Finite Group Databasebeta

← #75924

Subgroups — #75924

Order 384.

612 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#895C211yesmaximal, characteristic, complemented
2C2×#884C211yesmaximal, characteristic, complemented
3C3×#388C211yesmaximal, characteristic, complemented
4#4982C211yesmaximal, characteristic, complemented
5#4961C211yesmaximal, characteristic, complemented
6#4960C211yesmaximal, characteristic, complemented
7#4960C211yesmaximal, characteristic, complemented
8#4959C211yesmaximal, characteristic, complemented
9#4956C211yesmaximal, characteristic, complemented
10#4940C211yesmaximal, characteristic, complemented
11#4939C211yesmaximal, characteristic, complemented
12#4672C211yesmaximal, characteristic, complemented
13#4665C211yesmaximal, characteristic, complemented
14#4664C211yesmaximal, characteristic, complemented
15#4664C211yesmaximal, characteristic, complemented
16#2699—31nomaximal, Sylow
17C22×#67C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C22×D12C2211yescharacteristic
22C22×D12C2211yescharacteristic
23C22×C4×D3C2211yescharacteristic
24C22×C4×D3C2211yescharacteristic
25C2×C3×C4⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C42C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×C4⋊Dic3C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44#895—24no—
45#884—24no—
46C3×#95C2211yescharacteristic
47C3×#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#121—31no—
55C2×#121—31no—
56#398—31no—
57#394—31no—
58#394—31no—
59#394—31no—
60#392—31no—
61#392—31no—
62#390—31no—
63#388—31no—
64#388D311yescharacteristic, complemented
65#386—31no—
66#386—31no—
67#381—31no—
68#381—31no—
69C24×C3C2311yescharacteristic
70C23×D3C2311yescharacteristic
71C23×D3C2311yescharacteristic
72C23×D3C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C2×#67—22no—
80C2×#67—22no—
81C2×#67—22no—
82C2×#67—41no—
83C2×#67—41no—
84C2×#67—41no—
85C2×#67—41no—
86C2×#67—41no—
87C2×#67—41no—
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C2×D12—22no—
94C2×D12—22no—
95C2×D12—22no—
96C2×D12—41no—
97C2×D12—41no—
98C2×D12—41no—
99C2×D12—41no—
100C2×D12—41no—
101C2×D12D414yescomplemented
102C2×C4×D3—22no—
103C2×C4×D3—22no—
104C2×C4×D3—22no—
105C2×C4×D3—22no—
106C2×C4×D3—41no—
107C2×C4×D3—41no—
108C3×C4⋊C4—22no—
109C3×C22⋊C4—22no—
110C3×C22⋊C4—22no—
111C3×C22⋊C4—41no—
112C3×C42—41no—
113#217—41no—
114(C2×D3)⋊C4—22no—
115(C2×D3)⋊C4—22no—
116(C2×D3)⋊C4—22no—
117(C2×D3)⋊C4—22no—
118(C2×D3)⋊C4—41no—
119(C2×D3)⋊C4—41no—
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—41no—
123(C2×D3)⋊C4—41no—
124C4⋊Dic3—41no—
125Dic3⋊C4—22no—
126Dic3⋊C4—41no—
127C22×D4—31no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4—31no—
131C22×D4—31no—
132C22×D4—31no—
133C23×C4—31no—
134C23×C4—31no—
135#121—64no—
136#121—64no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4C2×D311yescharacteristic
141C2×C22⋊C4—31no—
142C2×C22⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4—31no—
145C2×C22⋊C4—31no—
146C2×C22⋊C4—31no—
147C2×C22⋊C4—31no—
148C2×C22⋊C4—31no—
149C2×C22⋊C4—31no—
150C2×C22⋊C4—31no—
151C2×C22⋊C4—31no—
152C2×C22⋊C4C2×D311yescharacteristic
153C2×C22⋊C4C2×D311yescharacteristic
154C2×C22⋊C4C2×D311yescharacteristic
155C2×C42C2×D311yescharacteristic
156#95—31no—
157#95—31no—
158#95—31no—
159#95—31no—
160#95—31no—
161#95—31no—
162#95C2×D311yescharacteristic
163#95C2×D311yescharacteristic
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3C2411yescharacteristic
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—21no—
179C22×D3—21no—
180C22×D3—21no—
181C22×D3—21no—
182C22×D3—21no—
183C22×D3—21no—
184C22×D3—21no—
185C22×D3—21no—
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—21no—
191C22×D3C2×D412yes—
192C22×D3C2×D412yes—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
201C2×C3×C4—21no—
202C2×C3×C4—21no—
203C2×C3×C4—21no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—21no—
207C2×C3×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4C2×D412yes—
210C2×C3×C4#4112yes—
211#67—44no—
212#67—44no—
213#67—44no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—21no—
222C2×Dic3—21no—
223C2×Dic3—21no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3#4112yes—
229D12—44no—
230D12—44no—
231D12—44no—
232C4×D3—44no—
233C4×D3—44no—
234C24—31no—
235C24—31no—
236C24—31no—
237C24C22×D311yescharacteristic
238C2×D4—34no—
239C2×D4—62no—
240C2×D4—62no—
241C2×D4—62no—
242C2×D4—62no—
243C2×D4—62no—
244C2×D4—62no—
245C2×D4—121no—
246C2×D4—121no—
247C2×D4—121no—
248C2×D4—121no—
249C2×D4—121no—
250C2×D4—121no—
251C2×D4—121no—
252C2×D4—121no—
253C2×D4—121no—
254C2×D4—121no—
255C2×D4—121no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—62no—
264C22×C4—62no—
265C22×C4—121no—
266C22×C4—121no—
267C22×C4C22×D311yescharacteristic
268C22×C4C22×D311yescharacteristic
269C22×C4C22×D311yescharacteristic
270C22×C4C22×D311yescharacteristic
271C22×C4C22×D311yescharacteristic
272C22×C4C22×D311yescharacteristic
273C4⋊C4—22no—
274C4⋊C4—62no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—41no—
280C22⋊C4—62no—
281C22⋊C4—62no—
282C22⋊C4—62no—
283C22⋊C4—62no—
284C22⋊C4—121no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C42—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3C22×D411yescharacteristic
300C22×C3C2×#4111yescharacteristic
301C22×C3C2×#4111yescharacteristic
302C22×C3H(5,𝔽2)11yescharacteristic
303C22×C3H(5,𝔽2)11yescharacteristic
304C22×C3H(5,𝔽2)11yescharacteristic
305C22×C3H(5,𝔽2)11yescharacteristic
306C2×D3—22no—
307C2×D3—22no—
308C2×D3—22no—
309C2×D3—22no—
310C2×D3—22no—
311C2×D3—22no—
312C2×D3—41no—
313C2×D3—41no—
314C2×D3—41no—
315C2×D3—41no—
316C2×D3—41no—
317C2×D3—41no—
318C2×D3—41no—
319C2×D3—41no—
320C2×D3—41no—
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C3×C4—22no—
328C3×C4—22no—
329C3×C4—41no—
330C3×C4—41no—
331C3×C4—41no—
332C3×C4—41no—
333Dic3—22no—
334Dic3—41no—
335Dic3—41no—
336Dic3—41no—
337Dic3—41no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—32no—
346C23—32no—
347C23—61no—
348C23—61no—
349C23—61no—
350C23—61no—
351C23—61no—
352C23—61no—
353C23—61no—
354C23—61no—
355C23—61no—
356C23—61no—
357C23—61no—
358C23—61no—
359C23—61no—
360C23—61no—
361C23—61no—
362C23—61no—
363C23—61no—
364C23—61no—
365C23—61no—
366C23C23×D311yescharacteristic
367D4—124no—
368D4—124no—
369D4—124no—
370D4—124no—
371D4—124no—
372D4—124no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—32no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—61no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
400C2×C4—61no—
401C2×C4—61no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—124no—
405C2×C4—124no—
406C2×C4#23512yes—
407C2×C4D3×D412yes—
408C2×C3—81no—
409C2×C3#53411yescharacteristic
410C2×C3#54011yescharacteristic
411C2×C3#54011yescharacteristic
412C2×C3#54611yescharacteristic
413C2×C3#54611yescharacteristic
414C2×C3#55511yescharacteristic
415C2×C3#56011yescharacteristic
416D3—42no—
417D3—42no—
418D3—81no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—62no—
427C22—62no—
428C22—62no—
429C22—62no—
430C22—62no—
431C22—62no—
432C22—121no—
433C22—121no—
434C22—121no—
435C22—121no—
436C22—121no—
437C22—121no—
438C22—121no—
439C22—121no—
440C22—121no—
441C22—121no—
442C22—121no—
443C22—121no—
444C22—121no—
445C22—121no—
446C22—121no—
447C22C2×#23511yescharacteristic
448C22C2×D3×D411yescharacteristic
449C22#100411yescharacteristic
450C22D3×#4111yescharacteristic
451C22#100911yescharacteristic
452C22#100911yescharacteristic
453C22#100911yescharacteristic
454C4—22no—
455C4—22no—
456C4—41no—
457C4—41no—
458C4—41no—
459C4—41no—
460C4—62no—
461C4—121no—
462C4—121no—
463C4—121no—
464C4—121no—
465C3#269911yescharacteristic, complemented, Sylow
466C2—81no—
467C2—122no—
468C2—122no—
469C2—241no—
470C2#553211yescharacteristic
471C2#555011yescharacteristic
472C2#556211yescharacteristic
473C2#560511yescharacteristic
474C2#566311yescharacteristic
475C2#567611yescharacteristic
476C2#571111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#388C2
hypercenter#388D3
hypocenterC3#2699
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#895C2semidirect
2. C2×#884C2semidirect
3. C3×#388C2semidirect
4. #4982C2semidirect
5. #4961C2semidirect
6. #4960C2semidirect
7. #4960C2semidirect
8. #4959C2semidirect
9. #4956C2semidirect
10. #4940C2semidirect
11. #4939C2semidirect
12. #4672C2semidirect
13. #4665C2semidirect
14. #4664C2semidirect
15. #4664C2semidirect
64. #388D3semidirect
101. C2×D12D4semidirect
465. C3#2699semidirect

Maximal subgroups

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

subgroupindex
C2×#8952
C2×#8842
C3×#3882
#49822
#49612
#49602
#49592
#49562
#49402
#49392
#46722
#46652
#46642
#26993