Finite Group Databasebeta

← #82873

Subgroups — #82873

Order 384.

668 conjugacy classes of subgroups.

The subgroup lattice has 490 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×#930C211yesmaximal, characteristic, complemented
4C2×#894C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#5210C211yesmaximal, characteristic, complemented
7#4994C211yesmaximal, characteristic, complemented
8#4989C211yesmaximal, characteristic, complemented
9#4984C211yesmaximal, characteristic, complemented
10#4961C211yesmaximal, characteristic, complemented
11#4961C211yesmaximal, characteristic, complemented
12#4960C211yesmaximal, characteristic, complemented
13#4957C211yesmaximal, characteristic, complemented
14#4957C211yesmaximal, characteristic, complemented
15#4948C211yesmaximal, characteristic, complemented
16#2682—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×D12C2211yescharacteristic
24C22×C4×D3C2211yescharacteristic
25C3×#123—24no—
26C2×C3×C4⋊C4C2211yescharacteristic
27C2×C3×C4⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C22⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35#938—24no—
36#930—24no—
37C2×(C2×D3)⋊C4C2211yescharacteristic
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×C4⋊Dic3C2211yescharacteristic
46C2×C4⋊Dic3C2211yescharacteristic
47C2×Dic3⋊C4C2211yescharacteristic
48C2×Dic3⋊C4C2211yescharacteristic
49C2×Dic3⋊C4C2211yescharacteristic
50C2×C4×Dic3C2211yescharacteristic
51#894—24no—
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×#123—31no—
57C2×#123D311yescharacteristic, complemented
58C2×#121—31no—
59C2×#121—31no—
60#396—31no—
61#396—31no—
62#394—31no—
63#394—31no—
64#392—31no—
65#392—31no—
66#390—31no—
67#390—31no—
68#389—31no—
69#385—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—41no—
90C2×#67—41no—
91C2×#67—41no—
92C2×#67—41no—
93C2×#67—41no—
94C2×#67—41no—
95C2×#67—41no—
96C2×#67—41no—
97C2×#67D414yescomplemented
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100C22×Dic3C2311yescharacteristic
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C2×D12—22no—
105C2×D12—22no—
106C2×D12—41no—
107C2×C4×D3—22no—
108C2×C4×D3—22no—
109C2×C4×D3—41no—
110C3×C4⋊C4—22no—
111C3×C4⋊C4D414yescomplemented
112C3×C22⋊C4—22no—
113C3×C22⋊C4—22no—
114C3×C22⋊C4—22no—
115#217—22no—
116#217—22no—
117#217—22no—
118#217—41no—
119(C2×D3)⋊C4—22no—
120(C2×D3)⋊C4—22no—
121(C2×D3)⋊C4—22no—
122(C2×D3)⋊C4—22no—
123(C2×D3)⋊C4—22no—
124(C2×D3)⋊C4—22no—
125(C2×D3)⋊C4—41no—
126C4⋊Dic3—22no—
127C4⋊Dic3—22no—
128C4⋊Dic3—41no—
129Dic3⋊C4—22no—
130Dic3⋊C4—41no—
131Dic3⋊C4—41no—
132C4×Dic3—41no—
133C22×D4—31no—
134C22×D4—31no—
135C22×D4—31no—
136C22×D4—31no—
137C22×D4—31no—
138C22×D4C2×D311yescharacteristic
139C23×C4—31no—
140C23×C4C2×D311yescharacteristic
141#123—24no—
142#123—64no—
143#121—64no—
144#121—64no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4C2×D311yescharacteristic
152C2×C4⋊C4C2×D311yescharacteristic
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⋊C4—31no—
162C2×C22⋊C4—31no—
163C2×C22⋊C4—31no—
164C2×C22⋊C4C2×D311yescharacteristic
165C2×C22⋊C4C2×D311yescharacteristic
166C2×C22⋊C4C2×D311yescharacteristic
167C2×C42—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—21no—
200C22×D3C2×D412yes—
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—44no—
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×Dic3—21no—
238C2×Dic3C2×D412yes—
239C2×Dic3#4112yes—
240D12—44no—
241C4×D3—44no—
242C24—31no—
243C24—31no—
244C24C22×D311yescharacteristic
245C24C22×D311yescharacteristic
246C2×D4—22no—
247C2×D4—22no—
248C2×D4—34no—
249C2×D4—41no—
250C2×D4—62no—
251C2×D4—62no—
252C2×D4—62no—
253C2×D4—62no—
254C2×D4—62no—
255C2×D4—121no—
256C2×D4—121no—
257C2×D4—121no—
258C2×D4—121no—
259C2×D4—121no—
260C2×D4—121no—
261C2×D4—121no—
262C2×D4—121no—
263C2×D4—121no—
264C22×C4—22no—
265C22×C4—22no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—31no—
272C22×C4—41no—
273C22×C4—62no—
274C22×C4—62no—
275C22×C4—121no—
276C22×C4C22×D311yescharacteristic
277C22×C4C22×D311yescharacteristic
278C22×C4C22×D311yescharacteristic
279C22×C4C22×D311yescharacteristic
280C22×C4C22×D311yescharacteristic
281C4⋊C4—22no—
282C4⋊C4—62no—
283C4⋊C4—62no—
284C4⋊C4—62no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C4⋊C4—121no—
288C4⋊C4#6714yescomplemented
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—62no—
293C22⋊C4—62no—
294C22⋊C4—62no—
295C22⋊C4—62no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—62no—
301C22⋊C4—121no—
302C22⋊C4—121no—
303C42—121no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3—22no—
307C22×C3—41no—
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×C3C22×D411yescharacteristic
319C22×C3C22×D411yescharacteristic
320C22×C3C2×#4111yescharacteristic
321C22×C3H(5,𝔽2)11yescharacteristic
322C22×C3H(5,𝔽2)11yescharacteristic
323C22×C3H(5,𝔽2)11yescharacteristic
324C22×C32−1+411yescharacteristic
325C2×D3—22no—
326C2×D3—22no—
327C2×D3—22no—
328C2×D3—41no—
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—41no—
332C2×D3—41no—
333C2×D3—41no—
334C2×D3—41no—
335C2×D3—41no—
336C2×D3—41no—
337C2×D3—41no—
338C2×D3—41no—
339C3×C4—22no—
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—41no—
343C3×C4—41no—
344Dic3—22no—
345Dic3—22no—
346Dic3—41no—
347Dic3—41no—
348Dic3—41no—
349Dic3—41no—
350C23—21no—
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—32no—
364C23—61no—
365C23—61no—
366C23—61no—
367C23—61no—
368C23—61no—
369C23—61no—
370C23—61no—
371C23—61no—
372C23—61no—
373C23—61no—
374C23—61no—
375C23—61no—
376C23—61no—
377C23D3×D412yes—
378C23C23×D311yescharacteristic
379D4—44no—
380D4—124no—
381D4—124no—
382D4—124no—
383D4—124no—
384D4—124no—
385C2×C4—21no—
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—32no—
398C2×C4—32no—
399C2×C4—44no—
400C2×C4—61no—
401C2×C4—61no—
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—124no—
417C2×C4C2×#6712yes—
418C2×C4C2×#6712yes—
419C2×C4C2×#6712yes—
420C2×C3—42no—
421C2×C3—81no—
422C2×C3#53411yescharacteristic
423C2×C3#53711yescharacteristic
424C2×C3#53811yescharacteristic
425C2×C3D4211yescharacteristic
426C2×C3#54711yescharacteristic
427C2×C3#55011yescharacteristic
428C2×C3#56211yescharacteristic
429D3—42no—
430D3—81no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—62no—
446C22—62no—
447C22—62no—
448C22—121no—
449C22—121no—
450C22—121no—
451C22—121no—
452C22—121no—
453C22—121no—
454C22—121no—
455C22—121no—
456C22—121no—
457C22—121no—
458C22—121no—
459C22C2×D3×D411yescharacteristic
460C22#100411yescharacteristic
461C22D3×#4111yescharacteristic
462C22#100911yescharacteristic
463C22#100911yescharacteristic
464C22#101011yescharacteristic
465C22C22×#6711yescharacteristic
466C4—22no—
467C4—22no—
468C4—22no—
469C4—41no—
470C4—41no—
471C4—62no—
472C4—62no—
473C4—121no—
474C4—121no—
475C4—121no—
476C4—121no—
477C3#268211yescharacteristic, complemented, Sylow
478C2—42no—
479C2—81no—
480C2—122no—
481C2—241no—
482C2#565311yescharacteristic
483C2#565511yescharacteristic
484C2#565711yescharacteristic
485C2#566711yescharacteristic
486C2D4×#6711yescharacteristic
487C2#583311yescharacteristic
488C2#583411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#123C2
hypercenterC2×#123D3
hypocenterC3#2682
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×#930C2semidirect
4. C2×#894C2semidirect
5. #5223C2semidirect
6. #5210C2semidirect
7. #4994C2semidirect
8. #4989C2semidirect
9. #4984C2semidirect
10. #4961C2semidirect
11. #4961C2semidirect
12. #4960C2semidirect
13. #4957C2semidirect
14. #4957C2semidirect
15. #4948C2semidirect
57. C2×#123D3semidirect
97. C2×#67D4semidirect
111. C3×C4⋊C4D4semidirect
288. C4⋊C4#67semidirect
477. C3#2682semidirect

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×#9302
C2×#8942
#52232
#52102
#49942
#49892
#49842
#49612
#49602
#49572
#49482
#26823