Finite Group Databasebeta

← #11778138

Subgroups — #11778138

Order 1152.

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

normal subgroups only

This group has 626 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
2#10592745C211yesmaximal, characteristic
3#10591321C211yesmaximal, characteristic
4#10591183C211yesmaximal, characteristic
5C3×#4538C211yesmaximal, characteristic, complemented
6#10589153C211yesmaximal, characteristic, complemented
7#10589147C211yesmaximal, characteristic, complemented
8#71902—31nomaximal
9#68325—31nomaximal
10C22×C4⋊(C3⋊D3)C2211yescharacteristic
11C32×C22⋊Q8C2211yescharacteristic
12C2×#3755—21no—
13C3×#944—21no—
14C3×#924C2211yescharacteristic
15C2×C3×Dic3⋊C4C2211yescharacteristic
16C3×#889—21no—
17C3×#889—21no—
18C3×#888—21no—
19C3×#878—21no—
20C3×#877—21no—
21C2×#3722—21no—
22C2×#3722—21no—
23C2×#3717—21no—
24#63657C2211yescharacteristic
25#63636—21no—
26C3×C3⋊#95C2211yescharacteristic
27C3×#830C2211yescharacteristic
28#63564—21no—
29C3×#544—31no—
30#5623D311yescharacteristic, complemented
31#5046—31no—
32#5046—31no—
33#5046—31no—
34#5046—61no—
35#4991—31no—
36#4731—31no—
37#4672—31no—
38C3×#328—31no—
39#4538—31no—
40#4538D311yescharacteristic, complemented
41#4528—31no—
42#4469—31no—
43#4452—31no—
44C23×C3⋊D3—21no—
45C2×C32×Q8D411yescharacteristic
46C22×C32×C4C2311yescharacteristic
47C2×C4⋊(C3⋊D3)—21no—
48C2×C4⋊(C3⋊D3)—21no—
49C2×C4⋊(C3⋊D3)—42no—
50C22×C3×Dic3—21no—
51C22×C3×Dic3D411yescharacteristic
52C22×C3×Dic3D411yescharacteristic
53C2×C3×Dic6—21no—
54C2×C3×Dic6D411yescharacteristic
55C32×C4⋊C4—21no—
56C32×C4⋊C4D411yescharacteristic
57C32×C22⋊C4—21no—
58#3755—42no—
59C2×C3⋊(C3⋊C8)—21no—
60C3×#217—21no—
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#3722—42no—
74#3722—42no—
75#3717—42no—
76#1599—91noSylow
77C22×D12—31no—
78C22×D12—31no—
79C22×D12—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—
91#944—21no—
92C2×#215—61no—
93C2×#215—62no—
94C2×#215—61no—
95C2×#215—61no—
96C2×(C2×D3)⋊C4—61no—
97C2×(C2×D3)⋊C4—61no—
98C2×(C2×D3)⋊C4—61no—
99C2×(C2×D3)⋊C4—61no—
100#924C2×D311yescharacteristic
101C2×Dic3⋊C4C2×D311yescharacteristic
102C2×#204—61no—
103#889—21no—
104#889—21no—
105#888—21no—
106#878—21no—
107#877—21no—
108C3×C22⋊C8—31no—
109C3×#95—31no—
110C3⋊#95C2×D311yescharacteristic
111#830—31no—
112#830—31no—
113#830—61no—
114#830C2×D311yescharacteristic
115#821—61no—
116#809—61no—
117#809—62no—
118#809—61no—
119#809—61no—
120C23×C32C2×D411yescharacteristic
121C22×C3⋊D3—22no—
122C22×C3⋊D3—41no—
123C22×C3⋊D3—41no—
124C22×C3⋊D3—41no—
125C22×C3⋊D3—41no—
126C22×C3⋊D3—41no—
127C22×C3⋊D3—41no—
128C32×Q8—41no—
129C2×C32×C4—21no—
130C2×C32×C4—21no—
131C2×C32×C4—41no—
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4C2×D411yescharacteristic
134C4⋊(C3⋊D3)—22no—
135C4⋊(C3⋊D3)—22no—
136C4⋊(C3⋊D3)—41no—
137C4⋊(C3⋊D3)—41no—
138C2×C3×Dic3—21no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—21no—
142C2×C3×Dic3—21no—
143C2×C3×Dic3—41no—
144C2×C3×Dic3—41no—
145C2×C3×Dic3—41no—
146C2×C3×Dic3—41no—
147C2×C3×Dic3SD1612yescomplemented
148C3×Dic6—41no—
149C3×Dic6—81no—
150C3⋊(C3⋊C8)—41no—
151C3×C3⋊C8—41no—
152#544—31no—
153#450—91no—
154#450—91no—
155#394—91no—
156#328—31no—
157#328—91no—
158#324—91no—
159C23×D3—61no—
160C23×D3—61no—
161C23×D3—121no—
162C2×C3×Q8—21no—
163C2×C3×Q8—31no—
164C2×C3×Q8—61no—
165C2×C3×Q8D1211yescharacteristic
166C2×C3×Q8#6711yescharacteristic
167C22×C3×C4—21no—
168C22×C3×C4—31no—
169C22×C3×C4—31no—
170C22×C3×C4—61no—
171C22×C3×C4C22×D311yescharacteristic
172C22×C3×C4C22×D311yescharacteristic
173C22×Dic3—21no—
174C22×Dic3#6711yescharacteristic
175C22×Dic3#6711yescharacteristic
176C2×D12—62no—
177C2×D12—61no—
178C2×D12—61no—
179C2×D12—61no—
180C2×D12—61no—
181C2×D12—121no—
182C2×D12—122no—
183C2×D12—122no—
184C2×D12—124no—
185C2×Dic6—21no—
186C2×Dic6#6711yescharacteristic
187C2×C3×C8—61no—
188C3×C4⋊C4—21no—
189C3×C4⋊C4—21no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—31no—
192C3×C4⋊C4—41no—
193C3×C4⋊C4—61no—
194C3×C4⋊C4—61no—
195C3×C4⋊C4—61no—
196C3×C4⋊C4—61no—
197C3×C4⋊C4—61no—
198C3×C4⋊C4—61no—
199C3×C4⋊C4—61no—
200C3×C4⋊C4—61no—
201C3×C4⋊C4D1211yescharacteristic
202C3×C4⋊C4#6711yescharacteristic
203C3×C22⋊C4—21no—
204C3×C22⋊C4—21no—
205C3×C22⋊C4—41no—
206C3×C22⋊C4—61no—
207C3×C42—61no—
208C3×C42—61no—
209#217—21no—
210#215—122no—
211#215—122no—
212#215—122no—
213#215—124no—
214(C2×D3)⋊C4—122no—
215(C2×D3)⋊C4—122no—
216(C2×D3)⋊C4—122no—
217(C2×D3)⋊C4—122no—
218C4⋊Dic3—21no—
219C4⋊Dic3#6711yescharacteristic
220Dic3⋊C4—21no—
221Dic3⋊C4—21no—
222Dic3⋊C4—21no—
223Dic3⋊C4—21no—
224Dic3⋊C4—21no—
225Dic3⋊C4—21no—
226Dic3⋊C4—21no—
227C4×Dic3—21no—
228C4×Dic3—21no—
229C2×C3⋊C8—21no—
230C2×C3⋊C8—62no—
231C2×C3⋊C8—61no—
232C2×C3⋊C8—61no—
233#204—122no—
234C22×C32—21no—
235C22×C32—21no—
236C22×C32#12011yescharacteristic
237C22×C32C2×SD1611yescharacteristic
238C22×C32#13711yescharacteristic
239C2×C3⋊D3—42no—
240C2×C3⋊D3—42no—
241C2×C3⋊D3—42no—
242C2×C3⋊D3—81no—
243C2×C3⋊D3—81no—
244C2×C3⋊D3—81no—
245C2×C3⋊D3—81no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—41no—
249C32×C4—41no—
250C3×Dic3—22no—
251C3×Dic3—41no—
252C3×Dic3—41no—
253C3×Dic3—41no—
254C3×Dic3—81no—
255C22×D4—91no—
256C2×SD16—181no—
257C2×SD16—181no—
258C4⋊Q8—61no—
259C4⋊Q8—61no—
260#125—61no—
261#125—61no—
262C22⋊Q8—31no—
263C22⋊Q8—61no—
264C22⋊Q8D3211yescharacteristic
265#117—61no—
266C2×C4⋊C4—31no—
267C2×C22⋊C4—181no—
268C2×C22⋊C4—181no—
269#102—181no—
270#102—181no—
271C22⋊C8—31no—
272C22⋊C8—91no—
273#95—31no—
274C23×C3—21no—
275C23×C3D3×D411yescharacteristic
276C23×C3C2×#6711yescharacteristic
277C22×D3—62no—
278C22×D3—62no—
279C22×D3—122no—
280C22×D3—122no—
281C22×D3—121no—
282C22×D3—121no—
283C22×D3—121no—
284C22×D3—121no—
285C22×D3—121no—
286C22×D3—121no—
287C22×D3—122no—
288C22×D3—121no—
289C22×D3—121no—
290C22×D3—121no—
291C22×D3—121no—
292C22×D3—121no—
293C22×D3—121no—
294C22×D3—241no—
295C22×D3—241no—
296C22×D3—241no—
297C22×D3—241no—
298C3×Q8—41no—
299C3×Q8—41no—
300C3×Q8—42no—
301C3×Q8—121no—
302C3×Q8—241no—
303C2×C3×C4—21no—
304C2×C3×C4—21no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—32no—
310C2×C3×C4—41no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—41no—
314C2×C3×C4—42no—
315C2×C3×C4—61no—
316C2×C3×C4—61no—
317C2×C3×C4—61no—
318C2×C3×C4—61no—
319C2×C3×C4—61no—
320C2×C3×C4—121no—
321C2×C3×C4—121no—
322C2×C3×C4—121no—
323C2×C3×C4—121no—
324C2×C3×C4C2×D1211yescharacteristic
325C2×C3×C4D3×D411yescharacteristic
326C2×C3×C4C2×#6711yescharacteristic
327C2×C3×C4C2×#6711yescharacteristic
328C2×Dic3—21no—
329C2×Dic3—21no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—41no—
337C2×Dic3#21512yescomplemented
338D12—62no—
339D12—64no—
340D12—62no—
341D12—62no—
342D12—62no—
343D12—122no—
344D12—122no—
345D12—121no—
346D12—121no—
347D12—121no—
348D12—121no—
349D12—122no—
350Dic6—41no—
351Dic6—81no—
352C3×C8—121no—
353C3⋊C8—41no—
354C3⋊C8—122no—
355C3⋊C8—121no—
356C3⋊C8—121no—
357C2×C32—21no—
358C2×C32—21no—
359C2×C32#44811yescharacteristic
360C2×C32#45411yescharacteristic
361C2×C32B(4,𝔽2)11yescharacteristic
362C3⋊D3—82no—
363C24—181no—
364C2×Q8—31no—
365C2×Q8—61no—
366C2×Q8#62611yescharacteristic
367C2×D4—181no—
368C2×D4—181no—
369C2×D4—362no—
370C22×C4—31no—
371C22×C4—31no—
372C22×C4—61no—
373C22×C4C2×D3211yescharacteristic
374SD16—362no—
375SD16—362no—
376C2×C8—61no—
377C2×C8—181no—
378C4⋊C4—21no—
379C4⋊C4—31no—
380C4⋊C4—61no—
381C4⋊C4—61no—
382C4⋊C4—61no—
383C4⋊C4—61no—
384C4⋊C4—61no—
385C4⋊C4—61no—
386C4⋊C4—61no—
387C4⋊C4—61no—
388C4⋊C4#62611yescharacteristic
389C22⋊C4—21no—
390C22⋊C4—61no—
391C22⋊C4—362no—
392C22⋊C4—362no—
393C42—61no—
394C42—61no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—41no—
403C22×C3—41no—
404C22×C3#88211yescharacteristic
405C22×C3D3×SD1611yescharacteristic
406C22×C3#91811yescharacteristic
407C22×C3C2×#21511yescharacteristic
408C22×C3#94211yescharacteristic
409C22×C3#95311yescharacteristic
410C2×D3—124no—
411C2×D3—122no—
412C2×D3—122no—
413C2×D3—122no—
414C2×D3—122no—
415C2×D3—122no—
416C2×D3—122no—
417C2×D3—242no—
418C2×D3—242no—
419C2×D3—241no—
420C2×D3—241no—
421C2×D3—241no—
422C2×D3—241no—
423C2×D3—242no—
424C2×D3—242no—
425C2×D3—242no—
426C2×D3—242no—
427C2×D3—241no—
428C2×D3—241no—
429C2×D3—241no—
430C2×D3—241no—
431C3×C4—22no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—21no—
435C3×C4—21no—
436C3×C4—41no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—41no—
440C3×C4—41no—
441C3×C4—62no—
442C3×C4—81no—
443C3×C4—81no—
444C3×C4—121no—
445C3×C4—121no—
446C3×C4—121no—
447C3×C4—241no—
448Dic3—22no—
449Dic3—41no—
450Dic3—41no—
451Dic3—41no—
452Dic3—81no—
453C32#159911yescharacteristic, complemented, Sylow
454C23—182no—
455C23—361no—
456C23—361no—
457C23—361no—
458C23—361no—
459C23—361no—
460C23—361no—
461C23D3×#6711yescharacteristic
462Q8—41no—
463Q8—121no—
464Q8—241no—
465D4—182no—
466D4—182no—
467D4—361no—
468D4—361no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—32no—
472C2×C4—41no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—121no—
479C2×C4—121no—
480C2×C4—121no—
481C2×C4—121no—
482C2×C4C2×#62611yescharacteristic
483C2×C4D3×#6711yescharacteristic
484C8—121no—
485C8—361no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—21no—
493C2×C3—41no—
494C2×C3—41no—
495C2×C3#474111yescharacteristic
496C2×C3#480611yescharacteristic
497C2×C3#482711yescharacteristic
498C2×C3#522711yescharacteristic
499C2×C3#524411yescharacteristic
500C2×C3#524511yescharacteristic
501D3—244no—
502D3—242no—
503D3—242no—
504C22—21no—
505C22—21no—
506C22—362no—
507C22—362no—
508C22—362no—
509C22—721no—
510C22—721no—
511C22—721no—
512C22—721no—
513C22D3×#21511yescharacteristic
514C22#6394811yescharacteristic
515C22#6397711yescharacteristic
516C4—21no—
517C4—21no—
518C4—41no—
519C4—41no—
520C4—62no—
521C4—121no—
522C4—121no—
523C4—121no—
524C4—241no—
525C3—21no—
526C3#6832511yescharacteristic, complemented
527C3#7190211yescharacteristic, complemented
528C2—21no—
529C2—21no—
530C2—722no—
531C2#1059153111yescharacteristic
532C2#1059158411yescharacteristic
533C2#1059168511yescharacteristic
subgroupG/H
centerC22#63977
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×C22⋊Q8C22
socleC22×C32#120
prime radicalC22×C4⋊(C3⋊D3)C22
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. C3×#4538C2semidirect
6. #10589153C2semidirect
7. #10589147C2semidirect
30. #5623D3semidirect
40. #4538D3semidirect
147. C2×C3×Dic3SD16semidirect
337. C2×Dic3#215semidirect
453. C32#1599semidirect
526. C3#68325semidirect
527. C3#71902semidirect

Maximal subgroups

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

subgroupindex
C3×#56232
#105927452
#105913212
#105911832
C3×#45382
#105891532
#105891472
#719023
#683253