Finite Group Databasebeta

← #11807698

Subgroups — #11807698

Order 1152.

1153 conjugacy classes of subgroups.

The subgroup lattice has 518 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#63897C211yesmaximal, characteristic
2#10592687C211yesmaximal, characteristic
3C3×#4976C212yesmaximal, complemented
4#10591380C211yesmaximal, characteristic
5#10591374C212yesmaximal, complemented
6#10591367C212yesmaximal, complemented
7#10591289C212yesmaximal, complemented
8#10591194C212yesmaximal, complemented
9#10591192C212yesmaximal, complemented
10#76207—32nomaximal
11C22×C4⋊(C3⋊D3)C2211yescharacteristic
12C22×#624C2211yescharacteristic
13C22×#624C2211yescharacteristic
14C2×C32×C4⋊C4C2211yescharacteristic
15C2×#3752C2212yes—
16C2×#3752C2212yes—
17C2×C4×C3⋊Dic3C2211yescharacteristic
18C2×C3×Dic3⋊C4C2212yes—
19C2×C3×Dic3⋊C4C2212yes—
20C2×C3×Dic3⋊C4C2212yes—
21C2×C3×C4×Dic3C2212yes—
22C2×Dic3⋊Dic3C2212yes—
23C2×#3722C2212yes—
24C2×#3722C2211yescharacteristic
25C2×#3722C2212yes—
26C2×#3722C2211yescharacteristic
27#63897C418yescomplemented
28C3×C3⋊#95C2212yes—
29C3×C3⋊#95C2212yes—
30#63579C2212yes—
31#63579C2212yes—
32#63579C2212yes—
33C2×#894—32no—
34C3×#389—32no—
35#4990—32no—
36#4989—32no—
37#4989—32no—
38#4988—32no—
39#4988—61no—
40#4987—32no—
41#4980—32no—
42#4976D312yescomplemented
43#4940—32no—
44#4940—32no—
45#4670—32no—
46#4667—32no—
47#4665—32no—
48#4652—32no—
49C23×C3⋊D3C2311yescharacteristic
50C23×C3⋊D3C2311yescharacteristic
51C22×C32×C4C2312yes—
52C22×C32×C4C2311yescharacteristic
53C22×C3⋊Dic3C2312yes—
54C2×C4⋊(C3⋊D3)—41no—
55C2×C4⋊(C3⋊D3)—41no—
56C2×C4⋊(C3⋊D3)C2×C414yes—
57C22×C3×Dic3C2312yes—
58C22×C3×Dic3C2312yes—
59C22×C3×Dic3C2312yes—
60C22×C3×Dic3C2312yes—
61C2×#624—41no—
62C2×#624—41no—
63C2×#624—42no—
64C2×#624—41no—
65C2×#624C2×C414yescomplemented
66C32×C4⋊C4—41no—
67#3752—42no—
68#3752—42no—
69C4×C3⋊Dic3—41no—
70C3×Dic3⋊C4—42no—
71C3×Dic3⋊C4—42no—
72C3×Dic3⋊C4C2×C418yescomplemented
73C3×C4×Dic3—42no—
74Dic3⋊Dic3—42no—
75#3722—41no—
76#3722—42no—
77#3722C2×C418yes—
78#3722C2×C414yescomplemented
79#2376—91noSylow
80C22×D12—32no—
81C22×D12—61no—
82C22×C4×D3—32no—
83C22×C4×D3—32no—
84C2×C3×C4⋊C4—21no—
85C2×C3×C4⋊C4—32no—
86C2×C3×C4⋊C4—32no—
87C2×C3×C4⋊C4—32no—
88C2×C3×C4⋊C4C2×D312yes—
89C2×C3×C42—32no—
90C2×(C2×D3)⋊C4—32no—
91C2×(C2×D3)⋊C4—32no—
92C2×(C2×D3)⋊C4—32no—
93C2×(C2×D3)⋊C4—32no—
94C2×(C2×D3)⋊C4—32no—
95C2×(C2×D3)⋊C4—32no—
96C2×(C2×D3)⋊C4—32no—
97C2×(C2×D3)⋊C4—32no—
98C2×(C2×D3)⋊C4—32no—
99C2×(C2×D3)⋊C4—32no—
100C2×(C2×D3)⋊C4—62no—
101C2×(C2×D3)⋊C4—62no—
102C2×C4⋊Dic3—32no—
103C2×Dic3⋊C4—32no—
104C2×Dic3⋊C4C2×D312yes—
105C2×Dic3⋊C4C2×D312yes—
106C2×Dic3⋊C4C2×D312yes—
107C2×C4×Dic3—32no—
108C2×C4×Dic3—61no—
109C2×C4×Dic3C2×D312yes—
110#894—316no—
111C3×#95—32no—
112C3×#95—32no—
113C3⋊#95—32no—
114C3⋊#95—32no—
115C3⋊#95—32no—
116C3⋊#95—32no—
117C3⋊#95—32no—
118C3⋊#95—32no—
119C3⋊#95C2×D312yes—
120C3⋊#95C2×D312yes—
121C23×C32C2411yescharacteristic
122C22×C3⋊D3—22no—
123C22×C3⋊D3—21no—
124C22×C3⋊D3—22no—
125C22×C3⋊D3—21no—
126C22×C3⋊D3—21no—
127C22×C3⋊D3—21no—
128C22×C3⋊D3—22no—
129C22×C3⋊D3—22no—
130C22×C3⋊D3C22×C412yes—
131C22×C3⋊D3C22×C412yes—
132C2×C32×C4—21no—
133C2×C32×C4—21no—
134C2×C32×C4—22no—
135C2×C32×C4—22no—
136C2×C32×C4—22no—
137C2×C32×C4C22×C412yes—
138C2×C3⋊Dic3—22no—
139C2×C3⋊Dic3—22no—
140C2×C3⋊Dic3—22no—
141C4⋊(C3⋊D3)—44no—
142C2×C3×Dic3—22no—
143C2×C3×Dic3—22no—
144C2×C3×Dic3—22no—
145C2×C3×Dic3—22no—
146C2×C3×Dic3—22no—
147C2×C3×Dic3—22no—
148C2×C3×Dic3—22no—
149C2×C3×Dic3—22no—
150C2×C3×Dic3—22no—
151C2×C3×Dic3—22no—
152C2×C3×Dic3C22×C414yes—
153C2×C3×Dic3C22×C414yes—
154#624—44no—
155#624—44no—
156C2×#123—91no—
157#390—91no—
158#389—32no—
159#388—92no—
160#388—92no—
161#387—92no—
162#386—91no—
163#385—92no—
164#380—92no—
165C23×D3—32no—
166C23×D3—32no—
167C23×D3—61no—
168C23×D3—61no—
169C22×C3×C4—21no—
170C22×C3×C4—22no—
171C22×C3×C4—32no—
172C22×C3×C4—32no—
173C22×C3×C4—32no—
174C22×C3×C4—32no—
175C22×C3×C4C22×D312yes—
176C22×C3×C4C22×D312yes—
177C22×C3×C4C22×D312yes—
178C22×Dic3—32no—
179C22×Dic3—32no—
180C22×Dic3—62no—
181C22×Dic3C22×D312yes—
182C22×Dic3C22×D312yes—
183C22×Dic3C22×D312yes—
184C22×Dic3C22×D312yes—
185C2×D12—38no—
186C2×D12—64no—
187C2×D12—122no—
188C2×D12—122no—
189C2×D12—122no—
190C2×D12—122no—
191C2×C4×D3—38no—
192C2×C4×D3—122no—
193C2×C4×D3—122no—
194C2×C4×D3—122no—
195C2×C4×D3—122no—
196C2×C4×D3—122no—
197C3×C4⋊C4—38no—
198C3×C4⋊C4—42no—
199C3×C4⋊C4—42no—
200C3×C4⋊C4—122no—
201C3×C4⋊C4—122no—
202C3×C42—122no—
203(C2×D3)⋊C4—38no—
204(C2×D3)⋊C4—38no—
205(C2×D3)⋊C4—38no—
206(C2×D3)⋊C4—122no—
207(C2×D3)⋊C4—122no—
208(C2×D3)⋊C4—122no—
209(C2×D3)⋊C4—122no—
210(C2×D3)⋊C4—122no—
211(C2×D3)⋊C4—122no—
212(C2×D3)⋊C4—122no—
213(C2×D3)⋊C4—124no—
214(C2×D3)⋊C4—124no—
215C4⋊Dic3—122no—
216Dic3⋊C4—42no—
217Dic3⋊C4—42no—
218Dic3⋊C4—122no—
219Dic3⋊C4C4×D318yescomplemented
220C4×Dic3—42no—
221C4×Dic3—122no—
222C4×Dic3—122no—
223C22×C32C23×C411yescharacteristic
224C22×C32H(5,𝔽2)11yescharacteristic
225C22×C32H(5,𝔽2)12yes—
226C22×C32H(5,𝔽2)11yescharacteristic
227C22×C322−1+412yes—
228C2×C3⋊D3—22no—
229C2×C3⋊D3—22no—
230C2×C3⋊D3—24no—
231C2×C3⋊D3—24no—
232C2×C3⋊D3—41no—
233C2×C3⋊D3—41no—
234C2×C3⋊D3—42no—
235C2×C3⋊D3—42no—
236C2×C3⋊D3—41no—
237C2×C3⋊D3—41no—
238C32×C4—22no—
239C32×C4—42no—
240C3⋊Dic3—42no—
241C3×Dic3—24no—
242C3×Dic3—24no—
243C3×Dic3—42no—
244C3×Dic3—42no—
245C22×D4—91no—
246C23×C4—91no—
247C23×C4—91no—
248#123—98no—
249C2×C4⋊C4—32no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—92no—
253C2×C4⋊C4D3211yescharacteristic
254C2×C22⋊C4—92no—
255C2×C22⋊C4—91no—
256C2×C22⋊C4—92no—
257C2×C22⋊C4—91no—
258C2×C22⋊C4—92no—
259C2×C22⋊C4—92no—
260C2×C42—32no—
261C2×C42—91no—
262#95—32no—
263#95—32no—
264#95—92no—
265#95—92no—
266#95—92no—
267C23×C3—21no—
268C23×C3C23×D312yes—
269C22×D3—34no—
270C22×D3—34no—
271C22×D3—62no—
272C22×D3—62no—
273C22×D3—62no—
274C22×D3—62no—
275C22×D3—62no—
276C22×D3—62no—
277C22×D3—62no—
278C22×D3—62no—
279C22×D3—62no—
280C22×D3—62no—
281C22×D3—62no—
282C22×D3—62no—
283C22×D3—62no—
284C22×D3—62no—
285C22×D3—122no—
286C22×D3—122no—
287C22×D3—121no—
288C22×D3—121no—
289C22×D3—121no—
290C22×D3—121no—
291C22×D3—122no—
292C22×D3—122no—
293C2×C3×C4—22no—
294C2×C3×C4—22no—
295C2×C3×C4—22no—
296C2×C3×C4—22no—
297C2×C3×C4—22no—
298C2×C3×C4—22no—
299C2×C3×C4—22no—
300C2×C3×C4—22no—
301C2×C3×C4—22no—
302C2×C3×C4—22no—
303C2×C3×C4—22no—
304C2×C3×C4—34no—
305C2×C3×C4—34no—
306C2×C3×C4—42no—
307C2×C3×C4—42no—
308C2×C3×C4—42no—
309C2×C3×C4—62no—
310C2×C3×C4—62no—
311C2×C3×C4—62no—
312C2×C3×C4—62no—
313C2×C3×C4—62no—
314C2×C3×C4—62no—
315C2×C3×C4—62no—
316C2×C3×C4—62no—
317C2×C3×C4—62no—
318C2×C3×C4—62no—
319C2×C3×C4C2×C4×D314yes—
320C2×Dic3—22no—
321C2×Dic3—22no—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—22no—
329C2×Dic3—22no—
330C2×Dic3—64no—
331C2×Dic3—62no—
332C2×Dic3—62no—
333C2×Dic3—62no—
334C2×Dic3—62no—
335C2×Dic3—62no—
336C2×Dic3—62no—
337C2×Dic3—122no—
338C2×Dic3—122no—
339C2×Dic3C2×C4×D314yes—
340C2×Dic3C2×C4×D314yes—
341D12—128no—
342D12—128no—
343C4×D3—128no—
344C4×D3—128no—
345C2×C32#51811yescharacteristic
346C2×C32#52012yes—
347C2×C32#56011yescharacteristic
348C2×C32#56212yes—
349C2×C32#56311yescharacteristic
350C3⋊D3—42no—
351C3⋊D3—42no—
352C24—91no—
353C24—91no—
354C2×D4—94no—
355C2×D4—361no—
356C2×D4—361no—
357C22×C4—32no—
358C22×C4—32no—
359C22×C4—32no—
360C22×C4—32no—
361C22×C4—94no—
362C22×C4—92no—
363C22×C4—361no—
364C22×C4—361no—
365C22×C4—362no—
366C22×C4—361no—
367C22×C4C2×D3212yes—
368C22×C4C2×D3211yescharacteristic
369C4⋊C4—38no—
370C4⋊C4—41no—
371C4⋊C4—122no—
372C4⋊C4—122no—
373C4⋊C4—362no—
374C22⋊C4—94no—
375C22⋊C4—98no—
376C22⋊C4—362no—
377C22⋊C4—362no—
378C22⋊C4—361no—
379C22⋊C4—362no—
380C42—122no—
381C42—361no—
382C22×C3—21no—
383C22×C3—21no—
384C22×C3—22no—
385C22×C3—22no—
386C22×C3—21no—
387C22×C3C22×C4×D312yes—
388C22×C3#100412yes—
389C22×C3#100412yes—
390C22×C3#100712yes—
391C22×C3#100712yes—
392C22×C3#100912yes—
393C22×C3#100912yes—
394C2×D3—64no—
395C2×D3—64no—
396C2×D3—64no—
397C2×D3—64no—
398C2×D3—64no—
399C2×D3—64no—
400C2×D3—122no—
401C2×D3—122no—
402C2×D3—124no—
403C2×D3—124no—
404C2×D3—122no—
405C2×D3—122no—
406C2×D3—122no—
407C2×D3—122no—
408C2×D3—122no—
409C2×D3—122no—
410C2×D3—122no—
411C2×D3—122no—
412C2×D3—241no—
413C2×D3—241no—
414C2×D3—242no—
415C2×D3—242no—
416C2×D3—241no—
417C2×D3—241no—
418C3×C4—24no—
419C3×C4—24no—
420C3×C4—42no—
421C3×C4—42no—
422C3×C4—64no—
423C3×C4—64no—
424C3×C4—82no—
425C3×C4—122no—
426C3×C4—122no—
427Dic3—24no—
428Dic3—24no—
429Dic3—42no—
430Dic3—42no—
431Dic3—124no—
432Dic3—122no—
433Dic3—122no—
434C32#237611yescharacteristic, complemented, Sylow
435C23—92no—
436C23—92no—
437C23—181no—
438C23—181no—
439C23—182no—
440C23—182no—
441C23—182no—
442C23—182no—
443C23—181no—
444C23—181no—
445C23C22×D3211yescharacteristic
446D4—364no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—22no—
452C2×C4—34no—
453C2×C4—34no—
454C2×C4—62no—
455C2×C4—62no—
456C2×C4—62no—
457C2×C4—62no—
458C2×C4—62no—
459C2×C4—62no—
460C2×C4—62no—
461C2×C4—62no—
462C2×C4—62no—
463C2×C4—62no—
464C2×C4—182no—
465C2×C4—182no—
466C2×C4—182no—
467C2×C4—364no—
468C2×C4—364no—
469C2×C4C4×D3212yes—
470C2×C3—22no—
471C2×C3—21no—
472C2×C3—22no—
473C2×C3—21no—
474C2×C3—21no—
475C2×C3#550412yes—
476C2×C3#554512yes—
477C2×C3#557412yes—
478C2×C3#562012yes—
479C2×C3#564712yes—
480C2×C3#570012yes—
481C2×C3#573512yes—
482D3—124no—
483D3—124no—
484D3—242no—
485D3—242no—
486C22—184no—
487C22—184no—
488C22—182no—
489C22—182no—
490C22—361no—
491C22—361no—
492C22—362no—
493C22—362no—
494C22—361no—
495C22—361no—
496C22C2×C4×D3211yescharacteristic
497C22#6430811yescharacteristic
498C22#6431412yes—
499C22#6431612yes—
500C22#6433011yescharacteristic
501C4—22no—
502C4—42no—
503C4—64no—
504C4—64no—
505C4—122no—
506C4—122no—
507C4—362no—
508C3—21no—
509C3#7620712yescomplemented
510C2—362no—
511C2—362no—
512C2#1059432712yes—
513C2#1059459312yes—
514C2#1059505811yescharacteristic
515C2#1059531211yescharacteristic
516C2#1059535211yescharacteristic
subgroupG/H
center, FrattiniC23C22×D32
commutatorC22×C32C23×C4
FittingC2×C32×C4⋊C4C22
socle, cosocleC23×C32C24
prime radicalC22×C4⋊(C3⋊D3)C22
hypercenterC2×C4⋊C4D32
hypocenterC32#2376
LCSG ▷ C22×C32 ▷ C32
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
3. C3×#4976C2semidirect
5. #10591374C2semidirect
6. #10591367C2semidirect
7. #10591289C2semidirect
8. #10591194C2semidirect
9. #10591192C2semidirect
27. #63897C4semidirect
42. #4976D3semidirect
65. C2×#624C2×C4semidirect
72. C3×Dic3⋊C4C2×C4semidirect
78. #3722C2×C4semidirect
219. Dic3⋊C4C4×D3semidirect
434. C32#2376semidirect
509. C3#76207semidirect

Maximal subgroups

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

subgroupindex
C2×#638972
#105926872
C3×#49762
#105913802
#105913742
#105913672
#105912892
#105911942
#105911922
#762073