Finite Group Databasebeta

← #11778635

Subgroups — #11778635

Order 1152.

513 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 513 conjugacy classes of subgroups — too many to draw. Only its 84 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×#5494C211yesmaximal, characteristic
2C3×#4953C211yesmaximal, characteristic, complemented
3#10591101C211yesmaximal, characteristic, complemented
4#10591089C211yesmaximal, characteristic, complemented
5#10589391C211yesmaximal, characteristic, complemented
6#10589153C211yesmaximal, characteristic, complemented
7#10589149C211yesmaximal, characteristic, complemented
8#71882—31nomaximal
9#69709—31nomaximal
10C2×C32×C22⋊C4C2211yescharacteristic
11C3×#953—21no—
12C2×C3×#217—21no—
13C3×#930C2211yescharacteristic
14C3×C4×#67—21no—
15C2×C3×C4⋊Dic3C2211yescharacteristic
16C3×#924C2211yescharacteristic
17C3×#885—21no—
18C3×#884—21no—
19C3×#883—21no—
20C3×#879—21no—
21#63657C2211yescharacteristic
22#63657C2211yescharacteristic
23C3×C3⋊#95—21no—
24C3×C3⋊#95C2211yescharacteristic
25#63576—21no—
26#63574—21no—
27#63565—21no—
28#63562—21no—
29C3×#538—31no—
30#5494D311yescharacteristic
31C3×#397—31no—
32#5043—31no—
33#5033—31no—
34#4971—31no—
35#4967—31no—
36#4953D311yescharacteristic, complemented
37#4538—31no—
38#4537—31no—
39#4525—31no—
40#4525—31no—
41#4525—61no—
42#4452—31no—
43#4451—31no—
44C24×C32D411yescharacteristic
45C22×C32×C4D411yescharacteristic
46C22×C32×C4C2311yescharacteristic
47C2×C3×#67—21no—
48C2×C3×#67—21no—
49C2×C3×#67—21no—
50C22×C3×Dic3—21no—
51C22×C3×Dic3—21no—
52C2×C3×D12D411yescharacteristic
53C2×C3×C4×D3—21no—
54C2×C3×Dic6D411yescharacteristic
55C32×C22⋊C4—21no—
56C32×C22⋊C4—21no—
57C2×C3⋊(C3⋊C8)—21no—
58C2×C3⋊(C3⋊C8)—21no—
59C3×#217—21no—
60C3×#217—21no—
61C3×#217—21no—
62C3×#217—42no—
63C3×(C2×D3)⋊C4—21no—
64C3×(C2×D3)⋊C4—21no—
65C3×C4⋊Dic3—21no—
66C3×C4⋊Dic3—21no—
67C3×C4⋊Dic3D411yescharacteristic
68C3×C4⋊Dic3D411yescharacteristic
69C3×Dic3⋊C4—21no—
70C3×Dic3⋊C4—21no—
71C3×C4×Dic3—21no—
72#1612—91noSylow
73C3×#126—61no—
74C3×#124—61no—
75C3×#123—61no—
76C3×C22⋊Q8—31no—
77C3×#121—31no—
78C3×#121—61no—
79C3×#120—61no—
80C3×C4×D4—61no—
81C2×C3×C4⋊C4—31no—
82C2×C3×C22⋊C4—21no—
83C2×C3×C22⋊C4—61no—
84C2×C3×C22⋊C4C2×D311yescharacteristic
85C2×C3×C22⋊C4C2×D311yescharacteristic
86#953—21no—
87C2×#217—21no—
88#930C2×D311yescharacteristic
89C4×#67—21no—
90C2×C4⋊Dic3C2×D311yescharacteristic
91#924C2×D311yescharacteristic
92#885—21no—
93#884—21no—
94#883—21no—
95#879—21no—
96C3×#95—31no—
97C3×#95—61no—
98#835—61no—
99#832—61no—
100C3⋊#95—21no—
101C3⋊#95C2×D311yescharacteristic
102#830—31no—
103#830—31no—
104#830—31no—
105#830—31no—
106#830—61no—
107#830—61no—
108#810—61no—
109#809—61no—
110#808—61no—
111#808—61no—
112#807—61no—
113#807—61no—
114C23×C32—21no—
115C23×C32—21no—
116C23×C32—21no—
117C23×C32—41no—
118C23×C32—41no—
119C23×C32C2×D411yescharacteristic
120C22×C3×D3—21no—
121C2×C32×C4—21no—
122C2×C32×C4—41no—
123C2×C32×C4—41no—
124C2×C32×C4C2×D411yescharacteristic
125C2×C32×C4C2×D411yescharacteristic
126C3×#67—41no—
127C3×#67—41no—
128C3×#67—81no—
129C3×#67—81no—
130C2×C3×Dic3—21no—
131C2×C3×Dic3—21no—
132C2×C3×Dic3—21no—
133C2×C3×Dic3—41no—
134C2×C3×Dic3—41no—
135C2×C3×Dic3—41no—
136C2×C3×Dic3—41no—
137C2×C3×Dic3—41no—
138C2×C3×Dic3—41no—
139C3×D12—41no—
140C3×C4×D3—81no—
141C3×Dic6—41no—
142C3⋊(C3⋊C8)—41no—
143C3⋊(C3⋊C8)—41no—
144#538—31no—
145#483—91no—
146#480—91no—
147#397—31no—
148#328—91no—
149#327—91no—
150#323—91no—
151C24×C3—21no—
152C24×C3#6711yescharacteristic
153C24×C3#6711yescharacteristic
154C2×C3×Q8—31no—
155C2×C3×D4—31no—
156C2×C3×D4—61no—
157C2×C3×D4—61no—
158C2×C3×D4—61no—
159C22×C3×C4—21no—
160C22×C3×C4—21no—
161C22×C3×C4—61no—
162C22×C3×C4—61no—
163C22×C3×C4—61no—
164C22×C3×C4#6711yescharacteristic
165C22×C3×C4#6711yescharacteristic
166C22×C3×C4C22×D311yescharacteristic
167C22×C3×C4C22×D311yescharacteristic
168C2×#67—21no—
169C2×#67—21no—
170C2×#67—21no—
171C22×Dic3—21no—
172C22×Dic3—21no—
173C2×D12#6711yescharacteristic
174C2×C4×D3—21no—
175C2×Dic6#6711yescharacteristic
176C3×C4⋊C4—31no—
177C3×C4⋊C4—31no—
178C3×C4⋊C4—61no—
179C3×C4⋊C4—61no—
180C3×C4⋊C4—61no—
181C3×C4⋊C4—61no—
182C3×C22⋊C4—21no—
183C3×C22⋊C4—21no—
184C3×C22⋊C4—21no—
185C3×C22⋊C4—21no—
186C3×C22⋊C4—41no—
187C3×C22⋊C4—41no—
188C3×C22⋊C4—61no—
189C3×C22⋊C4—61no—
190C3×C22⋊C4—61no—
191C3×C22⋊C4—61no—
192C3×C22⋊C4—61no—
193C3×C22⋊C4—122no—
194C3×C42—61no—
195#217—21no—
196#217—21no—
197#217—21no—
198#217—42no—
199(C2×D3)⋊C4—21no—
200(C2×D3)⋊C4—21no—
201C4⋊Dic3—21no—
202C4⋊Dic3—21no—
203C4⋊Dic3#6711yescharacteristic
204C4⋊Dic3#6711yescharacteristic
205Dic3⋊C4—21no—
206Dic3⋊C4—21no—
207C4×Dic3—21no—
208C2×C3⋊C8—61no—
209C2×C3⋊C8—61no—
210C2×C3⋊C8—61no—
211C2×C3⋊C8—61no—
212C2×C3⋊C8—121no—
213C2×C3⋊C8—121no—
214C22×C32—21no—
215C22×C32—21no—
216C22×C32—41no—
217C22×C32—42no—
218C22×C32—41no—
219C22×C32—41no—
220C22×C32—81no—
221C22×C32#12011yescharacteristic
222C22×C32#13511yescharacteristic
223C22×C32AGL(1,ℤ/8)11yescharacteristic
224C2×C3×D3—41no—
225C2×C3×D3—41no—
226C2×C3×D3—41no—
227C32×C4—21no—
228C32×C4—21no—
229C32×C4—41no—
230C3×Dic3—41no—
231C3×Dic3—41no—
232C3×Dic3—41no—
233C3×Dic3—81no—
234C3×Dic3—81no—
235#126—61no—
236#124—61no—
237#123—61no—
238C22⋊Q8—31no—
239#121—31no—
240#121—61no—
241#120—61no—
242C4×D4—61no—
243C2×C4⋊C4—31no—
244C2×C22⋊C4—61no—
245C2×C22⋊C4D3211yescharacteristic
246#107—181no—
247#106—181no—
248#103—181no—
249#102—181no—
250C22⋊C8—91no—
251C22⋊C8—91no—
252#95—31no—
253#95—61no—
254C23×C3—22no—
255C23×C3—21no—
256C23×C3—21no—
257C23×C3—21no—
258C23×C3—21no—
259C23×C3—21no—
260C23×C3—21no—
261C23×C3—21no—
262C23×C3—41no—
263C23×C3—41no—
264C23×C3—41no—
265C23×C3—41no—
266C23×C3—41no—
267C23×C3—41no—
268C23×C3—61no—
269C23×C3—81no—
270C23×C3—81no—
271C23×C3C2×#6711yescharacteristic
272C23×C3C2×#6711yescharacteristic
273C22×D3—21no—
274C3×Q8—121no—
275C3×D4—121no—
276C3×D4—121no—
277C3×D4—121no—
278C3×D4—241no—
279C3×D4—241no—
280C2×C3×C4—21no—
281C2×C3×C4—21no—
282C2×C3×C4—21no—
283C2×C3×C4—21no—
284C2×C3×C4—41no—
285C2×C3×C4—41no—
286C2×C3×C4—41no—
287C2×C3×C4—41no—
288C2×C3×C4—41no—
289C2×C3×C4—61no—
290C2×C3×C4—61no—
291C2×C3×C4—61no—
292C2×C3×C4—81no—
293C2×C3×C4—81no—
294C2×C3×C4—121no—
295C2×C3×C4—121no—
296C2×C3×C4—121no—
297C2×C3×C4—121no—
298C2×C3×C4—121no—
299C2×C3×C4—121no—
300C2×C3×C4—241no—
301C2×C3×C4D3×D411yescharacteristic
302C2×C3×C4D3×D411yescharacteristic
303C2×C3×C4C2×#6711yescharacteristic
304C2×C3×C4C2×#6711yescharacteristic
305#67—41no—
306#67—41no—
307#67—81no—
308#67—81no—
309C2×Dic3—21no—
310C2×Dic3—21no—
311C2×Dic3—21no—
312C2×Dic3—41no—
313C2×Dic3—41no—
314C2×Dic3—41no—
315C2×Dic3—41no—
316C2×Dic3—41no—
317C2×Dic3—41no—
318D12—41no—
319C4×D3—81no—
320Dic6—41no—
321C3⋊C8—121no—
322C3⋊C8—121no—
323C3⋊C8—121no—
324C3⋊C8—121no—
325C3⋊C8—241no—
326C3⋊C8—241no—
327C2×C32—21no—
328C2×C32—21no—
329C2×C32—81no—
330C2×C32#44911yescharacteristic
331C2×C32#45511yescharacteristic
332C2×C32#45811yescharacteristic
333C3×D3—81no—
334C24#62511yescharacteristic
335C2×Q8—31no—
336C2×D4—31no—
337C2×D4—61no—
338C2×D4—61no—
339C2×D4—61no—
340C22×C4—61no—
341C22×C4—61no—
342C22×C4—61no—
343C22×C4#62511yescharacteristic
344C22×C4C2×D3211yescharacteristic
345C2×C8—181no—
346C2×C8—181no—
347C4⋊C4—31no—
348C4⋊C4—31no—
349C4⋊C4—61no—
350C4⋊C4—61no—
351C4⋊C4—61no—
352C4⋊C4—61no—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—61no—
356C22⋊C4—61no—
357C22⋊C4—61no—
358C22⋊C4—61no—
359C22⋊C4—61no—
360C22⋊C4—122no—
361C42—61no—
362C22×C3—21no—
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—21no—
366C22×C3—21no—
367C22×C3—21no—
368C22×C3—21no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—42no—
372C22×C3—42no—
373C22×C3—42no—
374C22×C3—41no—
375C22×C3—41no—
376C22×C3—41no—
377C22×C3—41no—
378C22×C3—41no—
379C22×C3—41no—
380C22×C3—81no—
381C22×C3—81no—
382C22×C3—82no—
383C22×C3—81no—
384C22×C3—81no—
385C22×C3—82no—
386C22×C3—121no—
387C22×C3—121no—
388C22×C3—121no—
389C22×C3#93211yescharacteristic
390C22×C3#93711yescharacteristic
391C22×C3#94911yescharacteristic
392C22×C3#95011yescharacteristic
393C22×C3#95011yescharacteristic
394C22×C3#95311yescharacteristic
395C2×D3—41no—
396C2×D3—41no—
397C2×D3—41no—
398C3×C4—21no—
399C3×C4—21no—
400C3×C4—21no—
401C3×C4—21no—
402C3×C4—41no—
403C3×C4—41no—
404C3×C4—41no—
405C3×C4—41no—
406C3×C4—81no—
407C3×C4—121no—
408C3×C4—121no—
409C3×C4—121no—
410C3×C4—241no—
411C3×C4—241no—
412Dic3—41no—
413Dic3—41no—
414Dic3—41no—
415Dic3—81no—
416Dic3—81no—
417C32#161211yescharacteristic, complemented, Sylow
418C23—21no—
419C23—21no—
420C23—21no—
421C23—41no—
422C23—41no—
423C23—61no—
424C23C2×#62511yescharacteristic
425Q8—121no—
426D4—121no—
427D4—121no—
428D4—121no—
429D4—241no—
430D4—241no—
431C2×C4—21no—
432C2×C4—41no—
433C2×C4—41no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—121no—
438C2×C4—121no—
439C2×C4—121no—
440C2×C4—121no—
441C2×C4—121no—
442C2×C4—121no—
443C2×C4—241no—
444C2×C4D3×#6711yescharacteristic
445C2×C4D3×#6711yescharacteristic
446C8—361no—
447C8—361no—
448C2×C3—21no—
449C2×C3—21no—
450C2×C3—21no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—21no—
454C2×C3—21no—
455C2×C3—41no—
456C2×C3—41no—
457C2×C3—82no—
458C2×C3—81no—
459C2×C3—81no—
460C2×C3—241no—
461C2×C3#503711yescharacteristic
462C2×C3#505311yescharacteristic
463C2×C3#506111yescharacteristic
464C2×C3#523811yescharacteristic
465C2×C3#524211yescharacteristic
466C2×C3#524311yescharacteristic
467D3—81no—
468C22—21no—
469C22—21no—
470C22—42no—
471C22—41no—
472C22—41no—
473C22—41no—
474C22—81no—
475C22—121no—
476C22—121no—
477C22—121no—
478C22#6382211yescharacteristic
479C22#6382311yescharacteristic
480C22#6397611yescharacteristic
481C4—21no—
482C4—21no—
483C4—41no—
484C4—121no—
485C4—121no—
486C4—121no—
487C4—241no—
488C4—241no—
489C3—21no—
490C3#6970911yescharacteristic, complemented
491C3#7188211yescharacteristic, complemented
492C2—21no—
493C2—21no—
494C2—81no—
495C2—241no—
496C2#1059116111yescharacteristic
497C2#1059151211yescharacteristic
498C2#1059153311yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC2×C32×C22⋊C4C22
socleC22×C32#120
prime radicalC3×#5494C2
hypercenterC2×C22⋊C4D32
hypocenterC32#1612
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C3×#4953C2semidirect
3. #10591101C2semidirect
4. #10591089C2semidirect
5. #10589391C2semidirect
6. #10589153C2semidirect
7. #10589149C2semidirect
36. #4953D3semidirect
417. C32#1612semidirect
490. C3#69709semidirect
491. C3#71882semidirect

Maximal subgroups

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

subgroupindex
C3×#54942
C3×#49532
#105911012
#105910892
#105893912
#105891532
#105891492
#718823
#697093