Finite Group Databasebeta

← #11787223

Subgroups — #11787223

Order 1152.

917 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 917 conjugacy classes of subgroups — too many to draw. Only its 134 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#10596299C211yesmaximal, characteristic
2C2×#63818C211yesmaximal, characteristic, complemented
3C2×C3×#837C212yesmaximal, complemented
4C4×#3722C211yesmaximal, characteristic, complemented
5#10589162C212yesmaximal, complemented
6#70381—32nomaximal
7C2×C32×#41C2211yescharacteristic
8C22×C4×C3⋊D3C2211yescharacteristic
9#64155C411yescharacteristic, complemented
10#64155C411yescharacteristic, complemented
11#64147C411yescharacteristic, complemented
12#64147C411yescharacteristic, complemented
13#64146—21no—
14#64144—21no—
15C4×#638—21no—
16C4×#638—21no—
17#64135C2211yescharacteristic
18C2×C3×C4×Dic3C2212yes—
19C2×C3×#208C2212yes—
20C2×#3722—21no—
21C2×#3712—21no—
22#63818—24no—
23C2×#3710—21no—
24C3×#837—24no—
25#63579—21no—
26#5828—32no—
27#5828—61no—
28C2×D3×M16—32no—
29C2×C3×#104—32no—
30C2×#837D312yescomplemented
31C4×(C2×D3)⋊C4—32no—
32#4555—32no—
33#4523—32no—
34C23×C3⋊D3D411yescharacteristic
35C32×#41—22no—
36C32×#41—22no—
37C2×C32×Q8C2×C411yescharacteristic
38C2×C32×D4—21no—
39C2×C32×D4C2×C411yescharacteristic
40C22×C32×C4—21no—
41C22×C32×C4C2311yescharacteristic
42C2×#638—21no—
43C2×#638—21no—
44C22×C3⋊Dic3D411yescharacteristic
45C2×C4×C3⋊D3—22no—
46C2×C4×C3⋊D3—22no—
47C2×C4×C3⋊D3C2×C411yescharacteristic
48C2×C4×C3⋊D3C2×C411yescharacteristic
49C2×C4×C3⋊D3D411yescharacteristic
50C2×C4×C3⋊D3D411yescharacteristic
51C22×C3×Dic3—22no—
52#3757—21no—
53#3757—21no—
54#3757—21no—
55#3752—21no—
56#3752—21no—
57C4⋊(C3⋊Dic3)C2×C411yescharacteristic
58C4⋊(C3⋊Dic3)C2×C411yescharacteristic
59#3750—21no—
60#3750—21no—
61C4×C3⋊Dic3—21no—
62C3×C4×Dic3—22no—
63C3×C4×Dic3—22no—
64C3×#208—22no—
65C3×#208D414yescomplemented
66C2×C3×C3⋊C8—22no—
67#3722—24no—
68#3712—24no—
69#3710—24no—
70#1871—91noSylow
71C2×C3×#41—21no—
72C2×C3×#41C2×D312yes—
73C22×C4×D3—32no—
74C22×C4×D3—61no—
75C2×C3×M16—32no—
76C2×C3×C42—32no—
77#946—32no—
78#946—32no—
79#946—61no—
80#946—61no—
81#938—32no—
82#938—32no—
83#938—61no—
84#938—61no—
85#937—62no—
86#937—62no—
87#935—62no—
88#935—62no—
89C4×#67—62no—
90C4×#67—62no—
91C4×#67—62no—
92C4×#67—62no—
93C2×(C2×D3)⋊C4—62no—
94#926—32no—
95#926—61no—
96C2×C4×Dic3C2×D312yes—
97C2×#208C2×D312yes—
98D3×M16—64no—
99D3×M16—64no—
100C2×ΓL(1,𝔽25)—62no—
101C2×D3×C8—62no—
102C3×#104—64no—
103#837—24no—
104C3⋊#95—62no—
105C23×C32—21no—
106C23×C32C2×D411yescharacteristic
107C22×C3⋊D3—22no—
108C22×C3⋊D3—22no—
109C22×C3⋊D3—41no—
110C22×C3⋊D3C22⋊C411yescharacteristic
111C22×C3⋊D3C22⋊C411yescharacteristic
112C32×Q8—22no—
113C32×D4—22no—
114C32×D4—42no—
115C2×C32×C4—21no—
116C2×C32×C4—41no—
117C2×C32×C4—41no—
118C2×C32×C4C22×C411yescharacteristic
119C2×C32×C4C2×D411yescharacteristic
120C2×C32×C4C2×D412yes—
121C2×C32×C4#4112yes—
122#638—42no—
123#638—42no—
124C2×C3⋊Dic3—21no—
125C2×C3⋊Dic3—21no—
126C2×C3⋊Dic3—22no—
127C2×C3⋊Dic3C22⋊C411yescharacteristic
128C2×C3⋊Dic3C22⋊C411yescharacteristic
129C4×C3⋊D3—22no—
130C4×C3⋊D3—22no—
131C4×C3⋊D3—21no—
132C4×C3⋊D3—21no—
133C4×C3⋊D3—21no—
134C4×C3⋊D3—21no—
135C2×C3×Dic3—24no—
136C2×C3×Dic3—42no—
137C2×C3×Dic3—42no—
138C3×C3⋊C8—24no—
139C22×M16—91no—
140#525—91no—
141C2×#104—32no—
142C4×C22⋊C4—91no—
143#339—92no—
144C23×D3—32no—
145C23×D3—61no—
146C3×#41—22no—
147C3×#41—22no—
148C3×#41—22no—
149C3×#41—22no—
150C3×#41—42no—
151C3×#41—42no—
152C2×C3×Q8—21no—
153C2×C3×Q8C4×D312yes—
154C2×C3×D4—22no—
155C2×C3×D4—22no—
156C2×C3×D4—21no—
157C2×C3×D4C4×D312yes—
158C22×C3×C4—22no—
159C22×C3×C4—22no—
160C22×C3×C4—21no—
161C22×C3×C4—62no—
162C22×C3×C4C22×D312yes—
163C2×#67—62no—
164C2×#67—62no—
165C2×#67—62no—
166C2×#67—62no—
167C22×Dic3—22no—
168C22×Dic3—32no—
169C22×Dic3—61no—
170C2×C4×D3—32no—
171C2×C4×D3—32no—
172C2×C4×D3—32no—
173C2×C4×D3—32no—
174C2×C4×D3—61no—
175C2×C4×D3—61no—
176C2×C4×D3—61no—
177C2×C4×D3—61no—
178C2×C4×D3—62no—
179C2×C4×D3—62no—
180C2×C4×D3—62no—
181C2×C4×D3—62no—
182C2×C4×D3—122no—
183C2×C4×D3—122no—
184C3×M16—34no—
185C3×M16—62no—
186C2×C3×C8—62no—
187C3×C42—62no—
188C3×C42—62no—
189#217—62no—
190#217—62no—
191#217—62no—
192#217—62no—
193#217—62no—
194#217—62no—
195(C2×D3)⋊C4—68no—
196(C2×D3)⋊C4—62no—
197(C2×D3)⋊C4—62no—
198(C2×D3)⋊C4—62no—
199(C2×D3)⋊C4—62no—
200C4⋊Dic3—32no—
201C4⋊Dic3—32no—
202C4⋊Dic3—61no—
203C4⋊Dic3—61no—
204Dic3⋊C4—62no—
205Dic3⋊C4—62no—
206Dic3⋊C4—62no—
207Dic3⋊C4—62no—
208C4×Dic3—22no—
209C4×Dic3—22no—
210C4×Dic3—62no—
211C4×Dic3—62no—
212#208—22no—
213#208#6714yescomplemented
214C2×C3⋊C8—22no—
215ΓL(1,𝔽25)—68no—
216D3×C8—68no—
217C22×C32—21no—
218C22×C32—21no—
219C22×C32—41no—
220C22×C32—41no—
221C22×C32—41no—
222C22×C32C2×C22⋊C411yescharacteristic
223C22×C32#12112yes—
224C2×C3⋊D3—21no—
225C2×C3⋊D3—21no—
226C2×C3⋊D3—21no—
227C2×C3⋊D3—21no—
228C2×C3⋊D3—21no—
229C2×C3⋊D3—44no—
230C2×C3⋊D3#10412yescomplemented
231C32×C4—41no—
232C32×C4C4×D412yes—
233C32×C4#12011yescharacteristic
234C32×C4#12311yescharacteristic
235C3⋊Dic3—21no—
236C3⋊Dic3—21no—
237C3⋊Dic3—41no—
238C3⋊Dic3—41no—
239C3×Dic3—44no—
240C2×#41D3211yescharacteristic
241C23×C4—91no—
242C2×M16—32no—
243C2×M16—181no—
244C2×M16—184no—
245C22×C8—181no—
246#123—181no—
247C22⋊Q8—91no—
248C22⋊Q8—91no—
249#121—91no—
250#121—91no—
251#120—181no—
252C4×D4—181no—
253C4×D4—181no—
254#117—91no—
255C2×C22⋊C4—181no—
256C2×C42—32no—
257#104—64no—
258#95—181no—
259C23×C3—22no—
260C23×C3—22no—
261C23×C3—21no—
262C23×C3D3×D412yes—
263C22×D3—32no—
264C22×D3—32no—
265C22×D3—61no—
266C22×D3—61no—
267C22×D3—64no—
268C22×D3—62no—
269C22×D3—62no—
270C22×D3—122no—
271C22×D3—122no—
272C22×D3—122no—
273C22×D3—122no—
274C3×Q8—22no—
275C3×Q8—22no—
276C3×Q8—42no—
277C3×D4—22no—
278C3×D4—22no—
279C3×D4—42no—
280C3×D4—44no—
281C3×D4—42no—
282C3×D4—42no—
283C2×C3×C4—22no—
284C2×C3×C4—22no—
285C2×C3×C4—22no—
286C2×C3×C4—21no—
287C2×C3×C4—22no—
288C2×C3×C4—21no—
289C2×C3×C4—42no—
290C2×C3×C4—42no—
291C2×C3×C4—42no—
292C2×C3×C4—42no—
293C2×C3×C4—64no—
294C2×C3×C4—122no—
295C2×C3×C4—122no—
296C2×C3×C4C2×C4×D312yes—
297C2×C3×C4#23512yes—
298C2×C3×C4D3×D412yes—
299C2×C3×C4D3×D412yes—
300C2×C3×C4#23712yes—
301C2×C3×C4C2×#6712yes—
302#67—124no—
303#67—124no—
304#67—124no—
305#67—124no—
306C2×Dic3—24no—
307C2×Dic3—32no—
308C2×Dic3—32no—
309C2×Dic3—42no—
310C2×Dic3—42no—
311C2×Dic3—61no—
312C2×Dic3—61no—
313C2×Dic3—62no—
314C2×Dic3—62no—
315C2×Dic3—62no—
316C2×Dic3—62no—
317C2×Dic3—62no—
318C2×Dic3—62no—
319C2×Dic3—122no—
320C4×D3—62no—
321C4×D3—62no—
322C4×D3—62no—
323C4×D3—62no—
324C4×D3—62no—
325C4×D3—62no—
326C4×D3—62no—
327C4×D3—62no—
328C4×D3—122no—
329C4×D3—122no—
330C4×D3—121no—
331C4×D3—121no—
332C4×D3—121no—
333C4×D3—121no—
334C3×C8—64no—
335C3⋊C8—24no—
336C2×C32—22no—
337C2×C32—81no—
338C2×C32#38611yescharacteristic
339C2×C32C2×#10411yescharacteristic
340C2×C32#42111yescharacteristic
341C3⋊D3—22no—
342C3⋊D3—41no—
343C24—91no—
344#41—22no—
345#41—22no—
346C2×Q8#62411yescharacteristic
347C2×D4—21no—
348C2×D4—181no—
349C2×D4—181no—
350C2×D4#62411yescharacteristic
351C22×C4—21no—
352C22×C4—62no—
353C22×C4—91no—
354C22×C4—91no—
355C22×C4—91no—
356C22×C4—91no—
357C22×C4—91no—
358C22×C4—182no—
359C22×C4—182no—
360C22×C4C2×D3211yescharacteristic
361M16—34no—
362M16—62no—
363M16—184no—
364C2×C8—62no—
365C2×C8—184no—
366C4⋊C4—91no—
367C4⋊C4—91no—
368C4⋊C4—181no—
369C4⋊C4—181no—
370C22⋊C4—181no—
371C22⋊C4—181no—
372C22⋊C4—184no—
373C22⋊C4—181no—
374C22⋊C4—181no—
375C22⋊C4—181no—
376C42—62no—
377C42—62no—
378C42—181no—
379C22×C3—22no—
380C22×C3—22no—
381C22×C3—22no—
382C22×C3—21no—
383C22×C3—41no—
384C22×C3—41no—
385C22×C3—42no—
386C22×C3—42no—
387C22×C3—42no—
388C22×C3—42no—
389C22×C3—42no—
390C22×C3—42no—
391C22×C3D3×C22⋊C412yes—
392C22×C3#88412yes—
393C22×C3#93812yes—
394C2×D3—34no—
395C2×D3—62no—
396C2×D3—62no—
397C2×D3—62no—
398C2×D3—62no—
399C2×D3—62no—
400C2×D3—62no—
401C2×D3—62no—
402C2×D3—121no—
403C2×D3—121no—
404C2×D3—121no—
405C2×D3—121no—
406C2×D3—124no—
407C2×D3—124no—
408C2×D3—128no—
409C3×C4—22no—
410C3×C4—21no—
411C3×C4—21no—
412C3×C4—42no—
413C3×C4—42no—
414C3×C4—124no—
415C3×C4#67⋊C412yes—
416C3×C4#88312yes—
417C3×C4C4×#6712yes—
418C3×C4#93712yes—
419Dic3—44no—
420Dic3—62no—
421Dic3—62no—
422Dic3—121no—
423Dic3—121no—
424Dic3—122no—
425Dic3—122no—
426Dic3—122no—
427Dic3—122no—
428C32#187111yescharacteristic, complemented, Sylow
429C23—21no—
430C23—91no—
431C23—91no—
432C23—182no—
433C23—182no—
434C23—361no—
435C23#381111yescharacteristic
436Q8—22no—
437D4—22no—
438D4—42no—
439D4—362no—
440D4—362no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—64no—
445C2×C4—91no—
446C2×C4—91no—
447C2×C4—122no—
448C2×C4—122no—
449C2×C4—182no—
450C2×C4—182no—
451C2×C4—182no—
452C2×C4—181no—
453C2×C4—181no—
454C2×C4—181no—
455C2×C4—181no—
456C2×C4—181no—
457C2×C4—181no—
458C2×C4#380412yes—
459C2×C4C2×#62411yescharacteristic
460C2×C4D3×#6712yes—
461C2×C4#381111yescharacteristic
462C8—64no—
463C2×C3—22no—
464C2×C3—22no—
465C2×C3—21no—
466C2×C3—21no—
467C2×C3—21no—
468C2×C3—42no—
469C2×C3—82no—
470C2×C3—82no—
471C2×C3D3×#10412yes—
472C2×C3#482112yes—
473C2×C3#495612yes—
474D3—64no—
475D3—122no—
476D3—122no—
477D3—122no—
478C22—21no—
479C22—21no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—92no—
484C22—181no—
485C22—181no—
486C22—181no—
487C22—181no—
488C22—181no—
489C22—364no—
490C22#6395812yes—
491C22#6397411yescharacteristic
492C4—41no—
493C4—124no—
494C4—181no—
495C4—181no—
496C4—361no—
497C4—361no—
498C4#6395212yes—
499C4#6397511yescharacteristic
500C4#6398311yescharacteristic
501C3—21no—
502C3#7038112yescomplemented
503C2—22no—
504C2—81no—
505C2—182no—
506C2—361no—
507C2#1059167111yescharacteristic
508C2#1059167211yescharacteristic
509C2#1059172211yescharacteristic
subgroupG/H
centerC2×C4#3811
commutatorC2×C32×C4C22×C4
FrattiniC22×C4C2×D32
FittingC2×C32×#41C22
socleC22×C32C2×C22⋊C4
prime radical#10596299C2
cosocleC22×C32×C4C23
hypercenterC2×#41D32
hypocenterC32#1871
LCSG ▷ C2×C32×C4 ▷ C2×C32 ▷ C32
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. C2×#63818C2semidirect
3. C2×C3×#837C2semidirect
4. C4×#3722C2semidirect
5. #10589162C2semidirect
9. #64155C4semidirect
10. #64155C4semidirect
11. #64147C4semidirect
12. #64147C4semidirect
30. C2×#837D3semidirect
65. C3×#208D4semidirect
213. #208#67semidirect
230. C2×C3⋊D3#104semidirect
428. C32#1871semidirect
502. C3#70381semidirect

Maximal subgroups

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

subgroupindex
#105962992
C2×#638182
C2×C3×#8372
C4×#37222
#105891622
#703813