Finite Group Databasebeta

← #11897572

Subgroups — #11897572

Order 1152.

802 conjugacy classes of subgroups.

The subgroup lattice has 529 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5730C211yesmaximal, characteristic, complemented
2#10593903C211yesmaximal, characteristic, complemented
3#10592783C211yesmaximal, characteristic, complemented
4C3×#5080C211yesmaximal, characteristic, complemented
5#10590589C212yesmaximal
6#10590558C212yesmaximal, complemented
7#10590470C212yesmaximal, complemented
8#10590419C212yesmaximal, complemented
9C4×#3714C211yesmaximal, characteristic
10#10590269C211yesmaximal, characteristic, complemented
11#10590258C211yesmaximal, characteristic, complemented
12#83630—31nomaximal
13#81905—31nomaximal
14C2×#3797—21no—
15C32×C4⋊Q8C2211yescharacteristic
16C4×C4⋊(C3⋊D3)C2211yescharacteristic
17C3×#946—22no—
18C3×#897—21no—
19C3×#893C2212yes—
20C3×D3×C4⋊C4—21no—
21C3×#890C2212yes—
22C3×C4×D12C2211yescharacteristic
23C3×#869C2211yescharacteristic
24C3×C4×Dic6C2211yescharacteristic
25#63918C2212yes—
26#63895—22no—
27#63894C2212yes—
28#63871—21no—
29C2×#3714C2211yescharacteristic
30#63636C2212yes—
31#63635C2212yes—
32C4⋊(C3⋊(C3⋊C8))C2211yescharacteristic
33C3×#808C2212yes—
34C3×#808C2212yes—
35C3×C4×C3⋊C8C2211yescharacteristic
36#63576C2212yes—
37#63575C2211yescharacteristic
38Dic3⋊(C3⋊C8)C2212yes—
39#63564C2211yescharacteristic
40#63563C2211yescharacteristic
41#63562C2212yes—
42#63554C2212yes—
43C3×#554—31no—
44#5730D311yescharacteristic, complemented
45#5727—31no—
46Q8×D12—31no—
47C3×#498—31no—
48#5087—31no—
49#5084—31no—
50#5084—31no—
51#5084—61no—
52#5080D311yescharacteristic, complemented
53#5066—32no—
54#5064—32no—
55#5024—31no—
56#5014—32no—
57C4×#213—31no—
58#5008—32no—
59#5007—31no—
60#4871—32no—
61#4870—32no—
62#4869—32no—
63#4856—32no—
64C4×D24—31no—
65#4679—31no—
66C2×C32×Q8—21no—
67C2×C4⋊(C3⋊D3)C2311yescharacteristic
68C2×C4×C3⋊D3—21no—
69C2×C3×D12C2311yescharacteristic
70C2×C3×C4×D3—21no—
71C2×C3×C4×D3—22no—
72C2×C3×Dic6—21no—
73C2×C3×Dic6D411yescharacteristic
74C2×#627—22no—
75C2×#624—22no—
76#3797—44no—
77C32×C4⋊C4—21no—
78C32×C4⋊C4C2312yes—
79C32×C42C2311yescharacteristic
80#3752—21no—
81C4⋊(C3⋊Dic3)C2311yescharacteristic
82C2×C3⋊(C3⋊C8)C2312yes—
83C3×(C2×D3)⋊C4—21no—
84C3×(C2×D3)⋊C4—22no—
85C3×C4⋊Dic3—21no—
86C3×C4⋊Dic3—22no—
87C3×C4⋊Dic3D411yescharacteristic
88C3×C4⋊Dic3C2311yescharacteristic
89C3×C4⋊Dic3C2312yes—
90C3×Dic3⋊C4—22no—
91C3×Dic3⋊C4D412yes—
92C3×C4×Dic3C2312yes—
93C2×C3×C3⋊C8C2311yescharacteristic
94C2×C3×C3⋊C8C2311yescharacteristic
95Dic3⋊Dic3—22no—
96#3722—22no—
97#3714Q814yescomplemented
98#3384—91noSylow
99C2×D3×Q8—61no—
100C2×D3×Q8—61no—
101C3×C4⋊Q8—21no—
102C3×C4⋊Q8—31no—
103C3×C4⋊Q8C2×D311yescharacteristic
104C3×C4⋊Q8C2×D311yescharacteristic
105C3×#125—32no—
106C3×C22⋊Q8—61no—
107C3×C22⋊Q8—62no—
108C3×C4×Q8—31no—
109C3×C4×D4—32no—
110C3×C4×D4—31no—
111C2×C3×C4⋊C4—61no—
112#946—22no—
113#946—61no—
114C2×#213—31no—
115C2×D24—31no—
116#897—21no—
117#897—62no—
118#897—61no—
119#896—62no—
120#893—32no—
121#893C2×D312yes—
122D3×C4⋊C4—21no—
123#890C2×D312yes—
124#888—32no—
125C4×D12—31no—
126C4×D12—31no—
127C4×D12—32no—
128C4×D12—61no—
129C4×D12C2×D311yescharacteristic
130#869—32no—
131#869C2×D311yescharacteristic
132C4×Dic6C2×D311yescharacteristic
133C3×#106—32no—
134C3×#106—32no—
135C3×C4×C8—31no—
136#832—32no—
137#832—31no—
138#821—31no—
139#821—31no—
140#818—31no—
141#815—32no—
142Dic3⋊C8—32no—
143#809—32no—
144#809—32no—
145#809—32no—
146#809—31no—
147#809—62no—
148#808—32no—
149#808—32no—
150#808—62no—
151#808C2×D312yes—
152#808C2×D312yes—
153#807—32no—
154#807—32no—
155#807—31no—
156C4⋊(C3⋊C8)—31no—
157C4⋊(C3⋊C8)—31no—
158C4⋊(C3⋊C8)—32no—
159C4⋊(C3⋊C8)—61no—
160#803—32no—
161C4×C3⋊C8C2×D311yescharacteristic
162C22×C3⋊D3—21no—
163C22×C3×D3—21no—
164C32×Q8—81no—
165C2×C32×C4—22no—
166C2×C32×C4C2×D411yescharacteristic
167C2×C32×C4C2×D411yescharacteristic
168C2×C32×C4C2411yescharacteristic
169C2×C3⋊Dic3—21no—
170C4⋊(C3⋊D3)—21no—
171C4⋊(C3⋊D3)C2×Q812yes—
172C4×C3⋊D3—42no—
173C2×C3×Dic3—22no—
174C2×C3×Dic3—21no—
175C2×C3×Dic3C2×D412yes—
176C2×C3×Dic3C2×D412yes—
177C3×D12—21no—
178C3×D12C2×Q812yescomplemented
179C3×C4×D3—42no—
180C3×C4×D3—82no—
181C3×Dic6—22no—
182C3×Dic6—42no—
183#627—44no—
184#624—44no—
185C3⋊(C3⋊C8)—22no—
186C3×C3⋊C8—21no—
187C3×C3⋊C8C2×Q812yescomplemented
188#554—31no—
189D4×Q8—91no—
190#501—92no—
191#498—31no—
192#478—92no—
193#476—92no—
194#476—91no—
195#474—91no—
196#446—91no—
197#442—92no—
198C4×D8—91no—
199C2×C3×Q8—21no—
200C2×C3×Q8—21no—
201C2×C3×Q8—31no—
202C2×C3×Q8—41no—
203C2×C3×Q8—61no—
204C2×C3×D4—31no—
205C22×C3×C4—62no—
206C22×C3×C4—61no—
207D3×Q8—124no—
208D3×Q8—124no—
209C2×D12—31no—
210C2×D12—31no—
211C2×D12—61no—
212C2×D12C22×D311yescharacteristic
213C2×C4×D3—21no—
214C2×C4×D3—22no—
215C2×C4×D3—62no—
216C2×C4×D3—61no—
217C2×C4×D3—61no—
218C2×C4×D3—62no—
219C2×C4×D3—121no—
220C2×Dic6—21no—
221C2×Dic6—62no—
222C2×Dic6—62no—
223C2×Dic6#6711yescharacteristic
224C2×C3×C8—31no—
225C2×C3×C8—31no—
226C3×C4⋊C4—21no—
227C3×C4⋊C4—21no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—32no—
230C3×C4⋊C4—31no—
231C3×C4⋊C4—31no—
232C3×C4⋊C4—32no—
233C3×C4⋊C4—41no—
234C3×C4⋊C4—62no—
235C3×C4⋊C4—62no—
236C3×C4⋊C4—61no—
237C3×C4⋊C4C22×D312yes—
238C3×C4⋊C4C22×D312yes—
239C3×C22⋊C4—62no—
240C3×C22⋊C4—61no—
241C3×C42—21no—
242C3×C42—32no—
243C3×C42C22×D311yescharacteristic
244C3×C42C22×D311yescharacteristic
245#213—34no—
246(C2×D3)⋊C4—21no—
247(C2×D3)⋊C4—22no—
248(C2×D3)⋊C4—62no—
249(C2×D3)⋊C4—61no—
250(C2×D3)⋊C4—61no—
251(C2×D3)⋊C4—62no—
252(C2×D3)⋊C4—121no—
253C4⋊Dic3—21no—
254C4⋊Dic3—22no—
255C4⋊Dic3—31no—
256C4⋊Dic3—31no—
257C4⋊Dic3—61no—
258C4⋊Dic3—62no—
259C4⋊Dic3#6711yescharacteristic
260C4⋊Dic3C22×D311yescharacteristic
261C4⋊Dic3C22×D312yes—
262Dic3⋊C4—22no—
263Dic3⋊C4—62no—
264Dic3⋊C4#6712yes—
265C4×Dic3C22×D312yes—
266C2×C3⋊C8—32no—
267C2×C3⋊C8—32no—
268C2×C3⋊C8—62no—
269C2×C3⋊C8C22×D311yescharacteristic
270C2×C3⋊C8C22×D311yescharacteristic
271D24—34no—
272C22×C32C22×D411yescharacteristic
273C2×C3⋊D3—22no—
274C2×C3⋊D3—41no—
275C2×C3⋊D3—41no—
276C2×C3×D3—22no—
277C2×C3×D3—41no—
278C2×C3×D3—41no—
279C32×C4—21no—
280C32×C4—42no—
281C32×C4AGL(1,ℤ/8)12yes—
282C32×C4C22×Q811yescharacteristic
283C32×C42−1+411yescharacteristic
284C3⋊Dic3—41no—
285C3×Dic3—22no—
286C3×Dic3—22no—
287C3×Dic3—42no—
288C3×Dic3—41no—
289C22×Q8—181no—
290C2×D8—91no—
291C4⋊Q8—31no—
292C4⋊Q8—92no—
293C4⋊Q8D3211yescharacteristic
294#125—32no—
295C22⋊Q8—61no—
296C22⋊Q8—62no—
297C22⋊Q8—182no—
298C22⋊Q8—181no—
299C4×Q8—31no—
300C4×D4—32no—
301C4×D4—31no—
302C4×D4—92no—
303C4×D4—91no—
304C2×C4⋊C4—61no—
305#107—92no—
306#107—91no—
307#106—32no—
308#106—32no—
309#106—92no—
310C4⋊C8—92no—
311C4⋊C8—91no—
312#102—91no—
313#102—92no—
314#102—91no—
315#102—92no—
316#97—92no—
317C4×C8—31no—
318C23×C3—61no—
319C22×D3—21no—
320C22×D3—61no—
321C22×D3—61no—
322C22×D3—121no—
323C3×Q8—62no—
324C3×Q8—81no—
325C3×Q8—81no—
326C3×Q8—82no—
327C3×Q8—122no—
328C3×D4—32no—
329C3×D4—61no—
330C2×C3×C4—22no—
331C2×C3×C4—22no—
332C2×C3×C4—21no—
333C2×C3×C4—21no—
334C2×C3×C4—21no—
335C2×C3×C4—32no—
336C2×C3×C4—32no—
337C2×C3×C4—42no—
338C2×C3×C4—61no—
339C2×C3×C4—62no—
340C2×C3×C4—122no—
341C2×C3×C4—242no—
342C2×C3×C4D3×D411yescharacteristic
343C2×C3×C4D3×D411yescharacteristic
344C2×C3×C4C2×#6711yescharacteristic
345C2×C3×C4C2×#6711yescharacteristic
346C2×C3×C4C23×D311yescharacteristic
347C2×C3×C4C23×D311yescharacteristic
348C2×Dic3—22no—
349C2×Dic3—21no—
350C2×Dic3—61no—
351C2×Dic3—61no—
352C2×Dic3—121no—
353C2×Dic3C2×#6712yes—
354C2×Dic3C2×#6712yes—
355D12—21no—
356D12—32no—
357D12—32no—
358D12—62no—
359D12—61no—
360D12—61no—
361D12—121no—
362D12D3×Q812yescomplemented
363C4×D3—42no—
364C4×D3—82no—
365C4×D3—124no—
366C4×D3—124no—
367C4×D3—122no—
368C4×D3—122no—
369C4×D3—242no—
370Dic6—22no—
371Dic6—42no—
372Dic6—124no—
373Dic6—124no—
374C3×C8—32no—
375C3×C8—61no—
376C3⋊C8—21no—
377C3⋊C8—62no—
378C3⋊C8—62no—
379C3⋊C8—122no—
380C3⋊C8D3×Q812yescomplemented
381C2×C32D4×Q811yescharacteristic
382C2×C32C2×AGL(1,ℤ/8)11yescharacteristic
383C2×C32#57711yescharacteristic
384C3⋊D3—42no—
385C3×D3—42no—
386C2×Q8—21no—
387C2×Q8—31no—
388C2×Q8—61no—
389C2×Q8—182no—
390C2×Q8—364no—
391C2×D4—31no—
392C2×D4—91no—
393C22×C4—62no—
394C22×C4—61no—
395C22×C4—182no—
396C22×C4—181no—
397D8—94no—
398C2×C8—31no—
399C2×C8—31no—
400C2×C8—92no—
401C4⋊C4—21no—
402C4⋊C4—31no—
403C4⋊C4—32no—
404C4⋊C4—32no—
405C4⋊C4—31no—
406C4⋊C4—62no—
407C4⋊C4—62no—
408C4⋊C4—61no—
409C4⋊C4—91no—
410C4⋊C4—182no—
411C4⋊C4C2×D3212yes—
412C22⋊C4—62no—
413C22⋊C4—61no—
414C22⋊C4—182no—
415C22⋊C4—181no—
416C42—32no—
417C42C2×D3211yescharacteristic
418C22×C3—21no—
419C22×C3—62no—
420C22×C3—121no—
421C22×C3—121no—
422C22×C3C2×D3×D411yescharacteristic
423C22×C3C22×#6711yescharacteristic
424C2×D3—22no—
425C2×D3—41no—
426C2×D3—41no—
427C2×D3—62no—
428C2×D3—62no—
429C2×D3—122no—
430C2×D3—121no—
431C2×D3—121no—
432C2×D3—121no—
433C2×D3—121no—
434C2×D3—241no—
435C2×D3—241no—
436C3×C4—21no—
437C3×C4—21no—
438C3×C4—21no—
439C3×C4—21no—
440C3×C4—22no—
441C3×C4—41no—
442C3×C4—42no—
443C3×C4—42no—
444C3×C4—62no—
445C3×C4—62no—
446C3×C4—82no—
447C3×C4—121no—
448C3×C4—122no—
449C3×C4#91412yes—
450C3×C4#94912yes—
451C3×C4C2×D3×Q811yescharacteristic
452C3×C4C2×D3×Q811yescharacteristic
453C3×C4#100711yescharacteristic
454C3×C4#100711yescharacteristic
455Dic3—22no—
456Dic3—22no—
457Dic3—42no—
458Dic3—41no—
459Dic3—121no—
460Dic3—121no—
461Dic3—241no—
462C32#338411yescharacteristic, complemented, Sylow
463C23—61no—
464C23—181no—
465Q8—62no—
466Q8—81no—
467Q8—122no—
468Q8—364no—
469D4—32no—
470D4—61no—
471D4—92no—
472D4—181no—
473C2×C4—22no—
474C2×C4—32no—
475C2×C4—32no—
476C2×C4—61no—
477C2×C4—62no—
478C2×C4—122no—
479C2×C4—181no—
480C2×C4—242no—
481C2×C4—364no—
482C2×C4—362no—
483C2×C4D3×#6711yescharacteristic
484C2×C4D3×#6711yescharacteristic
485C2×C4C22×D3211yescharacteristic
486C8—32no—
487C8—61no—
488C8—182no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—122no—
493C2×C3#572711yescharacteristic
494C2×C3C2×#91411yescharacteristic
495C2×C3#576211yescharacteristic
496C2×C3Q8×#6711yescharacteristic
497C2×C3C2×#94911yescharacteristic
498C2×C3#583611yescharacteristic
499D3—42no—
500D3—122no—
501D3—122no—
502D3—242no—
503C22—62no—
504C22—121no—
505C22—121no—
506C22—182no—
507C22—361no—
508C22—361no—
509C22C2×D3×#6711yescharacteristic
510C4—21no—
511C4—42no—
512C4—62no—
513C4—62no—
514C4—121no—
515C4—122no—
516C4—361no—
517C4#6393912yes—
518C4#6431611yescharacteristic
519C4D32×Q811yescharacteristic
520C3—21no—
521C3#8190511yescharacteristic, complemented
522C3#8363011yescharacteristic, complemented
523C2—122no—
524C2—362no—
525C2#1059510511yescharacteristic
526C2C2×#6393911yescharacteristic
527C2#1059540311yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
prime radicalC2×#3714C22
hypercenterC4⋊Q8D32
hypocenterC32#3384
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×#5730C2semidirect
2. #10593903C2semidirect
3. #10592783C2semidirect
4. C3×#5080C2semidirect
6. #10590558C2semidirect
7. #10590470C2semidirect
8. #10590419C2semidirect
10. #10590269C2semidirect
11. #10590258C2semidirect
44. #5730D3semidirect
52. #5080D3semidirect
97. #3714Q8semidirect
178. C3×D12C2×Q8semidirect
187. C3×C3⋊C8C2×Q8semidirect
362. D12D3×Q8semidirect
380. C3⋊C8D3×Q8semidirect
462. C32#3384semidirect
521. C3#81905semidirect
522. C3#83630semidirect

Maximal subgroups

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

subgroupindex
C3×#57302
#105939032
#105927832
C3×#50802
#105905892
#105905582
#105904702
#105904192
C4×#37142
#105902692
#105902582
#836303
#819053