋊C4C22×D311yescharacteristic
231C3×C4⋊C4C22×D311yescharacteristic
232C3×C42—21no—
233C3×C42—21no—
234C3×C42—21no—
235C3×C42—32no—
236C3×C42—41no—
237C3×C42—62no—
238C3×C42C22×D311yescharacteristic
239C3×C42C22×D311yescharacteristic
240#216—24no—
241#215—64no—
242#215—62no—
243#215—62no—
244(C2×D3)⋊C4—62no—
245(C2×D3)⋊C4—62no—
246(C2×D3)⋊C4—61no—
247(C2×D3)⋊C4—61no—
248(C2×D3)⋊C4—62no—
249(C2×D3)⋊C4—62no—
250(C2×D3)⋊C4—121no—
251C4⋊Dic3—21no—
252C4⋊Dic3—21no—
253C4⋊Dic3—21no—
254C4⋊Dic3—22no—
255C4⋊Dic3—31no—
256C4⋊Dic3—31no—
257C4⋊Dic3—61no—
258C4⋊Dic3—62no—
259C4⋊Dic3#6711yescharacteristic
260C4⋊Dic3C22×D312yes—
261Dic3⋊C4—22no—
262Dic3⋊C4—62no—
263Dic3⋊C4#6712yes—
264C4×Dic3—22no—
265C4×Dic3—62no—
266C4×Dic3—62no—
267C4×Dic3C22×D312yes—
268C2×C3⋊C8—31no—
269C2×C3⋊C8—31no—
270C2×C3⋊C8—31no—
271C2×C3⋊C8—31no—
272C2×C3⋊C8—61no—
273C2×C3⋊C8—61no—
274C2×C3⋊C8C22×D312yes—
275C22×C32C22×D411yescharacteristic
276C2×C3⋊D3—41no—
277C2×C3⋊D3—41no—
278C2×C3⋊D3—41no—
279C32×C4—22no—
280C32×C4—21no—
281C32×C4—41no—
282C32×C4#13712yes—
283C32×C4C2×#4111yescharacteristic
284C32×C42−1+411yescharacteristic
285C3⋊Dic3—41no—
286C3×Dic3—22no—
287C3×Dic3—22no—
288C3×Dic3—42no—
289C3×Dic3—42no—
290C2×Q16—32no—
291C2×SD16—91no—
292C4⋊Q8—32no—
293C4⋊Q8—31no—
294C4⋊Q8—31no—
295C4⋊Q8—61no—
296C4⋊Q8—62no—
297#126—182no—
298#124—92no—
299C22⋊Q8—181no—
300C4×Q8—31no—
301C4×Q8—32no—
302C4×Q8—61no—
303C4×Q8—92no—
304C4×Q8D3211yescharacteristic
305C4×D4—91no—
306#117—181no—
307#106—92no—
308#106—91no—
309C4⋊C8—31no—
310C4⋊C8—92no—
311#103—32no—
312#103—92no—
313#103—92no—
314#103—91no—
315#102—92no—
316#102—91no—
317#97—92no—
318C4×C8—91no—
319C22×D3—61no—
320C22×D3—61no—
321C22×D3—121no—
322C3×Q8—21no—
323C3×Q8—21no—
324C3×Q8—22no—
325C3×Q8—41no—
326C3×Q8—62no—
327C3×Q8—62no—
328C3×Q8—62no—
329C3×Q8—241no—
330C3×Q8—242no—
331C3×Q8#23712yes—
332C3×Q8#23912yes—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—21no—
339C2×C3×C4—21no—
340C2×C3×C4—32no—
341C2×C3×C4—32no—
342C2×C3×C4—41no—
343C2×C3×C4—41no—
344C2×C3×C4—62no—
345C2×C3×C4—62no—
346C2×C3×C4D3×D411yescharacteristic
347C2×C3×C4D3×D411yescharacteristic
348C2×C3×C4C2×#6711yescharacteristic
349C2×C3×C4C2×#6711yescharacteristic
350C2×C3×C4C23×D311yescharacteristic
351C2×C3×C4C23×D311yescharacteristic
352C2×Dic3—22no—
353C2×Dic3—22no—
354C2×Dic3—61no—
355C2×Dic3—61no—
356C2×Dic3—121no—
357C2×Dic3C2×#6712yes—
358C2×Dic3C2×#6712yes—
359D12—62no—
360D12—61no—
361D12—61no—
362D12—61no—
363D12—61no—
364D12—121no—
365C4×D3—122no—
366C4×D3—122no—
367C4×D3—242no—
368Dic6—22no—
369Dic6—22no—
370Dic6—22no—
371Dic6—82no—
372Dic6—81no—
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#54811yescharacteristic
382C2×C32C2×#13711yescharacteristic
383C2×C32#57711yescharacteristic
384C3⋊D3—81no—
385C2×Q8—31no—
386C2×Q8—32no—
387C2×Q8—62no—
388C2×Q8—61no—
389C2×Q8C2×D3211yescharacteristic
390C2×D4—91no—
391C22×C4—181no—
392Q16—64no—
393SD16—182no—
394C2×C8—32no—
395C2×C8—91no—
396C2×C8—91no—
397C4⋊C4—21no—
398C4⋊C4—31no—
399C4⋊C4—32no—
400C4⋊C4—32no—
401C4⋊C4—62no—
402C4⋊C4—62no—
403C4⋊C4—61no—
404C4⋊C4—61no—
405C4⋊C4—61no—
406C4⋊C4—91no—
407C4⋊C4—182no—
408C4⋊C4C2×D3211yescharacteristic
409C22⋊C4—182no—
410C22⋊C4—182no—
411C22⋊C4—181no—
412C42—21no—
413C42—32no—
414C42—62no—
415C42—182no—
416C42C2×D3211yescharacteristic
417C22×C3—21no—
418C22×C3C2×D3×D411yescharacteristic
419C22×C3C22×#6711yescharacteristic
420C2×D3—122no—
421C2×D3—121no—
422C2×D3—121no—
423C2×D3—121no—
424C2×D3—121no—
425C2×D3—121no—
426C2×D3—121no—
427C2×D3—241no—
428C2×D3—241no—
429C3×C4—21no—
430C3×C4—22no—
431C3×C4—21no—
432C3×C4—22no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—22no—
436C3×C4—41no—
437C3×C4—42no—
438C3×C4—41no—
439C3×C4—41no—
440C3×C4—62no—
441C3×C4—62no—
442C3×C4—81no—
443C3×C4—122no—
444C3×C4—122no—
445C3×C4#91812yes—
446C3×C4#94212yes—
447C3×C4C2×#23711yescharacteristic
448C3×C4C2×#23911yescharacteristic
449C3×C4#100711yescharacteristic
450C3×C4#101011yescharacteristic
451Dic3—22no—
452Dic3—22no—
453Dic3—42no—
454Dic3—42no—
455Dic3—121no—
456Dic3—121no—
457Dic3—241no—
458C32#336911yescharacteristic, complemented, Sylow
459C23—181no—
460Q8—21no—
461Q8—62no—
462Q8—62no—
463Q8—62no—
464Q8—241no—
465Q8—242no—
466Q8#379612yes—
467D4—181no—
468D4—181no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—32no—
472C2×C4—32no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—181no—
476C2×C4—362no—
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#559811yescharacteristic
487C2×C3C2×#91811yescharacteristic
488C2×C3#577811yescharacteristic
489C2×C3C2×#94211yescharacteristic
490C2×C3#581711yescharacteristic
491C2×C3#583611yescharacteristic
492D3—242no—
493D3—241no—
494D3—241no—
495C22—361no—
496C22—361no—
497C22—361no—
498C22C2×D3×#6711yescharacteristic
499C4—22no—
500C4—21no—
501C4—41no—
502C4—62no—
503C4—62no—
504C4—122no—
505C4—122no—
506C4—361no—
507C4#6394812yes—
508C4C2×#379611yescharacteristic
509C4#6429711yescharacteristic
510C3—21no—
511C3#8046411yescharacteristic, complemented
512C3#8370411yescharacteristic, complemented
513C2—721no—
514C2#1059490011yescharacteristic
515C2C2×#6394811yescharacteristic
516C2#1059546411yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4×Q8C22
socleC22×C32C22×D4
prime radicalC2×C4⋊(C3⋊D3)C23
hypercenterC4×Q8D32
hypocenterC32#3369
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C22 ◁ C42 ◁ C4×Q8

Complements

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

NH ≅ G/Ntype
1. C3×Q8×Dic6C2semidirect
4. C3×#5031C2semidirect
6. #10591576C2semidirect
8. #10590486C2semidirect
10. #10590290C2semidirect
11. #10590259C2semidirect
46. Q8×Dic6D3semidirect
55. #5031D3semidirect
458. C32#3369semidirect
511. C3#80464semidirect
512. C3#83704semidirect

Maximal subgroups

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

subgroupindex
C3×Q8×Dic62
#105938652
C4×#37552
C3×#50312
#105916072
#105915762
#105905352
#105904862
#105903002
#105902902
#105902592
#837043
#804643