Finite Group Databasebeta

← #11778140

Subgroups — #11778140

Order 1152.

572 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 88 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#5623C211yesmaximal, characteristic, complemented
2C3×#5046C211yesmaximal, characteristic
3#10591300C211yesmaximal, characteristic
4#10591169C211yesmaximal, characteristic
5#10589404C211yesmaximal, characteristic, complemented
6#10589152C211yesmaximal, characteristic, complemented
7#10589147C211yesmaximal, characteristic, complemented
8#71902—31nomaximal
9#70979—31nomaximal
10C22×C3×D12C2211yescharacteristic
11C32×C22⋊Q8C2211yescharacteristic
12C3×#944—21no—
13C2×C3×#215—21no—
14C3×#924C2211yescharacteristic
15C2×C3×Dic3⋊C4C2211yescharacteristic
16C3×#889—21no—
17C3×#889—21no—
18C3×#888—21no—
19C3×#878—21no—
20C3×#877—21no—
21C2×(C2×D3)⋊Dic3—21no—
22C2×(C2×D3)⋊Dic3—21no—
23C2×#3715—21no—
24#63658C2211yescharacteristic
25#63657C2211yescharacteristic
26C3×#830C2211yescharacteristic
27C3×#809—21no—
28#63562—21no—
29C3×#544—31no—
30#5623D311yescharacteristic, complemented
31C3×#450—31no—
32#5222—31no—
33#5218—31no—
34#5046—31no—
35#5046D311yescharacteristic
36#4991—31no—
37#4538—31no—
38#4538—31no—
39#4538—31no—
40#4538—61no—
41#4528—31no—
42#4469—31no—
43#4467—31no—
44C23×C3×D3—21no—
45C2×C32×Q8D411yescharacteristic
46C22×C32×C4C2311yescharacteristic
47C22×C3⋊Dic3—21no—
48C22×C3×Dic3D411yescharacteristic
49C22×C3×Dic3D411yescharacteristic
50C2×C3×D12—21no—
51C2×C3×D12—21no—
52C2×C3×D12—42no—
53C2×C3×Dic6—21no—
54C2×C3×Dic6D411yescharacteristic
55C32×C4⋊C4—21no—
56C32×C4⋊C4D411yescharacteristic
57C32×C22⋊C4—21no—
58C2×C3⋊(C3⋊C8)—21no—
59C3×#217—21no—
60C3×#215—42no—
61C3×C4⋊Dic3—21no—
62C3×C4⋊Dic3D411yescharacteristic
63C3×Dic3⋊C4—21no—
64C3×Dic3⋊C4—21no—
65C3×Dic3⋊C4—21no—
66C3×Dic3⋊C4—21no—
67C3×Dic3⋊C4—21no—
68C3×Dic3⋊C4—21no—
69C3×Dic3⋊C4—21no—
70C3×C4×Dic3—21no—
71C3×C4×Dic3—21no—
72C2×C3×C3⋊C8—21no—
73(C2×D3)⋊Dic3—42no—
74(C2×D3)⋊Dic3—42no—
75#3715—42no—
76#1599—91noSylow
77C22×C3×D4—31no—
78C22×D12C2×D311yescharacteristic
79C2×C3×SD16—61no—
80C3×C4⋊Q8—61no—
81C3×C4⋊Q8—61no—
82C3×#125—61no—
83C3×#125—61no—
84C3×C22⋊Q8—21no—
85C3×C22⋊Q8—31no—
86C3×C22⋊Q8—61no—
87C3×C22⋊Q8C2×D311yescharacteristic
88C3×C22⋊Q8C2×D311yescharacteristic
89C3×#117—61no—
90C2×C3×C4⋊C4—31no—
91C2×#217—61no—
92C2×#217—61no—
93#944—21no—
94C2×#215—21no—
95C2×#215—61no—
96C2×#214—61no—
97C2×(C2×D3)⋊C4—61no—
98C2×(C2×D3)⋊C4—61no—
99#924C2×D311yescharacteristic
100C2×Dic3⋊C4C2×D311yescharacteristic
101#889—21no—
102#889—21no—
103#888—21no—
104#878—21no—
105#877—21no—
106C3×#102—61no—
107C3×C22⋊C8—31no—
108#832—61no—
109C3⋊#95—31no—
110C3⋊#95—31no—
111C3⋊#95—61no—
112#830—31no—
113#830—31no—
114#830—61no—
115#830C2×D311yescharacteristic
116#809—21no—
117#809—61no—
118C23×C32C2×D411yescharacteristic
119C22×C3×D3—22no—
120C22×C3×D3—41no—
121C22×C3×D3—41no—
122C22×C3×D3—41no—
123C22×C3×D3—41no—
124C22×C3×D3—41no—
125C22×C3×D3—41no—
126C32×Q8—41no—
127C2×C32×C4—21no—
128C2×C32×C4—21no—
129C2×C32×C4—41no—
130C2×C32×C4C2×D411yescharacteristic
131C2×C32×C4C2×D411yescharacteristic
132C2×C3⋊Dic3—41no—
133C2×C3⋊Dic3—41no—
134C2×C3⋊Dic3—41no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—21no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—21no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—41no—
141C2×C3×Dic3SD1612yescomplemented
142C3×D12—22no—
143C3×D12—22no—
144C3×D12—41no—
145C3×D12—41no—
146C3×Dic6—41no—
147C3×Dic6—81no—
148C3⋊(C3⋊C8)—41no—
149C3×C3⋊C8—41no—
150#544—31no—
151#450—31no—
152#450—91no—
153#394—91no—
154#328—91no—
155#328—91no—
156#324—91no—
157C24×C3—61no—
158C23×D3—21no—
159C2×C3×Q8—21no—
160C2×C3×Q8—31no—
161C2×C3×Q8—61no—
162C2×C3×Q8#6711yescharacteristic
163C2×C3×Q8#6711yescharacteristic
164C2×C3×D4—61no—
165C2×C3×D4—61no—
166C2×C3×D4—122no—
167C22×C3×C4—21no—
168C22×C3×C4—31no—
169C22×C3×C4—31no—
170C22×C3×C4C22×D311yescharacteristic
171C22×C3×C4C22×D311yescharacteristic
172C22×Dic3—61no—
173C22×Dic3—61no—
174C22×Dic3—121no—
175C22×Dic3#6711yescharacteristic
176C22×Dic3#6711yescharacteristic
177C2×D12—21no—
178C2×D12—21no—
179C2×D12—42no—
180C2×Dic6—21no—
181C2×Dic6#6711yescharacteristic
182C3×SD16—122no—
183C2×C3×C8—61no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4—31no—
188C3×C4⋊C4—41no—
189C3×C4⋊C4—61no—
190C3×C4⋊C4—61no—
191C3×C4⋊C4—61no—
192C3×C4⋊C4—61no—
193C3×C4⋊C4—61no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4—61no—
197C3×C4⋊C4#6711yescharacteristic
198C3×C4⋊C4#6711yescharacteristic
199C3×C22⋊C4—21no—
200C3×C22⋊C4—21no—
201C3×C22⋊C4—41no—
202C3×C22⋊C4—61no—
203C3×C42—61no—
204C3×C42—61no—
205#217—21no—
206#217—122no—
207#217—122no—
208#215—42no—
209#215—122no—
210#214—122no—
211(C2×D3)⋊C4—122no—
212(C2×D3)⋊C4—122no—
213C4⋊Dic3—21no—
214C4⋊Dic3#6711yescharacteristic
215Dic3⋊C4—21no—
216Dic3⋊C4—21no—
217Dic3⋊C4—21no—
218Dic3⋊C4—21no—
219Dic3⋊C4—21no—
220Dic3⋊C4—21no—
221Dic3⋊C4—21no—
222C4×Dic3—21no—
223C4×Dic3—21no—
224C2×C3⋊C8—21no—
225C2×C3⋊C8—61no—
226C2×C3⋊C8—61no—
227C2×C3⋊C8—121no—
228C22×C32—21no—
229C22×C32—21no—
230C22×C32#12011yescharacteristic
231C22×C32C2×SD1611yescharacteristic
232C22×C32#13711yescharacteristic
233C2×C3×D3—42no—
234C2×C3×D3—42no—
235C2×C3×D3—42no—
236C2×C3×D3—81no—
237C2×C3×D3—81no—
238C2×C3×D3—81no—
239C2×C3×D3—81no—
240C32×C4—21no—
241C32×C4—21no—
242C32×C4—41no—
243C32×C4—41no—
244C3⋊Dic3—81no—
245C3×Dic3—22no—
246C3×Dic3—41no—
247C3×Dic3—41no—
248C3×Dic3—41no—
249C22×D4—31no—
250C2×SD16—61no—
251C2×SD16—181no—
252C4⋊Q8—61no—
253C4⋊Q8—61no—
254#125—61no—
255#125—61no—
256C22⋊Q8—31no—
257C22⋊Q8—61no—
258C22⋊Q8D3211yescharacteristic
259#117—61no—
260C2×C4⋊C4—31no—
261C2×C22⋊C4—181no—
262C2×C22⋊C4—181no—
263#102—61no—
264#102—181no—
265C22⋊C8—31no—
266C22⋊C8—91no—
267#95—91no—
268C23×C3—21no—
269C23×C3—62no—
270C23×C3—121no—
271C23×C3—121no—
272C23×C3—121no—
273C23×C3—121no—
274C23×C3—121no—
275C23×C3—121no—
276C23×C3D3×D411yescharacteristic
277C23×C3C2×#6711yescharacteristic
278C22×D3—22no—
279C22×D3—41no—
280C22×D3—41no—
281C22×D3—41no—
282C22×D3—41no—
283C22×D3—41no—
284C22×D3—41no—
285C3×Q8—41no—
286C3×Q8—41no—
287C3×Q8—81no—
288C3×Q8—121no—
289C3×Q8—241no—
290C3×D4—62no—
291C3×D4—62no—
292C3×D4—121no—
293C3×D4—121no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—32no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—41no—
304C2×C3×C4—41no—
305C2×C3×C4—61no—
306C2×C3×C4—61no—
307C2×C3×C4—61no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—81no—
311C2×C3×C4—121no—
312C2×C3×C4D3×D411yescharacteristic
313C2×C3×C4C2×#6711yescharacteristic
314C2×C3×C4C2×#6711yescharacteristic
315C2×C3×C4C2×#6711yescharacteristic
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—21no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—41no—
322C2×Dic3—121no—
323C2×Dic3—121no—
324C2×Dic3—121no—
325C2×Dic3—121no—
326C2×Dic3—122no—
327C2×Dic3—121no—
328C2×Dic3—121no—
329C2×Dic3—241no—
330C2×Dic3—241no—
331C2×Dic3#21512yescomplemented
332D12—22no—
333D12—22no—
334D12—41no—
335D12—41no—
336Dic6—41no—
337Dic6—81no—
338C3×C8—121no—
339C3⋊C8—41no—
340C3⋊C8—121no—
341C3⋊C8—121no—
342C3⋊C8—241no—
343C2×C32—21no—
344C2×C32—21no—
345C2×C32#44811yescharacteristic
346C2×C32#45411yescharacteristic
347C2×C32B(4,𝔽2)11yescharacteristic
348C3×D3—82no—
349C24—61no—
350C2×Q8—31no—
351C2×Q8—61no—
352C2×Q8#62511yescharacteristic
353C2×D4—61no—
354C2×D4—61no—
355C2×D4—122no—
356C22×C4—31no—
357C22×C4—31no—
358C22×C4—181no—
359C22×C4C2×D3211yescharacteristic
360SD16—122no—
361SD16—362no—
362C2×C8—61no—
363C2×C8—181no—
364C4⋊C4—21no—
365C4⋊C4—31no—
366C4⋊C4—61no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—61no—
370C4⋊C4—61no—
371C4⋊C4—61no—
372C4⋊C4—61no—
373C4⋊C4—61no—
374C4⋊C4#62511yescharacteristic
375C22⋊C4—21no—
376C22⋊C4—61no—
377C22⋊C4—362no—
378C22⋊C4—362no—
379C42—61no—
380C42—61no—
381C22×C3—21no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—21no—
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—21no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—122no—
391C22×C3—122no—
392C22×C3—122no—
393C22×C3—241no—
394C22×C3—241no—
395C22×C3—241no—
396C22×C3—241no—
397C22×C3D3×SD1611yescharacteristic
398C22×C3#91511yescharacteristic
399C22×C3#93711yescharacteristic
400C22×C3C2×#21511yescharacteristic
401C22×C3#94211yescharacteristic
402C22×C3#95311yescharacteristic
403C2×D3—42no—
404C2×D3—42no—
405C2×D3—42no—
406C2×D3—81no—
407C2×D3—81no—
408C2×D3—81no—
409C2×D3—81no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—21no—
413C3×C4—21no—
414C3×C4—41no—
415C3×C4—41no—
416C3×C4—41no—
417C3×C4—41no—
418C3×C4—41no—
419C3×C4—41no—
420C3×C4—62no—
421C3×C4—81no—
422C3×C4—81no—
423C3×C4—121no—
424C3×C4—121no—
425C3×C4—121no—
426Dic3—22no—
427Dic3—41no—
428Dic3—41no—
429Dic3—41no—
430Dic3—241no—
431Dic3—241no—
432Dic3—242no—
433C32#159911yescharacteristic, complemented, Sylow
434C23—62no—
435C23—121no—
436C23—121no—
437C23—121no—
438C23—121no—
439C23—121no—
440C23—121no—
441C23D3×#6711yescharacteristic
442Q8—41no—
443Q8—121no—
444Q8—241no—
445D4—62no—
446D4—62no—
447D4—121no—
448D4—121no—
449C2×C4—21no—
450C2×C4—21no—
451C2×C4—32no—
452C2×C4—41no—
453C2×C4—61no—
454C2×C4—61no—
455C2×C4—61no—
456C2×C4—61no—
457C2×C4—61no—
458C2×C4—121no—
459C2×C4—361no—
460C2×C4—361no—
461C2×C4—361no—
462C2×C4C2×#62511yescharacteristic
463C2×C4D3×#6711yescharacteristic
464C8—121no—
465C8—361no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—21no—
470C2×C3—21no—
471C2×C3—21no—
472C2×C3—21no—
473C2×C3—41no—
474C2×C3—41no—
475C2×C3—242no—
476C2×C3#503211yescharacteristic
477C2×C3#517011yescharacteristic
478C2×C3#520111yescharacteristic
479C2×C3#522711yescharacteristic
480C2×C3#524411yescharacteristic
481C2×C3#524511yescharacteristic
482D3—82no—
483C22—21no—
484C22—21no—
485C22—122no—
486C22—122no—
487C22—122no—
488C22—241no—
489C22—241no—
490C22—241no—
491C22—241no—
492C22D3×#21511yescharacteristic
493C22#6394611yescharacteristic
494C22#6397611yescharacteristic
495C4—21no—
496C4—21no—
497C4—41no—
498C4—41no—
499C4—62no—
500C4—121no—
501C4—121no—
502C4—121no—
503C4—721no—
504C3—21no—
505C3#7097911yescharacteristic, complemented
506C3#7190211yescharacteristic, complemented
507C2—21no—
508C2—21no—
509C2—242no—
510C2#1059153011yescharacteristic
511C2#1059158311yescharacteristic
512C2#1059168411yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×C22⋊Q8C22
socleC22×C32#120
prime radicalC22×C3×D12C22
hypercenterC22⋊Q8D32
hypocenterC32#1599
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C3×#5623C2semidirect
5. #10589404C2semidirect
6. #10589152C2semidirect
7. #10589147C2semidirect
30. #5623D3semidirect
141. C2×C3×Dic3SD16semidirect
331. C2×Dic3#215semidirect
433. C32#1599semidirect
505. C3#70979semidirect
506. C3#71902semidirect

Maximal subgroups

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

subgroupindex
C3×#56232
C3×#50462
#105913002
#105911692
#105894042
#105891522
#105891472
#719023
#709793