Finite Group Databasebeta

← #11878717

Subgroups — #11878717

Order 1152.

905 conjugacy classes of subgroups.

The subgroup lattice has 495 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×D3×C4⋊Q8C211yesmaximal, characteristic, complemented
2#10593897C211yesmaximal, characteristic
3#10592778C211yesmaximal, characteristic, complemented
4C3×#5085C211yesmaximal, characteristic, complemented
5#10591583C214yesmaximal, complemented
6#10591564C214yesmaximal, complemented
7#10590375C211yesmaximal, characteristic, complemented
8#10590287C211yesmaximal, characteristic, complemented
9#10590261C211yesmaximal, characteristic, complemented
10#83213—31nomaximal
11#81764—31nomaximal
12C2×C3×D3×Q8C2212yes—
13C2×#3795—21no—
14C32×C4⋊Q8C2211yescharacteristic
15#64078C2211yescharacteristic
16C3×#944C2212yes—
17C2×C3×#215C2214yes—
18C3×D3×C4⋊C4—21no—
19C3×D3×C4⋊C4—21no—
20C3×#888—21no—
21C3×#888—21no—
22C3×#874C2211yescharacteristic
23C3×C42×D3C2211yescharacteristic
24C3×#869C2211yescharacteristic
25#63908—21no—
26#63908C2212yes—
27Dic3×D12—21no—
28Dic3×D12C2212yes—
29#63860—21no—
30C2×#3715C2214yes—
31#63662C2214yes—
32C4×C3⋊(C3⋊C8)C2211yescharacteristic
33C3×C4×C3⋊C8C2211yescharacteristic
34Dic3⋊(C3⋊C8)C2212yes—
35#63566C2214yes—
36(C2×D3)⋊(C3⋊C8)C2212yes—
37C2×C3×C4⋊Q8—31no—
38D3×C4⋊Q8D311yescharacteristic, complemented
39#5716—31no—
40#5477—31no—
41C3×#492—31no—
42#5227—34no—
43#5170—34no—
44#5162—34no—
45#5085—31no—
46#5085D311yescharacteristic, complemented
47#5083—34no—
48#5079—31no—
49#5079—31no—
50#5079—31no—
51#5079—61no—
52#5076—31no—
53#4925—31no—
54#4685—31no—
55#4681—31no—
56C2×C32×Q8C2312yes—
57C2×#634—21no—
58C3×D3×Q8—44no—
59C2×C3×D12—21no—
60C2×C3×D12—21no—
61C2×C3×D12C2312yes—
62C2×C3×C4×D3—22no—
63C2×C3×C4×D3D411yescharacteristic
64C2×C3×C4×D3D411yescharacteristic
65C2×C3×C4×D3C2311yescharacteristic
66C2×C3×Dic6—22no—
67C2×C3×Dic6C2312yes—
68C2×#625—22no—
69C2×D3×Dic3—22no—
70#3795—44no—
71C32×C4⋊C4—21no—
72C32×C4⋊C4—21no—
73C32×C42C2311yescharacteristic
74C4⋊(C3⋊Dic3)—21no—
75C4⋊(C3⋊Dic3)C2312yes—
76C2×C3⋊(C3⋊C8)C2312yes—
77C3×#215—44no—
78C3×C4⋊Dic3—21no—
79C3×C4⋊Dic3—21no—
80C3×Dic3⋊C4—22no—
81C3×Dic3⋊C4—22no—
82C3×C4×Dic3D411yescharacteristic
83C3×C4×Dic3D411yescharacteristic
84C3×C4×Dic3C2311yescharacteristic
85C2×C3×C3⋊C8C2312yes—
86(C2×D3)⋊Dic3—22no—
87(C2×D3)⋊Dic3—22no—
88#3715—44no—
89#3206—91noSylow
90C22×C3×Q8—32no—
91C2×D3×Q8C2×D312yes—
92C2×#237—61no—
93C2×#235—61no—
94C2×C3×SD16—34no—
95C3×C4⋊Q8—21no—
96C3×C4⋊Q8—32no—
97C3×C4⋊Q8—31no—
98C3×C4⋊Q8—61no—
99C3×C4⋊Q8—61no—
100C3×C4⋊Q8C2×D311yescharacteristic
101C3×C4⋊Q8C2×D311yescharacteristic
102C3×#127—31no—
103C2×C3×C4⋊C4—61no—
104C2×C3×C4⋊C4—61no—
105C2×C3×C42—31no—
106#944C2×D312yes—
107C2×#215—34no—
108C2×#215C2×D314yes—
109#938—32no—
110#938—61no—
111#936—61no—
112D4×Dic3—32no—
113D4×Dic3—61no—
114C2×#214—34no—
115#930—32no—
116#930—61no—
117D3×C4⋊C4—21no—
118D3×C4⋊C4—21no—
119#888—21no—
120#888—21no—
121#875—61no—
122#874C2×D311yescharacteristic
123C4×D12—32no—
124C4×D12—61no—
125C42×D3C2×D311yescharacteristic
126#869—31no—
127#869—31no—
128#869—61no—
129#869C2×D311yescharacteristic
130C3×C4×C8—31no—
131#835—34no—
132#835—34no—
133#835—34no—
134#835—64no—
135#830—32no—
136(C2×D3)⋊C8—32no—
137#816—34no—
138Dic3⋊C8—32no—
139C4⋊(C3⋊C8)—32no—
140C4×C3⋊C8—31no—
141C4×C3⋊C8—31no—
142C4×C3⋊C8—61no—
143C4×C3⋊C8C2×D311yescharacteristic
144C22×C3×D3—22no—
145C22×C3×D3C2×D411yescharacteristic
146C32×Q8—24no—
147C2×C32×C4—22no—
148C2×C32×C4C2×D411yescharacteristic
149C2×C32×C4C2×D411yescharacteristic
150C2×C32×C4C2411yescharacteristic
151C2×C3⋊Dic3—22no—
152#634—42no—
153C2×C3×Dic3—22no—
154C2×C3×Dic3C2×D411yescharacteristic
155C2×C3×Dic3C2×D411yescharacteristic
156C2×C3×Dic3C2×D411yescharacteristic
157C3×D12—24no—
158C3×D12—42no—
159C3×D12—42no—
160C3×C4×D3—22no—
161C3×C4×D3—22no—
162C3×C4×D3—44no—
163C3×C4×D3SD1614yescomplemented
164C3×Dic6—24no—
165C3×Dic6—44no—
166#625—44no—
167D3×Dic3—44no—
168C3⋊(C3⋊C8)—22no—
169C3×C3⋊C8—22no—
170#532—91no—
171C2×C4⋊Q8—31no—
172#492—31no—
173#492—91no—
174#487—91no—
175#487—91no—
176#461—94no—
177#448—94no—
178#431—91no—
179C2×C3×Q8—22no—
180C2×C3×Q8—32no—
181C2×C3×Q8—62no—
182C2×C3×Q8—124no—
183C2×C3×Q8C22×D312yes—
184C2×C3×Q8C22×D312yes—
185C2×C3×D4—32no—
186C2×C3×D4—61no—
187C2×C3×D4—61no—
188C22×C3×C4—31no—
189C22×C3×C4—31no—
190C22×C3×C4—31no—
191C22×C3×C4—62no—
192C2×#67—62no—
193C2×#67—62no—
194C22×Dic3—62no—
195D3×Q8—44no—
196#237—124no—
197#235—124no—
198C2×D12—21no—
199C2×D12—21no—
200C2×D12C22×D312yes—
201C2×C4×D3—22no—
202C2×C4×D3—62no—
203C2×C4×D3#6711yescharacteristic
204C2×C4×D3#6711yescharacteristic
205C2×C4×D3C22×D311yescharacteristic
206C2×Dic6—22no—
207C2×Dic6—61no—
208C2×Dic6—61no—
209C2×Dic6—121no—
210C2×Dic6C22×D312yes—
211C3×SD16—124no—
212C2×C3×C8—32no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—41no—
218C3×C4⋊C4—41no—
219C3×C4⋊C4—62no—
220C3×C4⋊C4—62no—
221C3×C4⋊C4—61no—
222C3×C4⋊C4—61no—
223C3×C42—21no—
224C3×C42—31no—
225C3×C42—31no—
226C3×C42—31no—
227C3×C42C22×D311yescharacteristic
228C3×C42C22×D311yescharacteristic
229#217—62no—
230#217—62no—
231#215—44no—
232#215—124no—
233#214—124no—
234(C2×D3)⋊C4—62no—
235(C2×D3)⋊C4—62no—
236C4⋊Dic3—21no—
237C4⋊Dic3—21no—
238C4⋊Dic3—32no—
239C4⋊Dic3—32no—
240C4⋊Dic3—62no—
241C4⋊Dic3—61no—
242C4⋊Dic3—61no—
243C4⋊Dic3—121no—
244Dic3⋊C4—22no—
245Dic3⋊C4—22no—
246C4×Dic3#6711yescharacteristic
247C4×Dic3#6711yescharacteristic
248C4×Dic3C22×D311yescharacteristic
249C2×C3⋊C8—32no—
250C2×C3⋊C8—32no—
251C2×C3⋊C8—62no—
252C2×C3⋊C8C22×D312yes—
253C22×C32C22×D411yescharacteristic
254C2×C3×D3—21no—
255C2×C3×D3—21no—
256C2×C3×D3—42no—
257C2×C3×D3—42no—
258C2×C3×D3—42no—
259C2×C3×D3C2×SD1612yes—
260C32×C4—42no—
261C32×C4C2×SD1612yes—
262C32×C4#13712yes—
263C32×C4H(5,𝔽2)12yes—
264C3⋊Dic3—42no—
265C3×Dic3—21no—
266C3×Dic3—21no—
267C3×Dic3—42no—
268C3×Dic3C2×SD1612yes—
269C2×#41—181no—
270C22×Q8—32no—
271C2×SD16—34no—
272C2×SD16—94no—
273C4⋊Q8—32no—
274C4⋊Q8—31no—
275C4⋊Q8—61no—
276C4⋊Q8—61no—
277C4⋊Q8—91no—
278C4⋊Q8D3211yescharacteristic
279#127—31no—
280#124—181no—
281#121—92no—
282#121—181no—
283C4×D4—92no—
284C4×D4—181no—
285C2×C4⋊C4—61no—
286C2×C4⋊C4—61no—
287C2×C42—31no—
288C4⋊C8—92no—
289#103—94no—
290#103—94no—
291C22⋊C8—92no—
292C4×C8—31no—
293C4×C8—91no—
294C23×C3—31no—
295C23×C3—62no—
296C22×D3—22no—
297C22×D3C2×#6711yescharacteristic
298C3×Q8—24no—
299C3×Q8—24no—
300C3×Q8—44no—
301C3×Q8—64no—
302C3×Q8—124no—
303C3×D4—64no—
304C3×D4—122no—
305C3×D4—122no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—21no—
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—31no—
312C2×C3×C4—34no—
313C2×C3×C4—31no—
314C2×C3×C4—31no—
315C2×C3×C4—42no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—62no—
319C2×C3×C4—124no—
320C2×C3×C4D3×D411yescharacteristic
321C2×C3×C4D3×D411yescharacteristic
322C2×C3×C4C2×#6711yescharacteristic
323C2×C3×C4C2×#6711yescharacteristic
324C2×C3×C4C23×D311yescharacteristic
325C2×C3×C4C23×D311yescharacteristic
326#67—124no—
327#67—124no—
328C2×Dic3—22no—
329C2×Dic3—62no—
330C2×Dic3—62no—
331C2×Dic3—122no—
332C2×Dic3—124no—
333C2×Dic3C2×#6711yescharacteristic
334C2×Dic3C2×#6711yescharacteristic
335C2×Dic3C2×#6711yescharacteristic
336D12—24no—
337D12—42no—
338D12—42no—
339C4×D3—22no—
340C4×D3—22no—
341C4×D3—44no—
342C4×D3—124no—
343C4×D3#21514yescomplemented
344Dic6—24no—
345Dic6—44no—
346Dic6—122no—
347Dic6—122no—
348Dic6—242no—
349C3×C8—62no—
350C3⋊C8—22no—
351C3⋊C8—62no—
352C3⋊C8—62no—
353C3⋊C8—122no—
354C2×C32#53511yescharacteristic
355C2×C32C22×SD1611yescharacteristic
356C2×C32C2×#13711yescharacteristic
357C3×D3—22no—
358C3×D3—82no—
359#41—364no—
360C2×Q8—32no—
361C2×Q8—62no—
362C2×Q8—124no—
363C2×Q8—181no—
364C2×Q8C2×D3212yes—
365C2×D4—32no—
366C2×D4—61no—
367C2×D4—61no—
368C2×D4—182no—
369C22×C4—31no—
370C22×C4—31no—
371C22×C4—31no—
372C22×C4—62no—
373C22×C4—182no—
374SD16—124no—
375SD16—364no—
376C2×C8—32no—
377C2×C8—92no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—62no—
381C4⋊C4—62no—
382C4⋊C4—61no—
383C4⋊C4—61no—
384C4⋊C4—92no—
385C4⋊C4—181no—
386C22⋊C4—182no—
387C22⋊C4—182no—
388C42—31no—
389C42—31no—
390C42—31no—
391C42C2×D3211yescharacteristic
392C22×C3—21no—
393C22×C3—32no—
394C22×C3—61no—
395C22×C3—61no—
396C22×C3—122no—
397C22×C3—122no—
398C22×C3—122no—
399C22×C3C2×D3×D411yescharacteristic
400C22×C3C22×#6711yescharacteristic
401C2×D3—21no—
402C2×D3—21no—
403C2×D3—42no—
404C2×D3—42no—
405C2×D3—42no—
406C2×D3C2×#21512yes—
407C3×C4—22no—
408C3×C4—22no—
409C3×C4—22no—
410C3×C4—32no—
411C3×C4—42no—
412C3×C4—42no—
413C3×C4—61no—
414C3×C4—61no—
415C3×C4—82no—
416C3×C4—122no—
417C3×C4D3×SD1612yes—
418C3×C4#91512yes—
419C3×C4C2×#21512yes—
420C3×C4#94212yes—
421C3×C4#100412yes—
422C3×C4#100412yes—
423Dic3—21no—
424Dic3—21no—
425Dic3—42no—
426Dic3—122no—
427Dic3—122no—
428Dic3—242no—
429Dic3C2×#21512yes—
430C32#320611yescharacteristic, complemented, Sylow
431C23—31no—
432C23—62no—
433Q8—24no—
434Q8—64no—
435Q8—124no—
436Q8—362no—
437D4—64no—
438D4—122no—
439D4—122no—
440D4—364no—
441C2×C4—22no—
442C2×C4—31no—
443C2×C4—34no—
444C2×C4—31no—
445C2×C4—31no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—124no—
450C2×C4—182no—
451C2×C4—364no—
452C2×C4D3×#6711yescharacteristic
453C2×C4D3×#6711yescharacteristic
454C2×C4C22×D3211yescharacteristic
455C8—62no—
456C8—182no—
457C2×C3—21no—
458C2×C3—21no—
459C2×C3—21no—
460C2×C3—62no—
461C2×C3—242no—
462C2×C3#571511yescharacteristic
463C2×C3C2×D3×SD1611yescharacteristic
464C2×C3C2×#91511yescharacteristic
465C2×C3#580511yescharacteristic
466C2×C3C22×#21511yescharacteristic
467C2×C3C2×#94211yescharacteristic
468D3—22no—
469D3—82no—
470C22—32no—
471C22—61no—
472C22—61no—
473C22—122no—
474C22—122no—
475C22—122no—
476C22C2×D3×#6711yescharacteristic
477C4—32no—
478C4—42no—
479C4—61no—
480C4—61no—
481C4—122no—
482C4—362no—
483C4D3×#21512yes—
484C4#6394612yes—
485C4#6431312yes—
486C3—21no—
487C3#8176411yescharacteristic, complemented
488C3#8321311yescharacteristic, complemented
489C2—62no—
490C2—242no—
491C2#1059523311yescharacteristic
492C2C2×D3×#21511yescharacteristic
493C2C2×#6394611yescharacteristic
subgroupG/H
centerC22C2×D3×#67
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
prime radical#10593897C2
hypercenterC4⋊Q8D32
hypocenterC32#3206
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×D3×C4⋊Q8C2semidirect
3. #10592778C2semidirect
4. C3×#5085C2semidirect
5. #10591583C2semidirect
6. #10591564C2semidirect
7. #10590375C2semidirect
8. #10590287C2semidirect
9. #10590261C2semidirect
38. D3×C4⋊Q8D3semidirect
46. #5085D3semidirect
163. C3×C4×D3SD16semidirect
343. C4×D3#215semidirect
430. C32#3206semidirect
487. C3#81764semidirect
488. C3#83213semidirect

Maximal subgroups

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

subgroupindex
C3×D3×C4⋊Q82
#105938972
#105927782
C3×#50852
#105915832
#105915642
#105903752
#105902872
#105902612
#832133
#817643