Finite Group Databasebeta

← #11898380

Subgroups — #11898380

Order 1152.

826 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#10596203C211yesmaximal, characteristic, complemented
2#10593863C211yesmaximal, characteristic, complemented
3C3×#5092C211yesmaximal, characteristic, complemented
4C3×#5083C211yesmaximal, characteristic, complemented
5#10591612C212yesmaximal, complemented
6#10591610C212yesmaximal
7#10591581C212yesmaximal, complemented
8#10591579C212yesmaximal, complemented
9#10590379C211yesmaximal, characteristic, complemented
10C4×#3714C211yesmaximal, characteristic
11#10590254C211yesmaximal, characteristic, complemented
12#81795—31nomaximal
13#80421—31nomaximal
14C32×C4⋊Q8C2211yescharacteristic
15#64156C2212yes—
16Q8×C3⋊Dic3C2212yes—
17#64107—22no—
18#64106—21no—
19#64101—21no—
20C4×C4⋊(C3⋊D3)C2211yescharacteristic
21C4×#634C2211yescharacteristic
22C2×C3×#216C2212yes—
23C2×C3×#216C2212yes—
24C2×C3×#215C2212yes—
25C3×C4×D12C2211yescharacteristic
26C3×#869C2211yescharacteristic
27#63863C2212yes—
28#63858—22no—
29#63857—21no—
30#63843C2212yes—
31#63841—21no—
32C2×#3717C2212yes—
33C2×#3714C2211yescharacteristic
34C3×#835C2212yes—
35C3×C4⋊(C3⋊C8)C2211yescharacteristic
36C3×C4×C3⋊C8C2211yescharacteristic
37#63575C2211yescharacteristic
38#63572C2212yes—
39#63566C2212yes—
40#63564C2211yescharacteristic
41#63563C2211yescharacteristic
42#63553C2212yes—
43#5732—31no—
44#5732—31no—
45#5732—31no—
46#5732—61no—
47#5542—31no—
48C3×#494—31no—
49C3×#461—31no—
50#5191—32no—
51#5185—32no—
52#5185—32no—
53Dic3×Q16—32no—
54#5168—32no—
55#5166—32no—
56#5092D311yescharacteristic, complemented
57#5083—31no—
58#5083D311yescharacteristic, complemented
59C4×#213—31no—
60#4841—31no—
61#4830—31no—
62#4798—32no—
63#4790—32no—
64C4×D24—31no—
65C8×Dic6—31no—
66C2×C32×Q8C2312yes—
67C2×C4⋊(C3⋊D3)C2311yescharacteristic
68C2×C4×C3⋊D3—21no—
69C2×#634D411yescharacteristic
70C2×C3×D12C2311yescharacteristic
71C2×C3×C4×D3—21no—
72C2×C3×Dic6C2312yes—
73C32×C4⋊C4—21no—
74C32×C4⋊C4—21no—
75C32×C42C2311yescharacteristic
76#3752—21no—
77#3752—22no—
78#3752—22no—
79C4⋊(C3⋊Dic3)—22no—
80C4⋊(C3⋊Dic3)D411yescharacteristic
81C4⋊(C3⋊Dic3)C2311yescharacteristic
82#3750D412yes—
83C4×C3⋊Dic3—22no—
84C4×C3⋊Dic3C2312yes—
85C3×#216—24no—
86C3×#216—42no—
87C3×#215—42no—
88C3×(C2×D3)⋊C4—21no—
89C3×C4⋊Dic3—21no—
90C3×C4⋊Dic3—21no—
91C3×C4⋊Dic3C2311yescharacteristic
92C2×C3×C3⋊C8C2312yes—
93C2×C3×C3⋊C8C2311yescharacteristic
94C2×C3×C3⋊C8C2311yescharacteristic
95Dic3⋊Dic3—22no—
96(C2×D3)⋊Dic3—22no—
97(C2×D3)⋊Dic3—22no—
98Dic32—22no—
99#3717—42no—
100#3714—22no—
101#3393—91noSylow
102C2×C3×Q16—32no—
103C2×C3×Q16—32no—
104C2×C3×SD16—32no—
105C3×C4⋊Q8—21no—
106C3×C4⋊Q8—31no—
107C3×C4⋊Q8C2×D311yescharacteristic
108C3×C4⋊Q8C2×D311yescharacteristic
109C3×C4×D4—31no—
110#947—32no—
111#947—32no—
112#947—32no—
113#947—62no—
114Q8×Dic3—32no—
115Q8×Dic3—32no—
116Q8×Dic3—32no—
117Q8×Dic3—62no—
118C2×#216C2×D312yes—
119C2×#216C2×D312yes—
120C2×#215—32no—
121C2×#215C2×D312yes—
122#936—32no—
123C2×#213—31no—
124#926—61no—
125#924—61no—
126C2×D24—31no—
127C2×#204—32no—
128#898—62no—
129#898—62no—
130#898—62no—
131#898—64no—
132#897—61no—
133#897—61no—
134#897—62no—
135#897—61no—
136#892—61no—
137#892—61no—
138#892—61no—
139#892—62no—
140#887—32no—
141#879—62no—
142C4×D12—31no—
143C4×D12—31no—
144C4×D12—61no—
145C4×D12C2×D311yescharacteristic
146#869C2×D311yescharacteristic
147C4×Dic6—31no—
148C4×Dic6—31no—
149C4×Dic6—61no—
150C3×C4⋊C8—31no—
151C3×#103—32no—
152C3×C4×C8—31no—
153#835—32no—
154#835C2×D312yes—
155#832—31no—
156#821—31no—
157#821—31no—
158#818—31no—
159#816—32no—
160Dic3⋊C8—32no—
161Dic3⋊C8—32no—
162C8×Dic3—32no—
163C8×Dic3—32no—
164#809—31no—
165#807—31no—
166C4⋊(C3⋊C8)C2×D311yescharacteristic
167C4×C3⋊C8C2×D311yescharacteristic
168C22×C3⋊D3—21no—
169C22×C3×D3—21no—
170C32×Q8—22no—
171C32×Q8—22no—
172C2×C32×C4—22no—
173C2×C32×C4C2×D411yescharacteristic
174C2×C32×C4C2×D411yescharacteristic
175C2×C32×C4C2411yescharacteristic
176C2×C3⋊Dic3—21no—
177C2×C3⋊Dic3C2×D412yes—
178C2×C3⋊Dic3C2×D412yes—
179C4⋊(C3⋊D3)—21no—
180C4⋊(C3⋊D3)—21no—
181C4×C3⋊D3—42no—
182#634—22no—
183C2×C3×Dic3—21no—
184C2×C3×Dic3—22no—
185C3×D12—21no—
186C3×D12—21no—
187C3×C4×D3—42no—
188C3×Dic6—22no—
189C3×Dic6—22no—
190C3×C3⋊C8—22no—
191C3×C3⋊C8—21no—
192C3×C3⋊C8#4112yes—
193#556—91no—
194#556—91no—
195#495—92no—
196#494—31no—
197#464—92no—
198#464—92no—
199#461—31no—
200#461—91no—
201C8×Q8—91no—
202C4×Q16—92no—
203C4×D8—91no—
204C2×C3×Q8—22no—
205C2×C3×Q8—32no—
206C2×C3×Q8C22×D312yes—
207C2×C3×Q8C22×D312yes—
208C2×C3×D4—31no—
209C22×C3×C4—61no—
210C2×D12—31no—
211C2×D12—31no—
212C2×D12—61no—
213C2×D12C22×D311yescharacteristic
214C2×C4×D3—21no—
215C2×C4×D3—62no—
216C2×C4×D3—61no—
217C2×C4×D3—61no—
218C2×Dic6—31no—
219C2×Dic6—31no—
220C2×Dic6—61no—
221C2×Dic6C22×D312yes—
222C3×Q16—64no—
223C3×Q16—122no—
224C3×SD16—122no—
225C2×C3×C8—31no—
226C2×C3×C8—32no—
227C2×C3×C8—31no—
228C3×C4⋊C4—22no—
229C3×C4⋊C4—22no—
230C3×C4⋊C4—21no—
231C3×C4⋊C4—21no—
232C3×C4⋊C4—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—31no—
235C3×C4⋊C4—61no—
236C3×C4⋊C4—61no—
237C3×C22⋊C4—61no—
238C3×C42—21no—
239C3×C42C22×D311yescharacteristic
240C3×C42C22×D311yescharacteristic
241#217—62no—
242#217—62no—
243#216—24no—
244#216—42no—
245#215—42no—
246#215—122no—
247#213—62no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—62no—
250(C2×D3)⋊C4—62no—
251(C2×D3)⋊C4—62no—
252(C2×D3)⋊C4—62no—
253(C2×D3)⋊C4—62no—
254(C2×D3)⋊C4—62no—
255(C2×D3)⋊C4—62no—
256(C2×D3)⋊C4—64no—
257(C2×D3)⋊C4—64no—
258(C2×D3)⋊C4—61no—
259(C2×D3)⋊C4—61no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4⋊Dic3—31no—
263C4⋊Dic3—31no—
264C4⋊Dic3—31no—
265C4⋊Dic3—31no—
266C4⋊Dic3—61no—
267C4⋊Dic3—61no—
268C4⋊Dic3—62no—
269C4⋊Dic3—62no—
270C4⋊Dic3—62no—
271C4⋊Dic3—64no—
272C4⋊Dic3C22×D311yescharacteristic
273Dic3⋊C4—32no—
274Dic3⋊C4—32no—
275Dic3⋊C4—62no—
276Dic3⋊C4—62no—
277C4×Dic3—32no—
278C4×Dic3—32no—
279C4×Dic3—62no—
280C4×Dic3—62no—
281C4×Dic3—62no—
282C4×Dic3—62no—
283C4×Dic3—62no—
284C4×Dic3—64no—
285C2×C3⋊C8C22×D311yescharacteristic
286C2×C3⋊C8C22×D312yes—
287C2×C3⋊C8C22×D311yescharacteristic
288D24—62no—
289#204—122no—
290C22×C32C22×D411yescharacteristic
291C2×C3⋊D3—41no—
292C2×C3⋊D3—41no—
293C2×C3⋊D3—41no—
294C2×C3×D3—41no—
295C2×C3×D3—41no—
296C2×C3×D3—41no—
297C32×C4—21no—
298C32×C4—42no—
299C32×C4#13512yes—
300C32×C4C2×#4111yescharacteristic
301C32×C4H(5,𝔽2)11yescharacteristic
302C3⋊Dic3—22no—
303C3⋊Dic3—22no—
304C3⋊Dic3—41no—
305C3×Dic3—42no—
306C3×Dic3—41no—
307C2×Q16—32no—
308C2×Q16—32no—
309C2×SD16—32no—
310C2×SD16—92no—
311C2×D8—91no—
312C4⋊Q8—31no—
313C4⋊Q8D3211yescharacteristic
314#126—182no—
315#126—182no—
316#124—92no—
317#124—92no—
318C22⋊Q8—181no—
319C22⋊Q8—181no—
320C4×Q8—91no—
321C4×Q8—92no—
322C4×Q8—92no—
323C4×D4—31no—
324C4×D4—91no—
325#117—181no—
326#117—181no—
327#107—91no—
328C4⋊C8—31no—
329C4⋊C8—92no—
330#103—32no—
331#103—92no—
332#102—91no—
333#102—91no—
334C4×C8—31no—
335C4×C8—92no—
336C23×C3—61no—
337C22×D3—21no—
338C22×D3—62no—
339C22×D3—61no—
340C22×D3—61no—
341C3×Q8—22no—
342C3×Q8—22no—
343C3×Q8—22no—
344C3×Q8—22no—
345C3×Q8—42no—
346C3×Q8—42no—
347C3×Q8—62no—
348C3×Q8—62no—
349C3×D4—61no—
350C3×D4—61no—
351C2×C3×C4—24no—
352C2×C3×C4—22no—
353C2×C3×C4—22no—
354C2×C3×C4—21no—
355C2×C3×C4—21no—
356C2×C3×C4—21no—
357C2×C3×C4—61no—
358C2×C3×C4—62no—
359C2×C3×C4—122no—
360C2×C3×C4D3×D411yescharacteristic
361C2×C3×C4D3×D411yescharacteristic
362C2×C3×C4D3×D411yescharacteristic
363C2×C3×C4D3×D411yescharacteristic
364C2×C3×C4C23×D311yescharacteristic
365C2×C3×C4C23×D311yescharacteristic
366C2×Dic3—21no—
367C2×Dic3—22no—
368C2×Dic3—32no—
369C2×Dic3—32no—
370C2×Dic3—32no—
371C2×Dic3—32no—
372C2×Dic3—62no—
373C2×Dic3—62no—
374C2×Dic3—62no—
375C2×Dic3—61no—
376C2×Dic3—61no—
377D12—21no—
378D12—21no—
379D12—61no—
380D12—61no—
381D12—61no—
382D12—61no—
383D12—121no—
384D12—121no—
385C4×D3—42no—
386C4×D3—124no—
387C4×D3—122no—
388C4×D3—122no—
389Dic6—22no—
390Dic6—22no—
391Dic6—62no—
392Dic6—62no—
393Dic6—122no—
394C3×C8—32no—
395C3×C8—62no—
396C3×C8—61no—
397C3⋊C8—22no—
398C3⋊C8—21no—
399C3⋊C8#23712yes—
400C2×C32#55011yescharacteristic
401C2×C32C2×#13511yescharacteristic
402C2×C32#57811yescharacteristic
403C3⋊D3—81no—
404C3×D3—81no—
405C2×Q8—32no—
406C2×Q8—91no—
407C2×Q8C2×D3212yes—
408C2×D4—31no—
409C2×D4—91no—
410C22×C4—61no—
411C22×C4—181no—
412Q16—64no—
413Q16—122no—
414SD16—122no—
415SD16—362no—
416D8—182no—
417C2×C8—31no—
418C2×C8—31no—
419C2×C8—32no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—31no—
423C4⋊C4—61no—
424C4⋊C4—61no—
425C4⋊C4—91no—
426C4⋊C4—92no—
427C4⋊C4—91no—
428C4⋊C4—182no—
429C4⋊C4—182no—
430C22⋊C4—61no—
431C22⋊C4—182no—
432C22⋊C4—182no—
433C22⋊C4—182no—
434C22⋊C4—182no—
435C22⋊C4—181no—
436C42—92no—
437C42—182no—
438C42—182no—
439C42C2×D3211yescharacteristic
440C22×C3—21no—
441C22×C3—121no—
442C22×C3—121no—
443C22×C3—121no—
444C22×C3C2×D3×D411yescharacteristic
445C22×C3C2×D3×D411yescharacteristic
446C2×D3—41no—
447C2×D3—41no—
448C2×D3—41no—
449C2×D3—122no—
450C2×D3—122no—
451C2×D3—122no—
452C2×D3—121no—
453C2×D3—121no—
454C2×D3—121no—
455C2×D3—121no—
456C2×D3—121no—
457C2×D3—121no—
458C3×C4—21no—
459C3×C4—21no—
460C3×C4—21no—
461C3×C4—21no—
462C3×C4—22no—
463C3×C4—41no—
464C3×C4—44no—
465C3×C4—42no—
466C3×C4—42no—
467C3×C4—121no—
468C3×C4—122no—
469C3×C4#91612yes—
470C3×C4#91912yes—
471C3×C4C2×#23711yescharacteristic
472C3×C4#100411yescharacteristic
473C3×C4#100411yescharacteristic
474C3×C4D3×#4111yescharacteristic
475Dic3—42no—
476Dic3—41no—
477Dic3—62no—
478Dic3—62no—
479Dic3—62no—
480Dic3—62no—
481Dic3—122no—
482Dic3—122no—
483Dic3—122no—
484Dic3—121no—
485Dic3—121no—
486C32#339311yescharacteristic, complemented, Sylow
487C23—61no—
488C23—181no—
489Q8—22no—
490Q8—22no—
491Q8—62no—
492Q8—62no—
493Q8—182no—
494D4—61no—
495D4—61no—
496D4—181no—
497D4—181no—
498C2×C4—22no—
499C2×C4—61no—
500C2×C4—62no—
501C2×C4—92no—
502C2×C4—92no—
503C2×C4—122no—
504C2×C4—181no—
505C2×C4—362no—
506C2×C4#381111yescharacteristic
507C2×C4#381111yescharacteristic
508C2×C4C22×D3211yescharacteristic
509C8—32no—
510C8—62no—
511C8—61no—
512C2×C3—21no—
513C2×C3—21no—
514C2×C3—21no—
515C2×C3—241no—
516C2×C3#559911yescharacteristic
517C2×C3#571811yescharacteristic
518C2×C3C2×#91611yescharacteristic
519C2×C3C2×#91911yescharacteristic
520C2×C3#576611yescharacteristic
521C2×C3#577911yescharacteristic
522D3—81no—
523D3—242no—
524D3—241no—
525D3—241no—
526C22—121no—
527C22—121no—
528C22—121no—
529C22—361no—
530C22—361no—
531C22—361no—
532C22C2×#381111yescharacteristic
533C4—21no—
534C4—42no—
535C4—121no—
536C4—122no—
537C4—182no—
538C4—182no—
539C4—361no—
540C4#6395012yes—
541C4D3×#23711yescharacteristic
542C4#6431311yescharacteristic
543C3—21no—
544C3#8042111yescharacteristic, complemented
545C3#8179511yescharacteristic, complemented
546C2—241no—
547C2—721no—
548C2#1059516411yescharacteristic
549C2C2×#6395011yescharacteristic
550C2#1059533711yescharacteristic
subgroupG/H
centerC22C2×#3811
commutator, cosocleC2×C32×C4C24
FrattiniC2×C4C22×D32
FittingC32×C4⋊Q8C22
socleC22×C32C22×D4
prime radicalC2×#3714C22
hypercenterC4⋊Q8D32
hypocenterC32#3393
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. #10596203C2semidirect
2. #10593863C2semidirect
3. C3×#5092C2semidirect
4. C3×#5083C2semidirect
5. #10591612C2semidirect
7. #10591581C2semidirect
8. #10591579C2semidirect
9. #10590379C2semidirect
11. #10590254C2semidirect
56. #5092D3semidirect
58. #5083D3semidirect
486. C32#3393semidirect
544. C3#80421semidirect
545. C3#81795semidirect

Maximal subgroups

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

subgroupindex
#105962032
#105938632
C3×#50922
C3×#50832
#105916122
#105916102
#105915812
#105915792
#105903792
C4×#37142
#105902542
#817953
#804213