Finite Group Databasebeta

← #83182

Subgroups — #83182

Order 384.

734 conjugacy classes of subgroups.

The subgroup lattice has 532 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#127C211yesmaximal, characteristic, complemented
2C2×#939C211yesmaximal, characteristic, complemented
3C2×#939C211yesmaximal, characteristic, complemented
4C2×#935C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#5223C211yesmaximal, characteristic, complemented
7#5223C211yesmaximal, characteristic, complemented
8#5223C211yesmaximal, characteristic, complemented
9#5222C211yesmaximal, characteristic, complemented
10#5222C211yesmaximal, characteristic, complemented
11#5220C211yesmaximal, characteristic, complemented
12#5220C211yesmaximal, characteristic, complemented
13#5220C211yesmaximal, characteristic, complemented
14#4940C211yesmaximal, characteristic, complemented
15#4933C211yesmaximal, characteristic, complemented
16#2673—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×C3×D4C2211yescharacteristic
21C22×C3×D4C2211yescharacteristic
22C22×C3×D4C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C22×#67C2211yescharacteristic
25C22×#67C2211yescharacteristic
26C22×#67C2211yescharacteristic
27C23×Dic3C2211yescharacteristic
28C23×Dic3C2211yescharacteristic
29C3×#127—24no—
30C2×C3×C42C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
37C2×#217C2211yescharacteristic
38C2×#217C2211yescharacteristic
39C2×#217C2211yescharacteristic
40C2×#217C2211yescharacteristic
41#939—24no—
42#939—24no—
43#935—24no—
44C2×(C2×D3)⋊C4C2211yescharacteristic
45C2×(C2×D3)⋊C4C2211yescharacteristic
46C2×(C2×D3)⋊C4C2211yescharacteristic
47C2×Dic3⋊C4C2211yescharacteristic
48C2×Dic3⋊C4C2211yescharacteristic
49C2×Dic3⋊C4C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×#127D311yescharacteristic, complemented
57C2×#123—31no—
58C2×#121—31no—
59C2×#121—31no—
60#394—31no—
61#394—31no—
62#392—31no—
63#392—31no—
64#392—31no—
65#392—31no—
66#388—31no—
67#387—31no—
68#386—31no—
69#386—31no—
70#386—31no—
71C24×C3C2311yescharacteristic
72C24×C3C2311yescharacteristic
73C24×C3C2311yescharacteristic
74C24×C3C2311yescharacteristic
75C23×D3C2311yescharacteristic
76C2×C3×D4—22no—
77C2×C3×D4—22no—
78C2×C3×D4—22no—
79C2×C3×D4—22no—
80C2×C3×D4—22no—
81C2×C3×D4—22no—
82C2×C3×D4—22no—
83C2×C3×D4—22no—
84C2×C3×D4—22no—
85C2×C3×D4—41no—
86C2×C3×D4—41no—
87C2×C3×D4—41no—
88C2×C3×D4—41no—
89C2×C3×D4—41no—
90C2×C3×D4—41no—
91C2×C3×D4—41no—
92C2×C3×D4—41no—
93C2×C3×D4D414yescomplemented
94C22×C3×C4C2311yescharacteristic
95C22×C3×C4C2311yescharacteristic
96C22×C3×C4C2311yescharacteristic
97C2×#67—22no—
98C2×#67—22no—
99C2×#67—22no—
100C2×#67—22no—
101C2×#67—41no—
102C2×#67—41no—
103C2×#67—41no—
104C2×#67—41no—
105C2×#67—41no—
106C2×#67—41no—
107C2×#67—41no—
108C2×#67—41no—
109C22×Dic3—22no—
110C22×Dic3—22no—
111C22×Dic3—22no—
112C22×Dic3—41no—
113C22×Dic3—41no—
114C22×Dic3—41no—
115C22×Dic3C2311yescharacteristic
116C22×Dic3C2311yescharacteristic
117C22×Dic3C2311yescharacteristic
118C22×Dic3C2311yescharacteristic
119C22×Dic3C2311yescharacteristic
120C22×Dic3C2311yescharacteristic
121C22×Dic3C2311yescharacteristic
122C3×C42—22no—
123#217—22no—
124#217—22no—
125#217—22no—
126#217—22no—
127#217—22no—
128#217—41no—
129#217—41no—
130#217—41no—
131#217—41no—
132#217—41no—
133(C2×D3)⋊C4—22no—
134(C2×D3)⋊C4—22no—
135(C2×D3)⋊C4—22no—
136Dic3⋊C4—22no—
137Dic3⋊C4—22no—
138Dic3⋊C4D414yescomplemented
139C22×D4—31no—
140C22×D4—31no—
141C22×D4—31no—
142C22×D4—31no—
143C22×D4C2×D311yescharacteristic
144C22×D4C2×D311yescharacteristic
145C22×D4C2×D311yescharacteristic
146C22×D4C2×D311yescharacteristic
147C22×D4C2×D311yescharacteristic
148C22×D4C2×D311yescharacteristic
149C23×C4—31no—
150C23×C4—31no—
151#127—24no—
152#123—64no—
153#121—64no—
154#121—64no—
155C2×C4⋊C4—31no—
156C2×C4⋊C4—31no—
157C2×C4⋊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⋊C4—31no—
165C2×C22⋊C4—31no—
166C2×C22⋊C4—31no—
167C2×C22⋊C4—31no—
168C2×C22⋊C4—31no—
169C2×C22⋊C4—31no—
170C2×C22⋊C4—31no—
171C2×C42C2×D311yescharacteristic
172#95—31no—
173#95—31no—
174#95—31no—
175#95—31no—
176#95—31no—
177#95—31no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3—21no—
182C23×C3—21no—
183C23×C3—21no—
184C23×C3—21no—
185C23×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3—21no—
190C23×C3—21no—
191C23×C3—21no—
192C23×C3—21no—
193C23×C3—21no—
194C23×C3—21no—
195C23×C3—21no—
196C23×C3—21no—
197C23×C3—21no—
198C23×C3—21no—
199C23×C3—21no—
200C23×C3—21no—
201C23×C3—21no—
202C23×C3—21no—
203C23×C3—21no—
204C23×C3C2×D412yes—
205C23×C3C2×D412yes—
206C23×C3C2411yescharacteristic
207C22×D3—21no—
208C22×D3—21no—
209C22×D3—21no—
210C22×D3—21no—
211C22×D3—21no—
212C22×D3—21no—
213C22×D3—21no—
214C3×D4—44no—
215C3×D4—44no—
216C3×D4—44no—
217C3×D4—44no—
218C3×D4—44no—
219C3×D4—44no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—21no—
226C2×C3×C4C2×D412yes—
227C2×C3×C4C2×D412yes—
228C2×C3×C4#4112yes—
229#67—44no—
230#67—44no—
231#67—44no—
232#67—44no—
233C2×Dic3—21no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240C2×Dic3—21no—
241C2×Dic3—21no—
242C2×Dic3—21no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246C2×Dic3—21no—
247C2×Dic3—21no—
248C2×Dic3—21no—
249C2×Dic3—21no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—44no—
253C2×Dic3—44no—
254C2×Dic3C2×D412yes—
255C2×Dic3C2×D412yes—
256C24—31no—
257C24C22×D311yescharacteristic
258C24C22×D311yescharacteristic
259C24C22×D311yescharacteristic
260C24C22×D311yescharacteristic
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—62no—
279C2×D4—62no—
280C2×D4—62no—
281C2×D4—62no—
282C2×D4—121no—
283C2×D4—121no—
284C2×D4—121no—
285C2×D4—121no—
286C2×D4—121no—
287C2×D4—121no—
288C2×D4—121no—
289C2×D4—121no—
290C2×D4#6714yescomplemented
291C22×C4—31no—
292C22×C4—31no—
293C22×C4—31no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—31no—
297C22×C4—31no—
298C22×C4—62no—
299C22×C4—62no—
300C22×C4—62no—
301C22×C4—121no—
302C22×C4—121no—
303C22×C4—121no—
304C22×C4C22×D311yescharacteristic
305C22×C4C22×D311yescharacteristic
306C22×C4C22×D311yescharacteristic
307C4⋊C4—34no—
308C4⋊C4—62no—
309C4⋊C4—62no—
310C22⋊C4—62no—
311C22⋊C4—62no—
312C22⋊C4—62no—
313C22⋊C4—62no—
314C22⋊C4—62no—
315C22⋊C4—62no—
316C22⋊C4—62no—
317C22⋊C4—62no—
318C22⋊C4—121no—
319C22⋊C4—121no—
320C22⋊C4—121no—
321C22⋊C4—121no—
322C22⋊C4—121no—
323C42—22no—
324C22×C3—22no—
325C22×C3—22no—
326C22×C3—22no—
327C22×C3—22no—
328C22×C3—22no—
329C22×C3—22no—
330C22×C3—41no—
331C22×C3—41no—
332C22×C3—41no—
333C22×C3—41no—
334C22×C3—41no—
335C22×C3—41no—
336C22×C3—41no—
337C22×C3—41no—
338C22×C3—41no—
339C22×C3—41no—
340C22×C3—41no—
341C22×C3—41no—
342C22×C3—41no—
343C22×C3—41no—
344C22×C3—41no—
345C22×C3—41no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—41no—
349C22×C3—41no—
350C22×C3—41no—
351C22×C3—41no—
352C22×C3C22×D411yescharacteristic
353C22×C3C22×D411yescharacteristic
354C22×C3C2×#4111yescharacteristic
355C22×C3H(5,𝔽2)11yescharacteristic
356C22×C3H(5,𝔽2)11yescharacteristic
357C22×C3H(5,𝔽2)11yescharacteristic
358C22×C3H(5,𝔽2)11yescharacteristic
359C2×D3—41no—
360C2×D3—41no—
361C2×D3—41no—
362C2×D3—41no—
363C2×D3—41no—
364C2×D3—41no—
365C2×D3—41no—
366C3×C4—22no—
367C3×C4—22no—
368C3×C4—22no—
369Dic3—22no—
370Dic3—22no—
371Dic3—41no—
372Dic3—41no—
373Dic3—41no—
374Dic3—41no—
375Dic3—41no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—21no—
385C23—21no—
386C23—21no—
387C23—21no—
388C23—21no—
389C23—21no—
390C23—21no—
391C23—21no—
392C23—21no—
393C23—21no—
394C23—21no—
395C23—21no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—21no—
401C23—21no—
402C23—61no—
403C23—61no—
404C23—61no—
405C23—61no—
406C23—61no—
407C23—61no—
408C23—61no—
409C23C2×#6712yes—
410C23C2×#6712yes—
411C23C23×D311yescharacteristic
412D4—44no—
413D4—44no—
414D4—44no—
415D4—44no—
416D4—44no—
417D4—44no—
418D4—124no—
419D4—124no—
420D4—124no—
421D4—124no—
422C2×C4—21no—
423C2×C4—21no—
424C2×C4—21no—
425C2×C4—21no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—32no—
429C2×C4—32no—
430C2×C4—61no—
431C2×C4—61no—
432C2×C4—61no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—61no—
437C2×C4—61no—
438C2×C4—61no—
439C2×C4—61no—
440C2×C4—61no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—61no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—61no—
449C2×C4—124no—
450C2×C4—124no—
451C2×C4D3×D412yes—
452C2×C4#23712yes—
453C2×C4C2×#6712yes—
454C2×C3—42no—
455C2×C3—42no—
456C2×C3—81no—
457C2×C3—81no—
458C2×C3#53411yescharacteristic
459C2×C3#53511yescharacteristic
460C2×C3#54011yescharacteristic
461C2×C3D4211yescharacteristic
462C2×C3#54611yescharacteristic
463C2×C3#55011yescharacteristic
464C2×C3#56011yescharacteristic
465D3—81no—
466C22—22no—
467C22—22no—
468C22—22no—
469C22—22no—
470C22—22no—
471C22—22no—
472C22—41no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22—41no—
488C22—41no—
489C22—41no—
490C22—41no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—121no—
495C22—121no—
496C22—121no—
497C22—121no—
498C22—121no—
499C22—121no—
500C22—121no—
501C22C2×D3×D411yescharacteristic
502C22C2×#23711yescharacteristic
503C22#100411yescharacteristic
504C22#100411yescharacteristic
505C22#100411yescharacteristic
506C22#100411yescharacteristic
507C22C22×#6711yescharacteristic
508C4—22no—
509C4—22no—
510C4—22no—
511C4—62no—
512C4—62no—
513C4—121no—
514C4—121no—
515C4—121no—
516C4—121no—
517C4—121no—
518C3#267311yescharacteristic, complemented, Sylow
519C2—42no—
520C2—42no—
521C2—81no—
522C2—81no—
523C2—241no—
524C2#571511yescharacteristic
525C2#571811yescharacteristic
526C2#571911yescharacteristic
527C2#572011yescharacteristic
528C2D4×#6711yescharacteristic
529C2#580411yescharacteristic
530C2#580611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#127C2
hypercenterC2×#127D3
hypocenterC3#2673
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#127

Complements

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

NH ≅ G/Ntype
1. C2×C3×#127C2semidirect
2. C2×#939C2semidirect
3. C2×#939C2semidirect
4. C2×#935C2semidirect
5. #5223C2semidirect
6. #5223C2semidirect
7. #5223C2semidirect
8. #5223C2semidirect
9. #5222C2semidirect
10. #5222C2semidirect
11. #5220C2semidirect
12. #5220C2semidirect
13. #5220C2semidirect
14. #4940C2semidirect
15. #4933C2semidirect
56. C2×#127D3semidirect
93. C2×C3×D4D4semidirect
138. Dic3⋊C4D4semidirect
290. C2×D4#67semidirect
518. C3#2673semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1272
C2×#9392
C2×#9352
#52232
#52222
#52202
#49402
#49332
#26733