Finite Group Databasebeta

← #82670

Subgroups — #82670

Order 384.

724 conjugacy classes of subgroups.

The subgroup lattice has 494 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#121C211yesmaximal, characteristic, complemented
2C2×#939C211yesmaximal, characteristic, complemented
3C2×#939C211yesmaximal, characteristic, complemented
4C2×D4×Dic3C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#5220C211yesmaximal, characteristic, complemented
7#5220C211yesmaximal, characteristic, complemented
8#5213C211yesmaximal, characteristic, complemented
9C2×C3⋊#95C211yesmaximal, characteristic, complemented
10#4990C211yesmaximal, characteristic, complemented
11#4982C211yesmaximal, characteristic, complemented
12#4961C211yesmaximal, characteristic, complemented
13#4958C211yesmaximal, characteristic, complemented
14#4952C211yesmaximal, characteristic, complemented
15#4945C211yesmaximal, characteristic, complemented
16#2543—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C23×C3×C4C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C23×Dic3C2211yescharacteristic
25C23×Dic3C2211yescharacteristic
26C23×Dic3C2211yescharacteristic
27C3×#121—24no—
28C2×C3×C4⋊C4C2211yescharacteristic
29C2×C3×C22⋊C4C2211yescharacteristic
30C2×C3×C22⋊C4C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
37#939—24no—
38#939—24no—
39D4×Dic3—24no—
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
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×#121—31no—
57C2×#121—31no—
58C2×#121D311yescharacteristic, complemented
59C2×C4×D4—31no—
60#398—31no—
61#397—31no—
62#394—31no—
63#392—31no—
64#388—31no—
65#388—31no—
66#386—31no—
67#386—31no—
68#386—31no—
69#386—31no—
70C2×#95—31no—
71C24×C3C2311yescharacteristic
72C24×C3C2311yescharacteristic
73C24×C3C2311yescharacteristic
74C23×D3C2311yescharacteristic
75C2×C3×D4—22no—
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—41no—
83C2×C3×D4D414yescomplemented
84C22×C3×C4—22no—
85C22×C3×C4—22no—
86C22×C3×C4—41no—
87C22×C3×C4C2311yescharacteristic
88C22×C3×C4C2311yescharacteristic
89C22×C3×C4C2311yescharacteristic
90C22×C3×C4C2311yescharacteristic
91C2×#67—22no—
92C2×#67—22no—
93C2×#67—22no—
94C2×#67—22no—
95C2×#67—22no—
96C2×#67—22no—
97C2×#67—41no—
98C2×#67—41no—
99C2×#67—41no—
100C22×Dic3—22no—
101C22×Dic3—22no—
102C22×Dic3—22no—
103C22×Dic3—41no—
104C22×Dic3—41no—
105C22×Dic3—41no—
106C22×Dic3—41no—
107C22×Dic3—41no—
108C22×Dic3D414yescomplemented
109C22×Dic3C2311yescharacteristic
110C22×Dic3C2311yescharacteristic
111C22×Dic3C2311yescharacteristic
112C22×Dic3C2311yescharacteristic
113C22×Dic3C2311yescharacteristic
114C22×Dic3C2311yescharacteristic
115C22×Dic3C2311yescharacteristic
116C3×C4⋊C4—22no—
117C3×C22⋊C4—22no—
118C3×C22⋊C4—22no—
119#217—22no—
120#217—22no—
121#217—22no—
122#217—22no—
123#217—22no—
124#217—41no—
125(C2×D3)⋊C4—22no—
126(C2×D3)⋊C4—22no—
127(C2×D3)⋊C4—22no—
128(C2×D3)⋊C4—41no—
129C4⋊Dic3—22no—
130Dic3⋊C4—22no—
131Dic3⋊C4—22no—
132C4×Dic3—22no—
133C22×D4—31no—
134C22×D4—31no—
135C22×D4—31no—
136C22×D4C2×D311yescharacteristic
137C22×D4C2×D311yescharacteristic
138C22×D4C2×D311yescharacteristic
139C23×C4—31no—
140C23×C4—31no—
141C23×C4—31no—
142C23×C4C2×D311yescharacteristic
143#121—24no—
144#121—64no—
145#121—64no—
146C4×D4—64no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—31no—
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×C42—31no—
164#95—31no—
165#95—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×C3—21no—
186C23×C3—21no—
187C23×C3—21no—
188C23×C3—21no—
189C23×C3C2×D412yes—
190C23×C3C2×D412yes—
191C23×C3C2×D412yes—
192C23×C3#4112yes—
193C23×C3C2411yescharacteristic
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—21no—
198C22×D3—21no—
199C22×D3—21no—
200C22×D3—21no—
201C3×D4—28no—
202C3×D4—44no—
203C3×D4—44no—
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×C4#4112yes—
217#67—44no—
218#67—44no—
219#67—44no—
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—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×Dic3—44no—
239C2×Dic3—44no—
240C2×Dic3—44no—
241C2×Dic3C2×D412yes—
242C2×Dic3C2×D412yes—
243C2×Dic3#4112yes—
244C24—31no—
245C24C22×D311yescharacteristic
246C24C22×D311yescharacteristic
247C24C22×D311yescharacteristic
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—22no—
255C2×D4—41no—
256C2×D4—62no—
257C2×D4—62no—
258C2×D4—62no—
259C2×D4—62no—
260C2×D4—62no—
261C2×D4—62no—
262C2×D4—121no—
263C2×D4—121no—
264C2×D4—121no—
265C2×D4#6714yescomplemented
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—31no—
271C22×C4—34no—
272C22×C4—31no—
273C22×C4—31no—
274C22×C4—31no—
275C22×C4—31no—
276C22×C4—41no—
277C22×C4—62no—
278C22×C4—62no—
279C22×C4—62no—
280C22×C4—121no—
281C22×C4—121no—
282C22×C4—121no—
283C22×C4—121no—
284C22×C4—121no—
285C22×C4C22×D311yescharacteristic
286C22×C4C22×D311yescharacteristic
287C22×C4C22×D311yescharacteristic
288C22×C4C22×D311yescharacteristic
289C4⋊C4—22no—
290C4⋊C4—62no—
291C4⋊C4—62no—
292C4⋊C4—62no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—62no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C22⋊C4—62no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C42—62no—
306C22×C3—22no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—22no—
312C22×C3—22no—
313C22×C3—22no—
314C22×C3—22no—
315C22×C3—41no—
316C22×C3—41no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3#12114yes—
327C22×C3C22×D411yescharacteristic
328C22×C3C22×D411yescharacteristic
329C22×C3C2×#4111yescharacteristic
330C22×C3C2×#4111yescharacteristic
331C22×C3C2×#4111yescharacteristic
332C22×C3H(5,𝔽2)11yescharacteristic
333C22×C3H(5,𝔽2)11yescharacteristic
334C2×D3—41no—
335C2×D3—41no—
336C2×D3—41no—
337C2×D3—41no—
338C2×D3—41no—
339C2×D3—41no—
340C2×D3—41no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—41no—
344C3×C4—41no—
345Dic3—22no—
346Dic3—22no—
347Dic3—22no—
348Dic3—41no—
349Dic3—41no—
350Dic3—41no—
351Dic3—41no—
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—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—61no—
370C23—61no—
371C23—61no—
372C23—61no—
373C23—61no—
374C23—61no—
375C23—61no—
376C23D3×D412yes—
377C23#23712yes—
378C23C2×#6712yes—
379C23C2×#6712yes—
380C23C23×D311yescharacteristic
381D4—28no—
382D4—44no—
383D4—44no—
384D4—124no—
385D4—124no—
386D4—124no—
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—32no—
400C2×C4—44no—
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—61no—
417C2×C4—61no—
418C2×C4—61no—
419C2×C4—124no—
420C2×C4—124no—
421C2×C4—124no—
422C2×C4#23712yes—
423C2×C4C2×#6712yes—
424C2×C3—24no—
425C2×C3—42no—
426C2×C3—81no—
427C2×C3C2×#12111yescharacteristic
428C2×C3#53211yescharacteristic
429C2×C3#53411yescharacteristic
430C2×C3#54611yescharacteristic
431C2×C3#54611yescharacteristic
432C2×C3#55311yescharacteristic
433C2×C3#55311yescharacteristic
434D3—81no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—22no—
440C22—22no—
441C22—22no—
442C22—22no—
443C22—22no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—41no—
455C22—121no—
456C22—121no—
457C22—121no—
458C22—121no—
459C22—121no—
460C22—121no—
461C22—121no—
462C22#93914yes—
463C22C2×D3×D411yescharacteristic
464C22C2×#23711yescharacteristic
465C22C2×#23711yescharacteristic
466C22#100411yescharacteristic
467C22D3×#4111yescharacteristic
468C22#100911yescharacteristic
469C22C22×#6711yescharacteristic
470C4—22no—
471C4—22no—
472C4—41no—
473C4—41no—
474C4—62no—
475C4—62no—
476C4—62no—
477C4—121no—
478C4—121no—
479C4—121no—
480C4—121no—
481C3#254311yescharacteristic, complemented, Sylow
482C2—24no—
483C2—42no—
484C2—81no—
485C2—241no—
486C2#559611yescharacteristic
487C2#560111yescharacteristic
488C2#560511yescharacteristic
489C2#561811yescharacteristic
490C2C2×#93911yescharacteristic
491C2#580611yescharacteristic
492C2#582911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#121C2
hypercenterC2×#121D3
hypocenterC3#2543
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×#121C2semidirect
2. C2×#939C2semidirect
3. C2×#939C2semidirect
4. C2×D4×Dic3C2semidirect
5. #5223C2semidirect
6. #5220C2semidirect
7. #5220C2semidirect
8. #5213C2semidirect
9. C2×C3⋊#95C2semidirect
10. #4990C2semidirect
11. #4982C2semidirect
12. #4961C2semidirect
13. #4958C2semidirect
14. #4952C2semidirect
15. #4945C2semidirect
58. C2×#121D3semidirect
83. C2×C3×D4D4semidirect
108. C22×Dic3D4semidirect
265. C2×D4#67semidirect
481. C3#2543semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×#9392
C2×D4×Dic32
#52232
#52202
#52132
C2×C3⋊#952
#49902
#49822
#49612
#49582
#49522
#49452
#25433