Finite Group Databasebeta

← #11778334

Subgroups — #11778334

Order 1152.

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

normal subgroups only

This group has 609 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×#5597C211yesmaximal, characteristic
2C3×#5040C211yesmaximal, characteristic, complemented
3#10591344C211yesmaximal, characteristic, complemented
4#10591171C211yesmaximal, characteristic, complemented
5#10589403C211yesmaximal, characteristic, complemented
6#10589150C211yesmaximal, characteristic, complemented
7#10589147C211yesmaximal, characteristic, complemented
8#71880—31nomaximal
9#70975—31nomaximal
10C2×C3×#237—21no—
11C22×C3×Dic6C2211yescharacteristic
12C32×#121C2211yescharacteristic
13C3×#939—21no—
14C3×#939—21no—
15C2×C3×#214—21no—
16C3×#930C2211yescharacteristic
17C2×C3×Dic3⋊C4C2211yescharacteristic
18C3×#896—21no—
19C3×#884—21no—
20C3×#878—21no—
21C2×#3724—21no—
22C2×#3715—21no—
23#63658C2211yescharacteristic
24#63657C2211yescharacteristic
25C3×#830C2211yescharacteristic
26C3×#810—21no—
27#63579—21no—
28#63565—21no—
29C3×#537—31no—
30#5597D311yescharacteristic
31C3×#448—31no—
32#5230—31no—
33#5227—31no—
34#5040—31no—
35#5040D311yescharacteristic, complemented
36#4979—31no—
37#4537—31no—
38#4537—31no—
39#4537—31no—
40#4537—61no—
41#4528—31no—
42#4470—31no—
43#4469—31no—
44C2×C32×D4—21no—
45C2×C32×D4D411yescharacteristic
46C22×C32×C4C2311yescharacteristic
47C22×C3⋊Dic3—21no—
48C2×C3×#67—21no—
49C2×C3×#67—21no—
50C2×C3×#67—21no—
51C22×C3×Dic3—21no—
52C22×C3×Dic3—21no—
53C22×C3×Dic3D411yescharacteristic
54C22×C3×Dic3D411yescharacteristic
55C3×#237—42no—
56C3×#237—42no—
57C2×C3×D12D411yescharacteristic
58C2×C3×C4×D3—21no—
59C2×C3×Dic6—21no—
60C2×C3×Dic6—21no—
61C2×C3×Dic6—21no—
62C2×C3×Dic6—42no—
63C32×C4⋊C4D411yescharacteristic
64C32×C22⋊C4—21no—
65C2×C3⋊(C3⋊C8)—21no—
66C3×#217—21no—
67C3×#214—42no—
68C3×(C2×D3)⋊C4—21no—
69C3×(C2×D3)⋊C4—21no—
70C3×C4⋊Dic3D411yescharacteristic
71C3×Dic3⋊C4—21no—
72C3×Dic3⋊C4—21no—
73C3×Dic3⋊C4—21no—
74C2×C3×C3⋊C8—21no—
75#3724—42no—
76#3715—42no—
77#1604—91noSylow
78C2×C3×#41—61no—
79C22×C3×Q8—31no—
80C2×#237—21no—
81C22×Dic6C2×D311yescharacteristic
82C2×C3×SD16—61no—
83C3×C22⋊Q8—61no—
84C3×C22⋊Q8—61no—
85C3×#121—21no—
86C3×#121—31no—
87C3×#121—61no—
88C3×#121—61no—
89C3×#121—61no—
90C3×#121C2×D311yescharacteristic
91C3×#121C2×D311yescharacteristic
92C2×C3×C4⋊C4—31no—
93C2×#215—61no—
94#939—21no—
95#939—21no—
96C2×#214—21no—
97C2×#214—61no—
98#930C2×D311yescharacteristic
99C2×Dic3⋊C4—61no—
100C2×Dic3⋊C4—61no—
101C2×Dic3⋊C4C2×D311yescharacteristic
102#896—21no—
103#884—21no—
104#878—21no—
105C3×#103—61no—
106C3×C22⋊C8—31no—
107#835—61no—
108C3⋊#95—31no—
109C3⋊#95—31no—
110C3⋊#95—61no—
111C3⋊#95—61no—
112C3⋊#95—61no—
113#830—31no—
114#830—31no—
115#830—61no—
116#830C2×D311yescharacteristic
117#810—21no—
118#810—61no—
119C23×C32—21no—
120C23×C32C2×D411yescharacteristic
121C22×C3×D3—21no—
122C32×D4—41no—
123C32×D4—41no—
124C32×D4—41no—
125C2×C32×C4—21no—
126C2×C32×C4—41no—
127C2×C32×C4C2×D411yescharacteristic
128C2×C32×C4C2×D411yescharacteristic
129C2×C3⋊Dic3—22no—
130C2×C3⋊Dic3—41no—
131C2×C3⋊Dic3—41no—
132C3×#67—42no—
133C3×#67—41no—
134C3×#67—41no—
135C3×#67—81no—
136C2×C3×Dic3—21no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—22no—
139C2×C3×Dic3—21no—
140C2×C3×Dic3—21no—
141C2×C3×Dic3—42no—
142C2×C3×Dic3—41no—
143C2×C3×Dic3—41no—
144C2×C3×Dic3—41no—
145C2×C3×Dic3SD1612yescomplemented
146C3×D12—41no—
147C3×C4×D3—42no—
148C3×Dic6—22no—
149C3×Dic6—22no—
150C3×Dic6—42no—
151C3×Dic6—41no—
152C3×Dic6—41no—
153C3⋊(C3⋊C8)—41no—
154C3×C3⋊C8—41no—
155#537—31no—
156#448—31no—
157#448—91no—
158#395—91no—
159#327—91no—
160#327—91no—
161#324—91no—
162C3×#41—122no—
163C3×#41—122no—
164C2×C3×Q8—61no—
165C2×C3×Q8—61no—
166C2×C3×Q8—61no—
167C2×C3×Q8—122no—
168C2×C3×D4—21no—
169C2×C3×D4—21no—
170C2×C3×D4—21no—
171C2×C3×D4—31no—
172C2×C3×D4—41no—
173C2×C3×D4—61no—
174C2×C3×D4—61no—
175C2×C3×D4—61no—
176C2×C3×D4#6711yescharacteristic
177C2×C3×D4#6711yescharacteristic
178C22×C3×C4—21no—
179C22×C3×C4—31no—
180C22×C3×C4—31no—
181C22×C3×C4—61no—
182C22×C3×C4—61no—
183C22×C3×C4—61no—
184C22×C3×C4C22×D311yescharacteristic
185C22×C3×C4C22×D311yescharacteristic
186C2×#67—21no—
187C2×#67—21no—
188C2×#67—21no—
189C22×Dic3—21no—
190C22×Dic3—21no—
191C22×Dic3—61no—
192C22×Dic3—61no—
193C22×Dic3—121no—
194C22×Dic3#6711yescharacteristic
195C22×Dic3#6711yescharacteristic
196#237—42no—
197#237—42no—
198C2×D12#6711yescharacteristic
199C2×C4×D3—21no—
200C2×Dic6—21no—
201C2×Dic6—21no—
202C2×Dic6—21no—
203C2×Dic6—42no—
204C3×SD16—122no—
205C2×C3×C8—61no—
206C3×C4⋊C4—21no—
207C3×C4⋊C4—31no—
208C3×C4⋊C4—61no—
209C3×C4⋊C4—61no—
210C3×C4⋊C4—61no—
211C3×C4⋊C4#6711yescharacteristic
212C3×C4⋊C4#6711yescharacteristic
213C3×C22⋊C4—21no—
214C3×C22⋊C4—21no—
215C3×C22⋊C4—41no—
216C3×C22⋊C4—61no—
217C3×C22⋊C4—61no—
218C3×C22⋊C4—61no—
219#217—21no—
220#215—122no—
221#214—42no—
222#214—122no—
223(C2×D3)⋊C4—21no—
224(C2×D3)⋊C4—21no—
225C4⋊Dic3#6711yescharacteristic
226Dic3⋊C4—21no—
227Dic3⋊C4—21no—
228Dic3⋊C4—21no—
229Dic3⋊C4—122no—
230Dic3⋊C4—122no—
231C2×C3⋊C8—21no—
232C2×C3⋊C8—61no—
233C2×C3⋊C8—61no—
234C2×C3⋊C8—121no—
235C22×C32—21no—
236C22×C32—21no—
237C22×C32—41no—
238C22×C32—41no—
239C22×C32—41no—
240C22×C32#12011yescharacteristic
241C22×C32C2×SD1611yescharacteristic
242C22×C32AGL(1,ℤ/8)11yescharacteristic
243C2×C3×D3—41no—
244C2×C3×D3—41no—
245C2×C3×D3—41no—
246C32×C4—21no—
247C32×C4—21no—
248C32×C4—41no—
249C3⋊Dic3—42no—
250C3×Dic3—22no—
251C3×Dic3—41no—
252C3×Dic3—42no—
253C3×Dic3—41no—
254C2×#41—61no—
255C22×Q8—31no—
256C2×SD16—61no—
257C2×SD16—181no—
258C22⋊Q8—61no—
259C22⋊Q8—61no—
260#121—31no—
261#121—61no—
262#121—61no—
263#121—61no—
264#121D3211yescharacteristic
265C2×C4⋊C4—31no—
266C2×C4⋊C4—181no—
267#103—61no—
268#103—181no—
269C22⋊C8—31no—
270C22⋊C8—91no—
271#95—91no—
272#95—181no—
273C23×C3—21no—
274C23×C3—21no—
275C23×C3—21no—
276C23×C3—41no—
277C23×C3—61no—
278C23×C3D3×D411yescharacteristic
279C23×C3C2×#6711yescharacteristic
280C22×D3—21no—
281C3×Q8—62no—
282C3×Q8—62no—
283C3×Q8—121no—
284C3×Q8—121no—
285C3×Q8—122no—
286C3×D4—41no—
287C3×D4—41no—
288C3×D4—41no—
289C3×D4—41no—
290C3×D4—41no—
291C3×D4—41no—
292C3×D4—81no—
293C3×D4—81no—
294C3×D4—81no—
295C3×D4—121no—
296C3×D4—122no—
297C3×D4—121no—
298C3×D4—121no—
299C3×D4—241no—
300C2×C3×C4—21no—
301C2×C3×C4—21no—
302C2×C3×C4—21no—
303C2×C3×C4—21no—
304C2×C3×C4—32no—
305C2×C3×C4—41no—
306C2×C3×C4—41no—
307C2×C3×C4—41no—
308C2×C3×C4—61no—
309C2×C3×C4—61no—
310C2×C3×C4—62no—
311C2×C3×C4—61no—
312C2×C3×C4—61no—
313C2×C3×C4—81no—
314C2×C3×C4—122no—
315C2×C3×C4—122no—
316C2×C3×C4—121no—
317C2×C3×C4—121no—
318C2×C3×C4—121no—
319C2×C3×C4D3×D411yescharacteristic
320C2×C3×C4C2×#6711yescharacteristic
321C2×C3×C4C2×#6711yescharacteristic
322C2×C3×C4C2×#6711yescharacteristic
323#67—42no—
324#67—41no—
325#67—41no—
326#67—81no—
327C2×Dic3—21no—
328C2×Dic3—21no—
329C2×Dic3—22no—
330C2×Dic3—21no—
331C2×Dic3—21no—
332C2×Dic3—42no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—41no—
336C2×Dic3—62no—
337C2×Dic3—62no—
338C2×Dic3—122no—
339C2×Dic3—121no—
340C2×Dic3—121no—
341C2×Dic3—121no—
342C2×Dic3—121no—
343C2×Dic3—241no—
344C2×Dic3—241no—
345C2×Dic3#21412yescomplemented
346D12—41no—
347C4×D3—42no—
348Dic6—22no—
349Dic6—22no—
350Dic6—42no—
351Dic6—41no—
352Dic6—41no—
353C3×C8—121no—
354C3⋊C8—41no—
355C3⋊C8—121no—
356C3⋊C8—121no—
357C3⋊C8—241no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32—81no—
361C2×C32#45011yescharacteristic
362C2×C32#45611yescharacteristic
363C2×C32B(4,𝔽2)11yescharacteristic
364C3×D3—81no—
365#41—122no—
366#41—122no—
367C2×Q8—61no—
368C2×Q8—61no—
369C2×Q8—61no—
370C2×Q8—122no—
371C2×D4—21no—
372C2×D4—31no—
373C2×D4—61no—
374C2×D4—61no—
375C2×D4—61no—
376C2×D4#62511yescharacteristic
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—61no—
380C22×C4—61no—
381C22×C4—61no—
382C22×C4—181no—
383C22×C4C2×D3211yescharacteristic
384SD16—122no—
385SD16—362no—
386C2×C8—61no—
387C2×C8—181no—
388C4⋊C4—31no—
389C4⋊C4—61no—
390C4⋊C4—61no—
391C4⋊C4—61no—
392C4⋊C4—362no—
393C4⋊C4#62511yescharacteristic
394C22⋊C4—21no—
395C22⋊C4—61no—
396C22⋊C4—61no—
397C22⋊C4—61no—
398C22×C3—21no—
399C22×C3—21no—
400C22×C3—21no—
401C22×C3—21no—
402C22×C3—21no—
403C22×C3—21no—
404C22×C3—21no—
405C22×C3—41no—
406C22×C3—41no—
407C22×C3—41no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3—41no—
412C22×C3—41no—
413C22×C3—81no—
414C22×C3—81no—
415C22×C3—81no—
416C22×C3—121no—
417C22×C3—121no—
418C22×C3—121no—
419C22×C3D3×SD1611yescharacteristic
420C22×C3#91411yescharacteristic
421C22×C3#93211yescharacteristic
422C22×C3C2×#21411yescharacteristic
423C22×C3#93711yescharacteristic
424C22×C3#95311yescharacteristic
425C2×D3—41no—
426C2×D3—41no—
427C2×D3—41no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—41no—
433C3×C4—41no—
434C3×C4—41no—
435C3×C4—41no—
436C3×C4—62no—
437C3×C4—81no—
438C3×C4—121no—
439C3×C4—122no—
440C3×C4—121no—
441Dic3—22no—
442Dic3—41no—
443Dic3—41no—
444Dic3—42no—
445Dic3—122no—
446Dic3—122no—
447Dic3—242no—
448C32#160411yescharacteristic, complemented, Sylow
449C23—21no—
450C23—61no—
451C23D3×#6711yescharacteristic
452Q8—62no—
453Q8—62no—
454Q8—121no—
455Q8—121no—
456Q8—122no—
457D4—41no—
458D4—41no—
459D4—41no—
460D4—121no—
461D4—122no—
462D4—121no—
463D4—121no—
464D4—241no—
465C2×C4—21no—
466C2×C4—32no—
467C2×C4—41no—
468C2×C4—61no—
469C2×C4—61no—
470C2×C4—62no—
471C2×C4—61no—
472C2×C4—61no—
473C2×C4—122no—
474C2×C4—122no—
475C2×C4—121no—
476C2×C4—121no—
477C2×C4—121no—
478C2×C4—182no—
479C2×C4—361no—
480C2×C4—361no—
481C2×C4C2×#62511yescharacteristic
482C2×C4D3×#6711yescharacteristic
483C8—121no—
484C8—361no—
485C2×C3—21no—
486C2×C3—21no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—21no—
491C2×C3—21no—
492C2×C3—41no—
493C2×C3—41no—
494C2×C3—81no—
495C2×C3—81no—
496C2×C3—161no—
497C2×C3—241no—
498C2×C3#503211yescharacteristic
499C2×C3#516911yescharacteristic
500C2×C3#520511yescharacteristic
501C2×C3#521811yescharacteristic
502C2×C3#524411yescharacteristic
503C2×C3#524611yescharacteristic
504D3—81no—
505C22—21no—
506C22—21no—
507C22—41no—
508C22—41no—
509C22—41no—
510C22—121no—
511C22—121no—
512C22—121no—
513C22D3×#21411yescharacteristic
514C22#6393411yescharacteristic
515C22#6397611yescharacteristic
516C4—21no—
517C4—21no—
518C4—41no—
519C4—62no—
520C4—122no—
521C4—121no—
522C4—121no—
523C4—362no—
524C3—21no—
525C3#7097511yescharacteristic, complemented
526C3#7188011yescharacteristic, complemented
527C2—21no—
528C2—21no—
529C2—81no—
530C2—241no—
531C2#1059145811yescharacteristic
532C2#1059153011yescharacteristic
533C2#1059168711yescharacteristic
subgroupG/H
centerC22#63976
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×#121C22
socleC22×C32#120
prime radicalC3×#5597C2
hypercenter#121D32
hypocenterC32#1604
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×#5040C2semidirect
3. #10591344C2semidirect
4. #10591171C2semidirect
5. #10589403C2semidirect
6. #10589150C2semidirect
7. #10589147C2semidirect
35. #5040D3semidirect
145. C2×C3×Dic3SD16semidirect
345. C2×Dic3#214semidirect
448. C32#1604semidirect
525. C3#70975semidirect
526. C3#71880semidirect

Maximal subgroups

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

subgroupindex
C3×#55972
C3×#50402
#105913442
#105911712
#105894032
#105891502
#105891472
#718803
#709753