Finite Group Databasebeta

← #11794072

Subgroups — #11794072

Order 1152.

749 conjugacy classes of subgroups.

The subgroup lattice has 589 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#3717C211yesmaximal, characteristic
2C2×#63636C211yesmaximal, characteristic
3C3×#4973C211yesmaximal, characteristic, complemented
4#10591320C211yesmaximal, characteristic
5C2×Dic3⋊(C3⋊C8)C211yesmaximal, characteristic, complemented
6C2×#63565C211yesmaximal, characteristic, complemented
7C3×#4529C211yesmaximal, characteristic, complemented
8#69680—31nomaximal
9#68863—31nomaximal
10C22×C4⋊(C3⋊D3)C2211yescharacteristic
11C22×C3×Dic6C2211yescharacteristic
12C2×C32×C4⋊C4C2211yescharacteristic
13C22×C3⋊(C3⋊C8)C2211yescharacteristic
14C2×C3×C4⋊Dic3C2211yescharacteristic
15C2×C3×C4×Dic3C2211yescharacteristic
16C22×C3×C3⋊C8C2211yescharacteristic
17C2×#3722—21no—
18C2×#3722—21no—
19C2×#3717—22no—
20C2×#3717—22no—
21C2×#3717—22no—
22#63636—22no—
23C3×C3⋊#95—21no—
24C3×C3⋊#95—21no—
25Dic3⋊(C3⋊C8)—22no—
26#63565—22no—
27C22×#215—31no—
28C22×#204—31no—
29C3×#391—31no—
30C2×#835—31no—
31C2×Dic3⋊C8—31no—
32#4988—31no—
33#4973D311yescharacteristic, complemented
34C2×#810—31no—
35C2×#809—31no—
36C2×#809—31no—
37C2×#809—61no—
38#4939—31no—
39C2×C4⋊(C3⋊C8)—31no—
40C3×#340—31no—
41#4529D311yescharacteristic, complemented
42C23×C3⋊D3—21no—
43C22×C32×C4—21no—
44C22×C32×C4C2311yescharacteristic
45C2×C4⋊(C3⋊D3)—21no—
46C2×C4⋊(C3⋊D3)—21no—
47C2×C4⋊(C3⋊D3)—21no—
48C2×C4⋊(C3⋊D3)—21no—
49C2×C4⋊(C3⋊D3)—21no—
50C2×C4⋊(C3⋊D3)D412yes—
51C22×C3×Dic3—21no—
52C22×C3×Dic3—21no—
53C22×C3×Dic3D411yescharacteristic
54C22×C3×Dic3D411yescharacteristic
55C2×C3×Dic6—21no—
56C2×C3×Dic6—21no—
57C2×C3×Dic6—21no—
58C2×C3×Dic6—21no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6D412yes—
61C32×C4⋊C4—21no—
62C32×C4⋊C4—21no—
63C2×C3⋊(C3⋊C8)—21no—
64C2×C3⋊(C3⋊C8)—21no—
65C2×C3⋊(C3⋊C8)—21no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3—21no—
68C3×C4×Dic3—21no—
69C3×C4×Dic3—21no—
70C2×C3×C3⋊C8—21no—
71C2×C3×C3⋊C8—21no—
72C2×C3×C3⋊C8D412yes—
73#3722—42no—
74#3722—42no—
75#3717—28no—
76#2055—91noSylow
77C22×C3×Q8—31no—
78C22×D12—31no—
79C22×D12—31no—
80C22×D12—61no—
81C22×Dic6C2×D311yescharacteristic
82C22×C3×C8—31no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4—31no—
85C2×C3×C4⋊C4C2×D311yescharacteristic
86C2×C3×C4⋊C4C2×D311yescharacteristic
87C2×C3×C42—31no—
88C2×#215—62no—
89C2×#215—62no—
90C2×#215—62no—
91C2×(C2×D3)⋊C4—61no—
92C2×(C2×D3)⋊C4—61no—
93C2×(C2×D3)⋊C4—61no—
94C2×(C2×D3)⋊C4—61no—
95C2×C4⋊Dic3C2×D311yescharacteristic
96C2×C4×Dic3C2×D311yescharacteristic
97C22×C3⋊C8—31no—
98C22×C3⋊C8—31no—
99C22×C3⋊C8—61no—
100C22×C3⋊C8C2×D311yescharacteristic
101C2×#204—62no—
102C2×#204—62no—
103C2×#204—62no—
104C3×#95—61no—
105C3×#95—61no—
106#835—62no—
107C3⋊#95—21no—
108C3⋊#95—21no—
109Dic3⋊C8—62no—
110#810—62no—
111#809—62no—
112#809—62no—
113#809—64no—
114C4⋊(C3⋊C8)—62no—
115C23×C32C2×D411yescharacteristic
116C22×C3⋊D3—22no—
117C22×C3⋊D3—41no—
118C22×C3⋊D3—41no—
119C22×C3⋊D3—41no—
120C22×C3⋊D3—41no—
121C22×C3⋊D3—41no—
122C22×C3⋊D3—41no—
123C2×C32×C4—41no—
124C2×C32×C4—41no—
125C2×C32×C4—41no—
126C2×C32×C4C2×D411yescharacteristic
127C2×C32×C4C2×D411yescharacteristic
128C2×C32×C4C2×D411yescharacteristic
129C2×C32×C4#4111yescharacteristic
130C2×C32×C4#4111yescharacteristic
131C2×C32×C4#4111yescharacteristic
132C4⋊(C3⋊D3)—22no—
133C4⋊(C3⋊D3)—22no—
134C4⋊(C3⋊D3)—41no—
135C4⋊(C3⋊D3)—41no—
136C2×C3×Dic3—21no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—21no—
139C2×C3×Dic3—22no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—41no—
143C2×C3×Dic3—41no—
144C2×C3×Dic3—41no—
145C2×C3×Dic3—41no—
146C2×C3×Dic3—41no—
147C2×C3×Dic3SD1612yescomplemented
148C3×Dic6—22no—
149C3×Dic6—22no—
150C3×Dic6—41no—
151C3×Dic6—41no—
152C3⋊(C3⋊C8)—41no—
153C3×C3⋊C8—22no—
154C22×SD16—91no—
155C2×C4⋊C8—91no—
156C2×#103—91no—
157C2×#102—91no—
158#391—31no—
159#390—91no—
160#340—31no—
161C23×D3—61no—
162C23×D3—61no—
163C23×D3—121no—
164C2×C3×Q8—32no—
165C2×C3×Q8—61no—
166C2×C3×Q8—61no—
167C2×C3×Q8—61no—
168C2×C3×Q8—61no—
169C2×C3×Q8—61no—
170C22×C3×C4—21no—
171C22×C3×C4—21no—
172C22×C3×C4—21no—
173C22×C3×C4—31no—
174C22×C3×C4—31no—
175C22×C3×C4—41no—
176C22×C3×C4—61no—
177C22×C3×C4—61no—
178C22×C3×C4C22×D311yescharacteristic
179C22×C3×C4C22×D311yescharacteristic
180C22×Dic3—21no—
181C22×Dic3—21no—
182C22×Dic3#6711yescharacteristic
183C22×Dic3#6711yescharacteristic
184C2×D12—32no—
185C2×D12—32no—
186C2×D12—62no—
187C2×D12—62no—
188C2×D12—62no—
189C2×D12—61no—
190C2×D12—61no—
191C2×D12—61no—
192C2×D12—61no—
193C2×D12—61no—
194C2×D12—61no—
195C2×D12—61no—
196C2×D12—61no—
197C2×D12—61no—
198C2×D12—61no—
199C2×D12—121no—
200C2×D12—121no—
201C2×D12—121no—
202C2×Dic6—21no—
203C2×Dic6—21no—
204C2×Dic6—21no—
205C2×Dic6—21no—
206C2×Dic6—21no—
207C2×Dic6D1212yes—
208C2×C3×C8—32no—
209C2×C3×C8—61no—
210C2×C3×C8—61no—
211C3×C4⋊C4—22no—
212C3×C4⋊C4—21no—
213C3×C4⋊C4—21no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—41no—
217C3×C4⋊C4—61no—
218C3×C4⋊C4—61no—
219C3×C42—61no—
220C3×C42—61no—
221#215—68no—
222(C2×D3)⋊C4—122no—
223(C2×D3)⋊C4—122no—
224(C2×D3)⋊C4—122no—
225(C2×D3)⋊C4—122no—
226C4⋊Dic3—21no—
227C4⋊Dic3—21no—
228C4×Dic3—21no—
229C4×Dic3—21no—
230C2×C3⋊C8—21no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—62no—
233C2×C3⋊C8—61no—
234C2×C3⋊C8—61no—
235C2×C3⋊C8—61no—
236C2×C3⋊C8—61no—
237C2×C3⋊C8—61no—
238C2×C3⋊C8—61no—
239C2×C3⋊C8—121no—
240C2×C3⋊C8—121no—
241C2×C3⋊C8D1212yes—
242#204—68no—
243C22×C32#12111yescharacteristic
244C22×C32#12111yescharacteristic
245C22×C32#12111yescharacteristic
246C22×C32C2×SD1611yescharacteristic
247C22×C32#13511yescharacteristic
248C22×C32AGL(1,ℤ/8)11yescharacteristic
249C22×C32#13711yescharacteristic
250C2×C3⋊D3—42no—
251C2×C3⋊D3—42no—
252C2×C3⋊D3—42no—
253C2×C3⋊D3—81no—
254C2×C3⋊D3—81no—
255C2×C3⋊D3—81no—
256C2×C3⋊D3—81no—
257C32×C4—81no—
258C32×C4#12011yescharacteristic
259C32×C4#12311yescharacteristic
260C32×C4#12311yescharacteristic
261C32×C4#12411yescharacteristic
262C3×Dic3—22no—
263C3×Dic3—42no—
264C3×Dic3—41no—
265C3×Dic3—81no—
266C22×Q8—31no—
267C22×D4—91no—
268C2×SD16—182no—
269C2×SD16—182no—
270C2×SD16—182no—
271C22×C8—31no—
272C22×C8—91no—
273C2×C4⋊C4—31no—
274C2×C4⋊C4D3211yescharacteristic
275C2×C22⋊C4—181no—
276C2×C22⋊C4—181no—
277C2×C42—31no—
278C4⋊C8—182no—
279#103—182no—
280#102—182no—
281#95—61no—
282#95—61no—
283C23×C3—21no—
284C23×C3D3×D411yescharacteristic
285C23×C3C2×#6711yescharacteristic
286C22×D3—62no—
287C22×D3—62no—
288C22×D3—122no—
289C22×D3—122no—
290C22×D3—122no—
291C22×D3—121no—
292C22×D3—121no—
293C22×D3—121no—
294C22×D3—121no—
295C22×D3—121no—
296C22×D3—121no—
297C22×D3—121no—
298C22×D3—121no—
299C22×D3—121no—
300C22×D3—121no—
301C22×D3—121no—
302C22×D3—121no—
303C22×D3—241no—
304C22×D3—241no—
305C22×D3—241no—
306C22×D3—241no—
307C3×Q8—62no—
308C3×Q8—62no—
309C3×Q8—121no—
310C3×Q8—121no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—21no—
316C2×C3×C4—21no—
317C2×C3×C4—32no—
318C2×C3×C4—42no—
319C2×C3×C4—41no—
320C2×C3×C4—41no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—61no—
326C2×C3×C4—61no—
327C2×C3×C4—62no—
328C2×C3×C4—61no—
329C2×C3×C4—61no—
330C2×C3×C4—61no—
331C2×C3×C4—81no—
332C2×C3×C4—81no—
333C2×C3×C4—121no—
334C2×C3×C4—121no—
335C2×C3×C4—121no—
336C2×C3×C4—121no—
337C2×C3×C4—121no—
338C2×C3×C4C2×D1211yescharacteristic
339C2×C3×C4C2×D1211yescharacteristic
340C2×C3×C4#23511yescharacteristic
341C2×C3×C4#23511yescharacteristic
342C2×C3×C4#23511yescharacteristic
343C2×C3×C4D3×D411yescharacteristic
344C2×C3×C4D3×D411yescharacteristic
345C2×C3×C4D3×D411yescharacteristic
346C2×C3×C4D3×D411yescharacteristic
347C2×C3×C4#23711yescharacteristic
348C2×C3×C4#23711yescharacteristic
349C2×C3×C4#23911yescharacteristic
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—22no—
353C2×Dic3—21no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—41no—
357C2×Dic3—41no—
358C2×Dic3—41no—
359C2×Dic3—41no—
360C2×Dic3—41no—
361C2×Dic3#21512yescomplemented
362D12—64no—
363D12—62no—
364D12—62no—
365D12—62no—
366D12—62no—
367D12—122no—
368D12—122no—
369D12—122no—
370D12—121no—
371D12—121no—
372D12—121no—
373D12—121no—
374Dic6—22no—
375Dic6—22no—
376Dic6—41no—
377Dic6—41no—
378C3×C8—62no—
379C3⋊C8—22no—
380C3⋊C8—122no—
381C3⋊C8—121no—
382C3⋊C8—121no—
383C2×C32#39411yescharacteristic
384C2×C32#46011yescharacteristic
385C2×C32#46111yescharacteristic
386C2×C32#46311yescharacteristic
387C2×C32#46411yescharacteristic
388C2×C32#46511yescharacteristic
389C2×C32#46811yescharacteristic
390C3⋊D3—82no—
391C24—181no—
392C2×Q8—32no—
393C2×Q8—61no—
394C2×Q8—61no—
395C2×Q8—61no—
396C2×Q8—61no—
397C2×Q8—61no—
398C2×D4—92no—
399C2×D4—181no—
400C2×D4—181no—
401C2×D4—181no—
402C2×D4—181no—
403C2×D4—181no—
404C22×C4—21no—
405C22×C4—31no—
406C22×C4—31no—
407C22×C4—61no—
408C22×C4—61no—
409C22×C4C2×D3211yescharacteristic
410SD16—188no—
411C2×C8—32no—
412C2×C8—61no—
413C2×C8—61no—
414C2×C8—181no—
415C2×C8—181no—
416C2×C8—181no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—61no—
420C4⋊C4—61no—
421C22⋊C4—362no—
422C22⋊C4—362no—
423C42—61no—
424C42—61no—
425C22×C3—21no—
426C22×C3—21no—
427C22×C3—21no—
428C22×C3—21no—
429C22×C3—21no—
430C22×C3—21no—
431C22×C3—21no—
432C22×C3#88411yescharacteristic
433C22×C3#88411yescharacteristic
434C22×C3#89511yescharacteristic
435C22×C3#91111yescharacteristic
436C22×C3D3×SD1611yescharacteristic
437C22×C3#91611yescharacteristic
438C22×C3#91811yescharacteristic
439C22×C3#93011yescharacteristic
440C22×C3#93811yescharacteristic
441C22×C3#93911yescharacteristic
442C22×C3C2×#21511yescharacteristic
443C22×C3#94211yescharacteristic
444C22×C3#94911yescharacteristic
445C22×C3#95011yescharacteristic
446C2×D3—124no—
447C2×D3—122no—
448C2×D3—122no—
449C2×D3—122no—
450C2×D3—122no—
451C2×D3—122no—
452C2×D3—122no—
453C2×D3—242no—
454C2×D3—242no—
455C2×D3—242no—
456C2×D3—242no—
457C2×D3—242no—
458C2×D3—242no—
459C2×D3—241no—
460C2×D3—241no—
461C2×D3—241no—
462C2×D3—241no—
463C2×D3—241no—
464C2×D3—241no—
465C2×D3—241no—
466C2×D3—241no—
467C3×C4—21no—
468C3×C4—21no—
469C3×C4—21no—
470C3×C4—21no—
471C3×C4—62no—
472C3×C4—82no—
473C3×C4—81no—
474C3×C4—81no—
475C3×C4—121no—
476C3×C4—122no—
477C3×C4—241no—
478C3×C4#87511yescharacteristic
479C3×C4#88211yescharacteristic
480C3×C4#88211yescharacteristic
481C3×C4#88311yescharacteristic
482C3×C4#88511yescharacteristic
483C3×C4#88611yescharacteristic
484C3×C4#89411yescharacteristic
485C3×C4#89411yescharacteristic
486Dic3—22no—
487Dic3—42no—
488Dic3—41no—
489Dic3—81no—
490C32#205511yescharacteristic, complemented, Sylow
491C23—182no—
492C23—361no—
493C23—361no—
494C23—361no—
495C23—361no—
496C23—361no—
497C23—361no—
498C23D3×#6711yescharacteristic
499Q8—62no—
500Q8—62no—
501Q8—121no—
502Q8—121no—
503D4—182no—
504D4—182no—
505D4—361no—
506D4—361no—
507C2×C4—32no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—61no—
512C2×C4—61no—
513C2×C4—61no—
514C2×C4—62no—
515C2×C4—61no—
516C2×C4—61no—
517C2×C4—121no—
518C2×C4—121no—
519C2×C4—121no—
520C2×C4—121no—
521C2×C4—121no—
522C2×C4#379611yescharacteristic
523C2×C4#379811yescharacteristic
524C2×C4D3×D1211yescharacteristic
525C2×C4D3×D1211yescharacteristic
526C2×C4#380411yescharacteristic
527C2×C4#381111yescharacteristic
528C8—62no—
529C8—361no—
530C2×C3—21no—
531C2×C3—21no—
532C2×C3—21no—
533C2×C3—21no—
534C2×C3—21no—
535C2×C3—21no—
536C2×C3—21no—
537C2×C3#467211yescharacteristic
538C2×C3#476211yescharacteristic
539C2×C3#478711yescharacteristic
540C2×C3#479811yescharacteristic
541C2×C3#481811yescharacteristic
542C2×C3#486411yescharacteristic
543C2×C3#488611yescharacteristic
544C2×C3#496111yescharacteristic
545C2×C3#502711yescharacteristic
546C2×C3#502811yescharacteristic
547C2×C3#504811yescharacteristic
548C2×C3#504911yescharacteristic
549C2×C3#505711yescharacteristic
550C2×C3#508311yescharacteristic
551D3—244no—
552D3—242no—
553D3—242no—
554C22—362no—
555C22—362no—
556C22—362no—
557C22—721no—
558C22—721no—
559C22—721no—
560C22—721no—
561C22#6388611yescharacteristic
562C22#6390611yescharacteristic
563C22#6392511yescharacteristic
564C22#6393511yescharacteristic
565C22D3×#21511yescharacteristic
566C22#6394811yescharacteristic
567C22#6395811yescharacteristic
568C4—62no—
569C4—81no—
570C4—122no—
571C4—121no—
572C4—241no—
573C4#6385211yescharacteristic
574C4#6385311yescharacteristic
575C4#6389011yescharacteristic
576C4#6392311yescharacteristic
577C3—21no—
578C3#6886311yescharacteristic, complemented
579C3#6968011yescharacteristic, complemented
580C2—722no—
581C2#1059053611yescharacteristic
582C2#1059054911yescharacteristic
583C2#1059057711yescharacteristic
584C2#1059058511yescharacteristic
585C2#1059131211yescharacteristic
586C2#1059143211yescharacteristic
587C2#1059156511yescharacteristic
subgroupG/H
centerC23D3×#67
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C4⋊C4C22
socleC23×C32C2×D4
prime radicalC22×C4⋊(C3⋊D3)C22
hypercenterC2×C4⋊C4D32
hypocenterC32#2055
LCSG ▷ C22×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×#4973C2semidirect
5. C2×Dic3⋊(C3⋊C8)C2semidirect
6. C2×#63565C2semidirect
7. C3×#4529C2semidirect
33. #4973D3semidirect
41. #4529D3semidirect
147. C2×C3×Dic3SD16semidirect
361. C2×Dic3#215semidirect
490. C32#2055semidirect
578. C3#68863semidirect
579. C3#69680semidirect

Maximal subgroups

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

subgroupindex
C22×#37172
C2×#636362
C3×#49732
#105913202
C2×Dic3⋊(C3⋊C8)2
C2×#635652
C3×#45292
#696803
#688633