Finite Group Databasebeta

← #11778340

Subgroups — #11778340

Order 1152.

609 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 609 conjugacy classes of subgroups — too many to draw. Only its 82 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#10594911C211yesmaximal, characteristic
2C3×#4738C211yesmaximal, characteristic, complemented
3#10591344C211yesmaximal, characteristic, complemented
4#10591174C211yesmaximal, characteristic, complemented
5#10589336C211yesmaximal, characteristic, complemented
6C3×#4469C211yesmaximal, characteristic, complemented
7#10589150C211yesmaximal, characteristic, complemented
8#70975—31nomaximal
9#68133—31nomaximal
10C22×C3×Dic6C2211yescharacteristic
11C2×#3804—21no—
12#64139C2211yescharacteristic
13C3×#930C2211yescharacteristic
14C2×C3×Dic3⋊C4C2211yescharacteristic
15C2×C3×#204—21no—
16#63978—21no—
17C2×#3724—21no—
18#63910—21no—
19#63895—21no—
20#63893—21no—
21#63886—21no—
22C2×#3717—21no—
23C32×C22⋊C8C2211yescharacteristic
24#63658C2211yescharacteristic
25C3×#830C2211yescharacteristic
26C3×#816—21no—
27#63579—21no—
28#63566—21no—
29#5597—31no—
30#5510—31no—
31C3×#448—31no—
32#5230—31no—
33#5227—31no—
34#4979—31no—
35#4738—31no—
36#4738D311yescharacteristic, complemented
37C3×#324—31no—
38#4537—31no—
39#4470—31no—
40#4470—31no—
41#4470—31no—
42#4470—61no—
43#4469D311yescharacteristic, complemented
44C22×C32×C4C2311yescharacteristic
45C2×#638—21no—
46C22×C3⋊Dic3—21no—
47C2×C4⋊(C3⋊D3)D411yescharacteristic
48C2×C3×#67—21no—
49C22×C3×Dic3—21no—
50C22×C3×Dic3D411yescharacteristic
51C22×C3×Dic3D411yescharacteristic
52C2×C3×D12D411yescharacteristic
53C2×C3×Dic6—21no—
54C2×C3×Dic6—21no—
55C2×C3×Dic6—42no—
56C2×#627—21no—
57C2×#626—21no—
58C2×#626—21no—
59C2×#624—21no—
60#3804—42no—
61#3804—42no—
62C2×D3×Dic3—21no—
63C2×C32×C8—21no—
64#3752—21no—
65C4⋊(C3⋊Dic3)D411yescharacteristic
66C3×(C2×D3)⋊C4—21no—
67C3×C4⋊Dic3D411yescharacteristic
68C3×Dic3⋊C4—21no—
69C3×Dic3⋊C4—21no—
70C2×C3×C3⋊C8—21no—
71C3×#204—42no—
72#3724—42no—
73Dic3⋊Dic3—21no—
74#3722—21no—
75(C2×D3)⋊Dic3—21no—
76#3717—42no—
77#1604—91noSylow
78C22×C3×Q8—31no—
79C2×#237—61no—
80C2×#235—61no—
81C22×Dic6C2×D311yescharacteristic
82C2×C3×SD16—61no—
83C3×#121—31no—
84C2×C3×C4⋊C4—31no—
85C2×#215—61no—
86#939—61no—
87#939—61no—
88#930—31no—
89#930—31no—
90#930—61no—
91#930—61no—
92#930C2×D311yescharacteristic
93C2×Dic3⋊C4—61no—
94C2×Dic3⋊C4—61no—
95C2×Dic3⋊C4C2×D311yescharacteristic
96C2×#204—21no—
97C2×#204—61no—
98#897—61no—
99#897—61no—
100#896—61no—
101#895—61no—
102#895—61no—
103#884—61no—
104#878—61no—
105C3×#103—61no—
106C3×C22⋊C8—21no—
107C3×C22⋊C8—31no—
108C3×C22⋊C8C2×D311yescharacteristic
109C3×C22⋊C8C2×D311yescharacteristic
110#835—61no—
111C3⋊#95—31no—
112C3⋊#95—31no—
113C3⋊#95—61no—
114C3⋊#95—61no—
115C3⋊#95—61no—
116#830C2×D311yescharacteristic
117#816—21no—
118#816—61no—
119C23×C32C2×D411yescharacteristic
120C22×C3⋊D3—21no—
121C22×C3×D3—21no—
122C2×C32×C4—41no—
123C2×C32×C4C2×D411yescharacteristic
124C2×C32×C4C2×D411yescharacteristic
125#638—41no—
126#638—41no—
127C2×C3⋊Dic3—22no—
128C2×C3⋊Dic3—21no—
129C2×C3⋊Dic3—41no—
130C2×C3⋊Dic3—41no—
131C4⋊(C3⋊D3)—41no—
132C3×#67—41no—
133C3×#67—41no—
134C2×C3×Dic3—21no—
135C2×C3×Dic3—21no—
136C2×C3×Dic3—22no—
137C2×C3×Dic3—21no—
138C2×C3×Dic3—21no—
139C2×C3×Dic3—41no—
140C2×C3×Dic3—41no—
141C2×C3×Dic3—41no—
142C2×C3×Dic3SD1612yescomplemented
143C3×D12—41no—
144C3×Dic6—22no—
145C3×Dic6—22no—
146C3×Dic6—41no—
147C3×Dic6—41no—
148#627—42no—
149#626—42no—
150#626—81no—
151#624—42no—
152D3×Dic3—42no—
153C32×C8—41no—
154C3×C3⋊C8—41no—
155#537—91no—
156#448—31no—
157#448—91no—
158#395—91no—
159#327—91no—
160#327—91no—
161#324—31no—
162C2×C3×Q8—61no—
163C2×C3×Q8—61no—
164C2×C3×Q8—122no—
165C2×C3×D4—31no—
166C2×C3×D4—61no—
167C22×C3×C4—21no—
168C22×C3×C4—31no—
169C22×C3×C4—31no—
170C22×C3×C4—61no—
171C22×C3×C4C22×D311yescharacteristic
172C22×C3×C4C22×D311yescharacteristic
173C2×#67—21no—
174C2×#67—61no—
175C2×#67—61no—
176C2×#67—61no—
177C2×#67—61no—
178C2×#67—121no—
179C22×Dic3—21no—
180C22×Dic3—61no—
181C22×Dic3—61no—
182C22×Dic3—61no—
183C22×Dic3—121no—
184C22×Dic3D1211yescharacteristic
185C22×Dic3D1211yescharacteristic
186#237—122no—
187#237—122no—
188#235—122no—
189#235—122no—
190C2×D12—31no—
191C2×D12—31no—
192C2×D12—61no—
193C2×D12—61no—
194C2×D12—61no—
195C2×D12#6711yescharacteristic
196C2×C4×D3—61no—
197C2×C4×D3—61no—
198C2×C4×D3—61no—
199C2×Dic6—21no—
200C2×Dic6—21no—
201C2×Dic6—42no—
202C2×Dic6—61no—
203C2×Dic6—61no—
204C3×SD16—122no—
205C2×C3×C8—21no—
206C2×C3×C8—21no—
207C2×C3×C8—41no—
208C2×C3×C8—61no—
209C3×C4⋊C4—31no—
210C3×C4⋊C4—61no—
211C3×C4⋊C4—61no—
212C3×C22⋊C4—61no—
213#217—61no—
214#215—122no—
215(C2×D3)⋊C4—21no—
216(C2×D3)⋊C4—61no—
217(C2×D3)⋊C4—61no—
218(C2×D3)⋊C4—61no—
219(C2×D3)⋊C4—61no—
220(C2×D3)⋊C4—61no—
221(C2×D3)⋊C4—121no—
222C4⋊Dic3—31no—
223C4⋊Dic3—31no—
224C4⋊Dic3—61no—
225C4⋊Dic3—61no—
226C4⋊Dic3#6711yescharacteristic
227Dic3⋊C4—21no—
228Dic3⋊C4—21no—
229Dic3⋊C4—61no—
230Dic3⋊C4—122no—
231Dic3⋊C4—122no—
232C2×C3⋊C8—21no—
233#204—42no—
234#204—122no—
235C22×C32—21no—
236C22×C32—21no—
237C22×C32#12011yescharacteristic
238C22×C32C2×SD1611yescharacteristic
239C22×C32AGL(1,ℤ/8)11yescharacteristic
240C2×C3⋊D3—41no—
241C2×C3⋊D3—41no—
242C2×C3⋊D3—41no—
243C2×C3×D3—41no—
244C2×C3×D3—41no—
245C2×C3×D3—41no—
246C32×C4—21no—
247C32×C4—21no—
248C3⋊Dic3—42no—
249C3⋊Dic3—41no—
250C3×Dic3—22no—
251C3×Dic3—41no—
252C3×Dic3—41no—
253C3×Dic3—42no—
254C2×#41—181no—
255C22×Q8—31no—
256C2×SD16—61no—
257C2×SD16—181no—
258C22⋊Q8—181no—
259C22⋊Q8—181no—
260#121—31no—
261#121—91no—
262#121—181no—
263#121—181no—
264#121—181no—
265C2×C4⋊C4—31no—
266C2×C4⋊C4—181no—
267#103—61no—
268#103—181no—
269C22⋊C8—31no—
270C22⋊C8D3211yescharacteristic
271#95—91no—
272#95—181no—
273C23×C3—21no—
274C23×C3—61no—
275C23×C3C2×D1211yescharacteristic
276C23×C3D3×D411yescharacteristic
277C22×D3—21no—
278C22×D3—61no—
279C22×D3—61no—
280C22×D3—121no—
281C3×Q8—62no—
282C3×Q8—62no—
283C3×Q8—121no—
284C3×Q8—121no—
285C3×D4—121no—
286C3×D4—121no—
287C3×D4—121no—
288C2×C3×C4—21no—
289C2×C3×C4—21no—
290C2×C3×C4—32no—
291C2×C3×C4—41no—
292C2×C3×C4—41no—
293C2×C3×C4—61no—
294C2×C3×C4—61no—
295C2×C3×C4—62no—
296C2×C3×C4—61no—
297C2×C3×C4—61no—
298C2×C3×C4—81no—
299C2×C3×C4—121no—
300C2×C3×C4—121no—
301C2×C3×C4—121no—
302C2×C3×C4D3×D411yescharacteristic
303C2×C3×C4D3×D411yescharacteristic
304C2×C3×C4D3×D411yescharacteristic
305C2×C3×C4C2×#6711yescharacteristic
306#67—41no—
307#67—41no—
308#67—122no—
309#67—121no—
310#67—121no—
311#67—121no—
312#67—121no—
313#67—241no—
314#67—241no—
315#67—241no—
316C2×Dic3—21no—
317C2×Dic3—21no—
318C2×Dic3—22no—
319C2×Dic3—21no—
320C2×Dic3—21no—
321C2×Dic3—41no—
322C2×Dic3—41no—
323C2×Dic3—41no—
324C2×Dic3—62no—
325C2×Dic3—62no—
326C2×Dic3—61no—
327C2×Dic3—61no—
328C2×Dic3—121no—
329C2×Dic3—122no—
330C2×Dic3—122no—
331C2×Dic3—121no—
332C2×Dic3—121no—
333C2×Dic3—121no—
334C2×Dic3—121no—
335C2×Dic3—241no—
336C2×Dic3—241no—
337C2×Dic3#20412yescomplemented
338D12—41no—
339D12—121no—
340D12—121no—
341D12—122no—
342D12—241no—
343D12—241no—
344C4×D3—122no—
345C4×D3—122no—
346C4×D3—122no—
347Dic6—22no—
348Dic6—22no—
349Dic6—41no—
350Dic6—41no—
351Dic6—122no—
352Dic6—122no—
353C3×C8—41no—
354C3×C8—41no—
355C3×C8—81no—
356C3×C8—121no—
357C3⋊C8—41no—
358C2×C32—21no—
359C2×C32—21no—
360C2×C32#45011yescharacteristic
361C2×C32#45611yescharacteristic
362C2×C32B(4,𝔽2)11yescharacteristic
363C3⋊D3—81no—
364C3×D3—81no—
365#41—362no—
366#41—362no—
367C2×Q8—61no—
368C2×Q8—61no—
369C2×Q8—122no—
370C2×Q8—181no—
371C2×D4—31no—
372C2×D4—61no—
373C2×D4—91no—
374C2×D4—181no—
375C2×D4—181no—
376C2×D4—181no—
377C22×C4—31no—
378C22×C4—31no—
379C22×C4—61no—
380C22×C4—181no—
381C22×C4—181no—
382C22×C4—181no—
383C22×C4C2×D3211yescharacteristic
384SD16—122no—
385SD16—362no—
386C2×C8—21no—
387C2×C8—61no—
388C4⋊C4—31no—
389C4⋊C4—61no—
390C4⋊C4—61no—
391C4⋊C4—91no—
392C4⋊C4—181no—
393C4⋊C4—362no—
394C22⋊C4—61no—
395C22⋊C4—181no—
396C22⋊C4—181no—
397C22⋊C4—181no—
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—121no—
408C22×C3—121no—
409C22×C3—121no—
410C22×C3#88211yescharacteristic
411C22×C3C2×#20411yescharacteristic
412C22×C3#90811yescharacteristic
413C22×C3D3×SD1611yescharacteristic
414C22×C3#91411yescharacteristic
415C22×C3#93711yescharacteristic
416C2×D3—41no—
417C2×D3—41no—
418C2×D3—41no—
419C2×D3—121no—
420C2×D3—121no—
421C2×D3—121no—
422C2×D3—121no—
423C2×D3—121no—
424C2×D3—121no—
425C2×D3—241no—
426C2×D3—241no—
427C2×D3—241no—
428C3×C4—21no—
429C3×C4—21no—
430C3×C4—21no—
431C3×C4—21no—
432C3×C4—41no—
433C3×C4—41no—
434C3×C4—62no—
435C3×C4—121no—
436C3×C4—122no—
437C3×C4—121no—
438Dic3—22no—
439Dic3—41no—
440Dic3—41no—
441Dic3—42no—
442Dic3—122no—
443Dic3—122no—
444Dic3—121no—
445Dic3—121no—
446Dic3—241no—
447Dic3—242no—
448C32#160411yescharacteristic, complemented, Sylow
449C23—61no—
450C23—181no—
451C23D3×D1211yescharacteristic
452Q8—62no—
453Q8—62no—
454Q8—121no—
455Q8—121no—
456Q8—362no—
457D4—121no—
458D4—121no—
459D4—121no—
460D4—361no—
461D4—362no—
462D4—361no—
463D4—361no—
464D4—721no—
465C2×C4—32no—
466C2×C4—41no—
467C2×C4—61no—
468C2×C4—61no—
469C2×C4—62no—
470C2×C4—61no—
471C2×C4—61no—
472C2×C4—121no—
473C2×C4—121no—
474C2×C4—121no—
475C2×C4—182no—
476C2×C4—181no—
477C2×C4—362no—
478C2×C4—362no—
479C2×C4—361no—
480C2×C4—361no—
481C2×C4#380211yescharacteristic
482C2×C4D3×#6711yescharacteristic
483C8—41no—
484C8—121no—
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—241no—
495C2×C3#473111yescharacteristic
496C2×C3#474111yescharacteristic
497C2×C3#475211yescharacteristic
498C2×C3#503211yescharacteristic
499C2×C3#516911yescharacteristic
500C2×C3#520511yescharacteristic
501D3—81no—
502D3—241no—
503D3—241no—
504D3—481no—
505C22—21no—
506C22—21no—
507C22—121no—
508C22—121no—
509C22—121no—
510C22—361no—
511C22—361no—
512C22—361no—
513C22D3×#20411yescharacteristic
514C22#6379411yescharacteristic
515C22#6392211yescharacteristic
516C4—21no—
517C4—21no—
518C4—62no—
519C4—122no—
520C4—121no—
521C4—121no—
522C4—362no—
523C4—361no—
524C3—21no—
525C3#6813311yescharacteristic, complemented
526C3#7097511yescharacteristic, complemented
527C2—21no—
528C2—21no—
529C2—241no—
530C2—721no—
531C2#1059040711yescharacteristic
532C2#1059094211yescharacteristic
533C2#1059101811yescharacteristic
subgroupG/H
centerC22#63922
commutator, cosocleC22×C32×C4C23
FrattiniC22×C4C2×D32
FittingC32×C22⋊C8C22
socleC22×C32#120
prime radical#10594911C2
hypercenterC22⋊C8D32
hypocenterC32#1604
LCSG ▷ C22×C32×C4 ▷ C22×C32 ▷ C32
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22⋊C8

Complements

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

NH ≅ G/Ntype
2. C3×#4738C2semidirect
3. #10591344C2semidirect
4. #10591174C2semidirect
5. #10589336C2semidirect
6. C3×#4469C2semidirect
7. #10589150C2semidirect
36. #4738D3semidirect
43. #4469D3semidirect
142. C2×C3×Dic3SD16semidirect
337. C2×Dic3#204semidirect
448. C32#1604semidirect
525. C3#68133semidirect
526. C3#70975semidirect

Maximal subgroups

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

subgroupindex
#105949112
C3×#47382
#105913442
#105911742
#105893362
C3×#44692
#105891502
#709753
#681333