Finite Group Databasebeta

← #11778631

Subgroups — #11778631

Order 1152.

589 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 589 conjugacy classes of subgroups — too many to draw. Only its 80 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
1#10596088C211yesmaximal, characteristic
2C3×#5033C211yesmaximal, characteristic, complemented
3#10591326C211yesmaximal, characteristic, complemented
4#10591124C211yesmaximal, characteristic, complemented
5C3×#4537C211yesmaximal, characteristic, complemented
6#10589154C211yesmaximal, characteristic, complemented
7#10589146C211yesmaximal, characteristic, complemented
8#71165—31nomaximal
9#68289—31nomaximal
10C32×#121C2211yescharacteristic
11#64148—21no—
12#64146—21no—
13D4×C3⋊Dic3—21no—
14C2×#3752C2211yescharacteristic
15#64133C2211yescharacteristic
16#64107—21no—
17#64095—21no—
18#64094—21no—
19C2×C3×C4⋊Dic3C2211yescharacteristic
20C2×#3722—21no—
21C3×#832—21no—
22C3×C3⋊#95C2211yescharacteristic
23C3×#830C2211yescharacteristic
24C3×#830C2211yescharacteristic
25C3×#807—21no—
26#63579—21no—
27#63573—21no—
28#63567—21no—
29#5617—31no—
30#5617—31no—
31#5617—61no—
32C3×#480—31no—
33#5033D311yescharacteristic, complemented
34#4992—31no—
35#4971—31no—
36#4721—31no—
37#4674—31no—
38C3×#327—31no—
39#4537D311yescharacteristic, complemented
40#4468—31no—
41#4468—31no—
42#4467—31no—
43#4452—31no—
44C23×C3⋊D3D411yescharacteristic
45C2×C32×D4—21no—
46C2×C32×D4D411yescharacteristic
47C22×C32×C4C2311yescharacteristic
48C2×#638—21no—
49C2×#638—21no—
50C22×C3⋊Dic3—21no—
51C22×C3⋊Dic3D411yescharacteristic
52C2×#634D411yescharacteristic
53C22×C3×Dic3—21no—
54C22×C3×Dic3—21no—
55C32×C4⋊C4D411yescharacteristic
56C32×C22⋊C4—21no—
57#3757—21no—
58#3757—21no—
59#3752—21no—
60#3752—21no—
61#3752—21no—
62#3752—21no—
63C4⋊(C3⋊Dic3)—21no—
64C4⋊(C3⋊Dic3)D411yescharacteristic
65#3750—21no—
66C4×C3⋊Dic3—21no—
67C3×C4⋊Dic3—21no—
68C3×C4⋊Dic3—21no—
69C2×C3×C3⋊C8—21no—
70C2×C3×C3⋊C8—21no—
71#3722—42no—
72#1612—91noSylow
73C3×#121—21no—
74C3×#121C2×D311yescharacteristic
75C3×#121C2×D311yescharacteristic
76C2×C3×C4⋊C4—31no—
77#939—62no—
78#939—61no—
79#939—61no—
80#937—61no—
81#937—61no—
82#937—62no—
83D4×Dic3—61no—
84D4×Dic3—61no—
85D4×Dic3—62no—
86C2×(C2×D3)⋊C4—31no—
87C2×(C2×D3)⋊C4—31no—
88C2×(C2×D3)⋊C4—61no—
89C2×(C2×D3)⋊C4—61no—
90C2×(C2×D3)⋊C4—61no—
91C2×C4⋊Dic3C2×D311yescharacteristic
92#924—31no—
93#924—31no—
94#924—61no—
95#898—62no—
96#898—61no—
97#898—61no—
98#886—61no—
99#886—61no—
100#886—62no—
101#885—61no—
102#885—61no—
103#885—62no—
104C3×#107—61no—
105C3×#102—61no—
106C3×C22⋊C8—31no—
107C3×C22⋊C8—31no—
108C3×#95—31no—
109#832—21no—
110C3⋊#95—61no—
111C3⋊#95—61no—
112C3⋊#95C2×D311yescharacteristic
113#830C2×D311yescharacteristic
114#830C2×D311yescharacteristic
115#817—61no—
116#816—61no—
117#810—61no—
118#808—61no—
119#807—21no—
120C23×C32—21no—
121C23×C32C2×D411yescharacteristic
122C22×C3⋊D3—21no—
123C22×C3⋊D3—21no—
124C22×C3⋊D3—21no—
125C22×C3⋊D3—41no—
126C22×C3⋊D3—41no—
127C32×D4—41no—
128C32×D4—41no—
129C32×D4—41no—
130C2×C32×C4—21no—
131C2×C32×C4—41no—
132C2×C32×C4C2×D411yescharacteristic
133C2×C32×C4C2×D411yescharacteristic
134#638—81no—
135#638—81no—
136C2×C3⋊Dic3—21no—
137C2×C3⋊Dic3—21no—
138C2×C3⋊Dic3—21no—
139C2×C3⋊Dic3—41no—
140C2×C3⋊Dic3—81no—
141#634—41no—
142C2×C3×Dic3—41no—
143C2×C3×Dic3—41no—
144C2×C3×Dic3—41no—
145C2×C3×Dic3—41no—
146C2×C3×Dic3—41no—
147C2×C3×Dic3—41no—
148C3×C3⋊C8—41no—
149C3×C3⋊C8—41no—
150#538—91no—
151#483—91no—
152#480—31no—
153#397—91no—
154#328—91no—
155#327—31no—
156#323—91no—
157C23×D3—31no—
158C23×D3—31no—
159C23×D3—61no—
160C2×C3×D4—22no—
161C2×C3×D4—21no—
162C2×C3×D4—21no—
163C2×C3×D4—21no—
164C2×C3×D4D1211yescharacteristic
165C2×C3×D4#6711yescharacteristic
166C22×C3×C4—21no—
167C22×C3×C4—61no—
168C22×C3×C4—61no—
169C22×C3×C4C22×D311yescharacteristic
170C22×C3×C4C22×D311yescharacteristic
171C2×#67—62no—
172C2×#67—61no—
173C2×#67—61no—
174C2×#67—62no—
175C2×#67—61no—
176C2×#67—61no—
177C22×Dic3—21no—
178C22×Dic3—21no—
179C22×Dic3—31no—
180C22×Dic3—31no—
181C22×Dic3—61no—
182C22×Dic3—62no—
183C22×Dic3—61no—
184C22×Dic3—61no—
185C2×Dic6—31no—
186C2×Dic6—31no—
187C2×Dic6—61no—
188C2×C3×C8—61no—
189C2×C3×C8—61no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—61no—
192C3×C4⋊C4—61no—
193C3×C4⋊C4D1211yescharacteristic
194C3×C4⋊C4#6711yescharacteristic
195C3×C22⋊C4—22no—
196C3×C22⋊C4—21no—
197C3×C22⋊C4—21no—
198#217—62no—
199#217—61no—
200#217—61no—
201#217—62no—
202#217—61no—
203#217—61no—
204(C2×D3)⋊C4—62no—
205(C2×D3)⋊C4—62no—
206(C2×D3)⋊C4—61no—
207(C2×D3)⋊C4—61no—
208(C2×D3)⋊C4—62no—
209(C2×D3)⋊C4—61no—
210(C2×D3)⋊C4—61no—
211(C2×D3)⋊C4—62no—
212(C2×D3)⋊C4—61no—
213(C2×D3)⋊C4—61no—
214(C2×D3)⋊C4—61no—
215(C2×D3)⋊C4—61no—
216(C2×D3)⋊C4—122no—
217(C2×D3)⋊C4—122no—
218C4⋊Dic3—21no—
219C4⋊Dic3—21no—
220C4⋊Dic3—31no—
221C4⋊Dic3—31no—
222C4⋊Dic3—61no—
223C4⋊Dic3—62no—
224C4⋊Dic3—61no—
225C4⋊Dic3—61no—
226Dic3⋊C4—62no—
227Dic3⋊C4—61no—
228Dic3⋊C4—61no—
229C4×Dic3—62no—
230C4×Dic3—61no—
231C4×Dic3—61no—
232C2×C3⋊C8—21no—
233C2×C3⋊C8—21no—
234C22×C32—21no—
235C22×C32—21no—
236C22×C32—41no—
237C22×C32—41no—
238C22×C32—41no—
239C22×C32#12011yescharacteristic
240C22×C32#13511yescharacteristic
241C22×C32AGL(1,ℤ/8)11yescharacteristic
242C2×C3⋊D3—41no—
243C2×C3⋊D3—41no—
244C2×C3⋊D3—41no—
245C2×C3⋊D3—42no—
246C2×C3⋊D3—81no—
247C32×C4—21no—
248C32×C4—21no—
249C32×C4—41no—
250C3⋊Dic3—41no—
251C3⋊Dic3—41no—
252C3⋊Dic3—41no—
253C3×Dic3—81no—
254C3×Dic3—81no—
255#126—181no—
256#124—181no—
257#123—181no—
258C22⋊Q8—91no—
259#121—181no—
260#121D3211yescharacteristic
261#120—181no—
262C4×D4—181no—
263C2×C4⋊C4—31no—
264C2×C22⋊C4—91no—
265C2×C22⋊C4—181no—
266#107—61no—
267#106—181no—
268#103—181no—
269#102—61no—
270C22⋊C8—31no—
271C22⋊C8—31no—
272#95—31no—
273#95—181no—
274C23×C3—22no—
275C23×C3—21no—
276C23×C3—21no—
277C23×C3—21no—
278C23×C3D3×D411yescharacteristic
279C23×C3D3×D411yescharacteristic
280C22×D3—62no—
281C22×D3—61no—
282C22×D3—61no—
283C22×D3—61no—
284C22×D3—61no—
285C22×D3—61no—
286C22×D3—61no—
287C22×D3—121no—
288C22×D3—121no—
289C22×D3—121no—
290C22×D3—121no—
291C22×D3—121no—
292C22×D3—121no—
293C22×D3—241no—
294C22×D3—241no—
295C3×D4—41no—
296C3×D4—41no—
297C3×D4—42no—
298C3×D4—42no—
299C3×D4—41no—
300C3×D4—41no—
301C3×D4—41no—
302C3×D4—41no—
303C3×D4—81no—
304C2×C3×C4—22no—
305C2×C3×C4—21no—
306C2×C3×C4—21no—
307C2×C3×C4—21no—
308C2×C3×C4—21no—
309C2×C3×C4—41no—
310C2×C3×C4—41no—
311C2×C3×C4—81no—
312C2×C3×C4—121no—
313C2×C3×C4—121no—
314C2×C3×C4—121no—
315C2×C3×C4—121no—
316C2×C3×C4—121no—
317C2×C3×C4—121no—
318C2×C3×C4C2×D1211yescharacteristic
319C2×C3×C4D3×D411yescharacteristic
320C2×C3×C4D3×D411yescharacteristic
321C2×C3×C4C2×#6711yescharacteristic
322#67—242no—
323#67—242no—
324#67—241no—
325#67—241no—
326#67—241no—
327#67—241no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—41no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—62no—
335C2×Dic3—62no—
336C2×Dic3—62no—
337C2×Dic3—61no—
338C2×Dic3—61no—
339C2×Dic3—61no—
340C2×Dic3—61no—
341C2×Dic3—61no—
342C2×Dic3—61no—
343C2×Dic3—121no—
344C2×Dic3—121no—
345C2×Dic3—241no—
346C2×Dic3—242no—
347C2×Dic3—241no—
348C2×Dic3—241no—
349Dic6—121no—
350Dic6—121no—
351Dic6—241no—
352C3×C8—121no—
353C3×C8—121no—
354C3⋊C8—41no—
355C3⋊C8—41no—
356C2×C32—21no—
357C2×C32—21no—
358C2×C32—81no—
359C2×C32#44911yescharacteristic
360C2×C32#45511yescharacteristic
361C2×C32#45811yescharacteristic
362C3⋊D3—81no—
363C24—91no—
364C2×Q8—91no—
365C2×D4—21no—
366C2×D4—181no—
367C2×D4—181no—
368C2×D4#62611yescharacteristic
369C22×C4—61no—
370C22×C4—61no—
371C22×C4—91no—
372C22×C4—181no—
373C22×C4C2×D3211yescharacteristic
374C2×C8—61no—
375C2×C8—61no—
376C4⋊C4—61no—
377C4⋊C4—61no—
378C4⋊C4—91no—
379C4⋊C4—181no—
380C4⋊C4—181no—
381C4⋊C4#62611yescharacteristic
382C22⋊C4—21no—
383C22⋊C4—181no—
384C22⋊C4—181no—
385C22⋊C4—181no—
386C22⋊C4—181no—
387C22⋊C4—181no—
388C22⋊C4—181no—
389C22⋊C4—362no—
390C42—181no—
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—21no—
396C22×C3—21no—
397C22×C3—21no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—42no—
401C22×C3—42no—
402C22×C3—42no—
403C22×C3—41no—
404C22×C3—41no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3—41no—
408C22×C3—41no—
409C22×C3#88211yescharacteristic
410C22×C3#91111yescharacteristic
411C22×C3#91111yescharacteristic
412C22×C3#91211yescharacteristic
413C22×C3#91611yescharacteristic
414C22×C3#93711yescharacteristic
415C2×D3—122no—
416C2×D3—122no—
417C2×D3—122no—
418C2×D3—122no—
419C2×D3—122no—
420C2×D3—121no—
421C2×D3—121no—
422C2×D3—121no—
423C2×D3—121no—
424C2×D3—121no—
425C2×D3—121no—
426C2×D3—242no—
427C2×D3—242no—
428C2×D3—241no—
429C2×D3—241no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—21no—
433C3×C4—21no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—42no—
437C3×C4—41no—
438C3×C4—41no—
439C3×C4—241no—
440C3×C4—241no—
441Dic3—81no—
442Dic3—81no—
443Dic3—122no—
444Dic3—122no—
445Dic3—122no—
446Dic3—121no—
447Dic3—121no—
448Dic3—121no—
449Dic3—121no—
450Dic3—121no—
451Dic3—121no—
452C32#161211yescharacteristic, complemented, Sylow
453C23—21no—
454C23—181no—
455C23—181no—
456C23—181no—
457C23—361no—
458C23—361no—
459C23#381111yescharacteristic
460Q8—361no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—721no—
465D4—721no—
466C2×C4—21no—
467C2×C4—41no—
468C2×C4—121no—
469C2×C4—121no—
470C2×C4—121no—
471C2×C4—121no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—181no—
475C2×C4—181no—
476C2×C4—181no—
477C2×C4—361no—
478C2×C4—721no—
479C2×C4C2×#62611yescharacteristic
480C2×C4#381111yescharacteristic
481C8—121no—
482C8—121no—
483C2×C3—21no—
484C2×C3—21no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—41no—
491C2×C3—41no—
492C2×C3—82no—
493C2×C3—81no—
494C2×C3—81no—
495C2×C3#474211yescharacteristic
496C2×C3#478111yescharacteristic
497C2×C3#482511yescharacteristic
498C2×C3#505311yescharacteristic
499C2×C3#515811yescharacteristic
500C2×C3#520211yescharacteristic
501D3—242no—
502D3—241no—
503D3—241no—
504C22—21no—
505C22—21no—
506C22—41no—
507C22—41no—
508C22—41no—
509C22—362no—
510C22—361no—
511C22—361no—
512C22—361no—
513C22—721no—
514C22#6392711yescharacteristic
515C22#6393611yescharacteristic
516C22#6398111yescharacteristic
517C4—21no—
518C4—21no—
519C4—41no—
520C4—241no—
521C4—241no—
522C4—361no—
523C4—361no—
524C4—361no—
525C3—21no—
526C3#6828911yescharacteristic, complemented
527C3#7116511yescharacteristic, complemented
528C2—21no—
529C2—21no—
530C2—81no—
531C2—721no—
532C2#1059145711yescharacteristic
533C2#1059153511yescharacteristic
534C2#1059169011yescharacteristic
subgroupG/H
centerC22#63981
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×#121C22
socleC22×C32#120
prime radical#10596088C2
hypercenter#121D32
hypocenterC32#1612
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
2. C3×#5033C2semidirect
3. #10591326C2semidirect
4. #10591124C2semidirect
5. C3×#4537C2semidirect
6. #10589154C2semidirect
7. #10589146C2semidirect
33. #5033D3semidirect
39. #4537D3semidirect
452. C32#1612semidirect
526. C3#68289semidirect
527. C3#71165semidirect

Maximal subgroups

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

subgroupindex
#105960882
C3×#50332
#105913262
#105911242
C3×#45372
#105891542
#105891462
#711653
#682893