Finite Group Databasebeta

← #82899

Subgroups — #82899

Order 384.

718 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#123C211yesmaximal, characteristic, complemented
2C2×#938C211yesmaximal, characteristic, complemented
3C2×C4×#67C211yesmaximal, characteristic, complemented
4C2×#893C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#5213C211yesmaximal, characteristic, complemented
7#4994C211yesmaximal, characteristic, complemented
8#4989C211yesmaximal, characteristic, complemented
9#4980C211yesmaximal, characteristic, complemented
10#4960C211yesmaximal, characteristic, complemented
11#4960C211yesmaximal, characteristic, complemented
12#4957C211yesmaximal, characteristic, complemented
13#4956C211yesmaximal, characteristic, complemented
14#4944C211yesmaximal, characteristic, complemented
15#4944C211yesmaximal, characteristic, complemented
16#2456—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×D12C2211yescharacteristic
24C22×C4×D3C2211yescharacteristic
25C22×C4×D3C2211yescharacteristic
26C3×#123—24no—
27C2×C3×C4⋊C4C2211yescharacteristic
28C2×C3×C4⋊C4C2211yescharacteristic
29C2×C3×C22⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36#938—24no—
37C4×#67—24no—
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×(C2×D3)⋊C4C2211yescharacteristic
44C2×C4⋊Dic3C2211yescharacteristic
45C2×Dic3⋊C4C2211yescharacteristic
46C2×Dic3⋊C4C2211yescharacteristic
47C2×C4×Dic3C2211yescharacteristic
48C2×C4×Dic3C2211yescharacteristic
49#893—24no—
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×#123D311yescharacteristic, complemented
57C2×#121—31no—
58C2×C4×D4—31no—
59C2×C4×D4—31no—
60#396—31no—
61#392—31no—
62#392—31no—
63#392—31no—
64#390—31no—
65#388—31no—
66#388—31no—
67#387—31no—
68#386—31no—
69#386—31no—
70#385—31no—
71C24×C3C2311yescharacteristic
72C24×C3C2311yescharacteristic
73C23×D3C2311yescharacteristic
74C23×D3C2311yescharacteristic
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C2×C3×D4—41no—
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4—41no—
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C2×#67—22no—
87C2×#67—22no—
88C2×#67—22no—
89C2×#67—22no—
90C2×#67—22no—
91C2×#67—22no—
92C2×#67—22no—
93C2×#67—22no—
94C2×#67—41no—
95C2×#67—41no—
96C2×#67—41no—
97C2×#67—41no—
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100C22×Dic3C2311yescharacteristic
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C2×D12—22no—
105C2×D12—22no—
106C2×D12—22no—
107C2×C4×D3—22no—
108C2×C4×D3—22no—
109C2×C4×D3—22no—
110C2×C4×D3—22no—
111C2×C4×D3—41no—
112C2×C4×D3—41no—
113C3×C4⋊C4—22no—
114C3×C4⋊C4D414yescomplemented
115C3×C22⋊C4—22no—
116C3×C22⋊C4—22no—
117C3×C22⋊C4—22no—
118#217—22no—
119#217—22no—
120#217—41no—
121#217D414yescomplemented
122(C2×D3)⋊C4—22no—
123(C2×D3)⋊C4—22no—
124(C2×D3)⋊C4—22no—
125(C2×D3)⋊C4—22no—
126(C2×D3)⋊C4—41no—
127(C2×D3)⋊C4—41no—
128C4⋊Dic3—22no—
129Dic3⋊C4—22no—
130Dic3⋊C4—41no—
131C4×Dic3—22no—
132C4×Dic3—22no—
133C22×D4—31no—
134C22×D4—31no—
135C22×D4—31no—
136C22×D4—31no—
137C22×D4—31no—
138C22×D4C2×D311yescharacteristic
139C23×C4—31no—
140C23×C4—31no—
141C23×C4C2×D311yescharacteristic
142#123—24no—
143#121—64no—
144C4×D4—64no—
145C4×D4—64no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4C2×D311yescharacteristic
150C2×C4⋊C4C2×D311yescharacteristic
151C2×C22⋊C4—31no—
152C2×C22⋊C4—31no—
153C2×C22⋊C4—31no—
154C2×C22⋊C4—31no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4—31no—
157C2×C22⋊C4—31no—
158C2×C22⋊C4—31no—
159C2×C22⋊C4—31no—
160C2×C22⋊C4—31no—
161C2×C22⋊C4C2×D311yescharacteristic
162C2×C22⋊C4C2×D311yescharacteristic
163C2×C22⋊C4C2×D311yescharacteristic
164C2×C42—31no—
165C2×C42—31no—
166#95—31no—
167#95—31no—
168#95—31no—
169#95—31no—
170#95—31no—
171#95—31no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3C2×D412yes—
186C23×C3C2411yescharacteristic
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—21no—
192C22×D3—21no—
193C22×D3—21no—
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—21no—
198C22×D3—21no—
199C22×D3#4112yes—
200C22×D3#4112yes—
201C3×D4—44no—
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×C4—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—21no—
214C2×C3×C4—44no—
215C2×C3×C4C2×D412yes—
216C2×C3×C4C2×D412yes—
217C2×C3×C4C2×D412yes—
218#67—28no—
219#67—44no—
220#67—44no—
221#67—44no—
222C2×Dic3—21no—
223C2×Dic3—21no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—21no—
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231C2×Dic3—21no—
232C2×Dic3—21no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3C2×D412yes—
238C2×Dic3C2×D412yes—
239C2×Dic3#4112yes—
240D12—28no—
241C4×D3—44no—
242C4×D3—44no—
243C24—31no—
244C24—31no—
245C24C22×D311yescharacteristic
246C24C22×D311yescharacteristic
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—41no—
250C2×D4—62no—
251C2×D4—62no—
252C2×D4—62no—
253C2×D4—62no—
254C2×D4—62no—
255C2×D4—62no—
256C2×D4—62no—
257C2×D4—62no—
258C2×D4—62no—
259C2×D4—62no—
260C2×D4—62no—
261C2×D4—121no—
262C2×D4—121no—
263C2×D4—121no—
264C2×D4—121no—
265C22×C4—22no—
266C22×C4—22no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—31no—
273C22×C4—41no—
274C22×C4—62no—
275C22×C4—62no—
276C22×C4—62no—
277C22×C4—62no—
278C22×C4—121no—
279C22×C4—121no—
280C22×C4C22×D311yescharacteristic
281C22×C4C22×D311yescharacteristic
282C22×C4C22×D311yescharacteristic
283C22×C4C22×D311yescharacteristic
284C22×C4C22×D311yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—62no—
287C4⋊C4—62no—
288C4⋊C4—121no—
289C4⋊C4#6714yescomplemented
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—34no—
294C22⋊C4—62no—
295C22⋊C4—62no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C42—62no—
304C42—62no—
305C22×C3—22no—
306C22×C3—22no—
307C22×C3—22no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3C22×D411yescharacteristic
320C22×C3C22×D411yescharacteristic
321C22×C3C2×#4111yescharacteristic
322C22×C3C2×#4111yescharacteristic
323C22×C3C2×#4111yescharacteristic
324C22×C3H(5,𝔽2)11yescharacteristic
325C22×C3H(5,𝔽2)11yescharacteristic
326C2×D3—22no—
327C2×D3—22no—
328C2×D3—22no—
329C2×D3—22no—
330C2×D3—22no—
331C2×D3—22no—
332C2×D3—41no—
333C2×D3—41no—
334C2×D3—41no—
335C2×D3—41no—
336C2×D3—41no—
337C2×D3—41no—
338C2×D3—41no—
339C2×D3—41no—
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—41no—
344C3×C4—41no—
345Dic3—22no—
346Dic3—22no—
347Dic3—22no—
348Dic3—41no—
349Dic3—41no—
350Dic3—41no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—21no—
356C23—21no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—21no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—32no—
365C23—32no—
366C23—61no—
367C23—61no—
368C23—61no—
369C23—61no—
370C23—61no—
371C23—61no—
372C23—61no—
373C23—61no—
374C23—61no—
375C23—61no—
376C23—61no—
377C23—61no—
378C23D3×D412yes—
379C23C23×D311yescharacteristic
380D4—44no—
381D4—68no—
382D4—68no—
383D4—124no—
384D4—124no—
385D4—124no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—32no—
399C2×C4—32no—
400C2×C4—32no—
401C2×C4—44no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—61no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—61no—
415C2×C4—61no—
416C2×C4—61no—
417C2×C4—124no—
418C2×C4—124no—
419C2×C4C2×#6712yes—
420C2×C4C2×#6712yes—
421C2×C4C2×#6712yes—
422C2×C3—42no—
423C2×C3—81no—
424C2×C3#52511yescharacteristic
425C2×C3#52511yescharacteristic
426C2×C3#53811yescharacteristic
427C2×C3D4211yescharacteristic
428C2×C3#54611yescharacteristic
429C2×C3#55011yescharacteristic
430C2×C3#55311yescharacteristic
431D3—42no—
432D3—42no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—62no—
448C22—62no—
449C22—62no—
450C22—62no—
451C22—62no—
452C22—62no—
453C22—121no—
454C22—121no—
455C22—121no—
456C22—121no—
457C22—121no—
458C22—121no—
459C22—121no—
460C22—121no—
461C22C2×D3×D411yescharacteristic
462C22#100411yescharacteristic
463C22D3×#4111yescharacteristic
464C22D3×#4111yescharacteristic
465C22D3×#4111yescharacteristic
466C22#100911yescharacteristic
467C22C22×#6711yescharacteristic
468C4—22no—
469C4—22no—
470C4—22no—
471C4—41no—
472C4—41no—
473C4—62no—
474C4—62no—
475C4—62no—
476C4—121no—
477C4—121no—
478C4—121no—
479C3#245611yescharacteristic, complemented, Sylow
480C2—42no—
481C2—81no—
482C2—122no—
483C2—122no—
484C2#565411yescharacteristic
485C2#565611yescharacteristic
486C2#565711yescharacteristic
487C2#566111yescharacteristic
488C2D4×#6711yescharacteristic
489C2#582811yescharacteristic
490C2#583411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#123C2
hypercenterC2×#123D3
hypocenterC3#2456
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×C3×#123C2semidirect
2. C2×#938C2semidirect
3. C2×C4×#67C2semidirect
4. C2×#893C2semidirect
5. #5223C2semidirect
6. #5213C2semidirect
7. #4994C2semidirect
8. #4989C2semidirect
9. #4980C2semidirect
10. #4960C2semidirect
11. #4960C2semidirect
12. #4957C2semidirect
13. #4956C2semidirect
14. #4944C2semidirect
15. #4944C2semidirect
56. C2×#123D3semidirect
114. C3×C4⋊C4D4semidirect
121. #217D4semidirect
289. C4⋊C4#67semidirect
479. C3#2456semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1232
C2×#9382
C2×C4×#672
C2×#8932
#52232
#52132
#49942
#49892
#49802
#49602
#49572
#49562
#49442
#24563