Finite Group Databasebeta

← #82716

Subgroups — #82716

Order 384.

682 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×#121C211yesmaximal, characteristic, complemented
2C2×#939C211yesmaximal, characteristic, complemented
3C2×#938C211yesmaximal, characteristic, complemented
4C2×#883C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#5222C211yesmaximal, characteristic, complemented
7#5222C211yesmaximal, characteristic, complemented
8#5220C211yesmaximal, characteristic, complemented
9#5213C211yesmaximal, characteristic, complemented
10#5212C211yesmaximal, characteristic, complemented
11#4989C211yesmaximal, characteristic, complemented
12#4982C211yesmaximal, characteristic, complemented
13#4961C211yesmaximal, characteristic, complemented
14#4950C211yesmaximal, characteristic, complemented
15#4949C211yesmaximal, characteristic, complemented
16#2671—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C23×C3×C4C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C22×#67C2211yescharacteristic
24C23×Dic3C2211yescharacteristic
25C22×C4×D3C2211yescharacteristic
26C3×#121—24no—
27C2×C3×C4⋊C4C2211yescharacteristic
28C2×C3×C22⋊C4C2211yescharacteristic
29C2×C3×C22⋊C4C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
37C2×#217C2211yescharacteristic
38#939—24no—
39#938—24no—
40C2×(C2×D3)⋊C4C2211yescharacteristic
41C2×(C2×D3)⋊C4C2211yescharacteristic
42C2×(C2×D3)⋊C4C2211yescharacteristic
43C2×C4⋊Dic3C2211yescharacteristic
44C2×C4⋊Dic3C2211yescharacteristic
45C2×C4⋊Dic3C2211yescharacteristic
46C2×Dic3⋊C4C2211yescharacteristic
47C2×Dic3⋊C4C2211yescharacteristic
48C2×Dic3⋊C4C2211yescharacteristic
49#883—24no—
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×#123—31no—
57C2×#121—31no—
58C2×#121—31no—
59C2×#121D311yescharacteristic, complemented
60#399—31no—
61#398—31no—
62#394—31no—
63#394—31no—
64#394—31no—
65#392—31no—
66#386—31no—
67#386—31no—
68#385—31no—
69#385—31no—
70#380—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—41no—
82C2×C3×D4—41no—
83C2×C3×D4—41no—
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—41no—
95C2×#67—41no—
96C2×#67—41no—
97C2×#67—41no—
98C2×#67—41no—
99C2×#67D414yescomplemented
100C22×Dic3—22no—
101C22×Dic3—41no—
102C22×Dic3—41no—
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C22×Dic3C2311yescharacteristic
106C22×Dic3C2311yescharacteristic
107C22×Dic3C2311yescharacteristic
108C22×Dic3C2311yescharacteristic
109C22×Dic3C2311yescharacteristic
110C2×C4×D3—22no—
111C2×C4×D3—22no—
112C2×C4×D3—41no—
113C3×C4⋊C4—22no—
114C3×C22⋊C4—22no—
115C3×C22⋊C4D414yescomplemented
116#217—22no—
117#217—22no—
118#217—22no—
119#217—22no—
120#217—41no—
121#217—41no—
122#217—41no—
123#217—41no—
124(C2×D3)⋊C4—22no—
125(C2×D3)⋊C4—22no—
126(C2×D3)⋊C4—22no—
127C4⋊Dic3—22no—
128C4⋊Dic3—22no—
129C4⋊Dic3—41no—
130Dic3⋊C4—22no—
131Dic3⋊C4—22no—
132Dic3⋊C4—41no—
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×C4C2×D311yescharacteristic
142#123—64no—
143#121—24no—
144#121—64no—
145#121—64no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
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
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×C3—21no—
190C23×C3—21no—
191C23×C3C2×D412yes—
192C23×C3C2×D412yes—
193C23×C3C2411yescharacteristic
194C22×D3—21no—
195C22×D3—21no—
196C22×D3—21no—
197C22×D3—21no—
198C22×D3—21no—
199C22×D3—21no—
200C22×D3C2×D412yes—
201C3×D4—44no—
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—44no—
214C2×C3×C4C2×D412yes—
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—21no—
239C2×Dic3—21no—
240C2×Dic3—44no—
241C2×Dic3C2×D412yes—
242C4×D3—44no—
243C24—31no—
244C24C22×D311yescharacteristic
245C24C22×D311yescharacteristic
246C24C22×D311yescharacteristic
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—22no—
252C2×D4—22no—
253C2×D4—34no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—62no—
258C2×D4—62no—
259C2×D4—62no—
260C2×D4—121no—
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—31no—
274C22×C4—41no—
275C22×C4—62no—
276C22×C4—62no—
277C22×C4—62no—
278C22×C4—121no—
279C22×C4—121no—
280C22×C4—121no—
281C22×C4C22×D311yescharacteristic
282C22×C4C22×D311yescharacteristic
283C22×C4C22×D311yescharacteristic
284C22×C4C22×D311yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—62no—
287C4⋊C4—62no—
288C4⋊C4—62no—
289C4⋊C4—62no—
290C4⋊C4—121no—
291C4⋊C4—121no—
292C22⋊C4—22no—
293C22⋊C4—62no—
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—
303C22⋊C4—121no—
304C22⋊C4#6714yescomplemented
305C22×C3—22no—
306C22×C3—22no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—22no—
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×C3—41no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3C22×D411yescharacteristic
327C22×C3C22×D411yescharacteristic
328C22×C3C2×#4111yescharacteristic
329C22×C3H(5,𝔽2)11yescharacteristic
330C22×C3H(5,𝔽2)11yescharacteristic
331C22×C3H(5,𝔽2)11yescharacteristic
332C22×C32−1+411yescharacteristic
333C2×D3—22no—
334C2×D3—22no—
335C2×D3—22no—
336C2×D3—41no—
337C2×D3—41no—
338C2×D3—41no—
339C2×D3—41no—
340C3×C4—22no—
341C3×C4—22no—
342C3×C4—22no—
343C3×C4—41no—
344Dic3—22no—
345Dic3—41no—
346Dic3—41no—
347Dic3—41no—
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—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—32no—
371C23—61no—
372C23—61no—
373C23—61no—
374C23—61no—
375C23—61no—
376C23—61no—
377C23D3×D412yes—
378C23C2×#6712yes—
379C23C23×D311yescharacteristic
380D4—44no—
381D4—44no—
382D4—44no—
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—32no—
396C2×C4—44no—
397C2×C4—61no—
398C2×C4—61no—
399C2×C4—61no—
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—61no—
417C2×C4—124no—
418C2×C4—124no—
419C2×C4#23712yes—
420C2×C4C2×#6712yes—
421C2×C4C2×#6712yes—
422C2×C3—42no—
423C2×C3—42no—
424C2×C3—81no—
425C2×C3#53411yescharacteristic
426C2×C3#53711yescharacteristic
427C2×C3#54011yescharacteristic
428C2×C3D4211yescharacteristic
429C2×C3#54611yescharacteristic
430C2×C3#54711yescharacteristic
431C2×C3#56211yescharacteristic
432D3—42no—
433C22—22no—
434C22—22no—
435C22—22no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—41no—
447C22—41no—
448C22—41no—
449C22—41no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—41no—
454C22—62no—
455C22—62no—
456C22—62no—
457C22—121no—
458C22—121no—
459C22—121no—
460C22—121no—
461C22C2×D3×D411yescharacteristic
462C22C2×#23711yescharacteristic
463C22#100411yescharacteristic
464C22#100411yescharacteristic
465C22#100911yescharacteristic
466C22#101011yescharacteristic
467C22C22×#6711yescharacteristic
468C4—22no—
469C4—22no—
470C4—22no—
471C4—41no—
472C4—62no—
473C4—121no—
474C4—121no—
475C4—121no—
476C4—121no—
477C4—121no—
478C4—121no—
479C3#267111yescharacteristic, complemented, Sylow
480C2—42no—
481C2—42no—
482C2—81no—
483C2—122no—
484C2#560511yescharacteristic
485C2#561111yescharacteristic
486C2#561911yescharacteristic
487C2#562111yescharacteristic
488C2D4×#6711yescharacteristic
489C2#580611yescharacteristic
490C2#583311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#121C2
hypercenterC2×#121D3
hypocenterC3#2671
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×#938C2semidirect
4. C2×#883C2semidirect
5. #5223C2semidirect
6. #5222C2semidirect
7. #5222C2semidirect
8. #5220C2semidirect
9. #5213C2semidirect
10. #5212C2semidirect
11. #4989C2semidirect
12. #4982C2semidirect
13. #4961C2semidirect
14. #4950C2semidirect
15. #4949C2semidirect
59. C2×#121D3semidirect
99. C2×#67D4semidirect
115. C3×C22⋊C4D4semidirect
304. C22⋊C4#67semidirect
479. C3#2671semidirect

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×#9382
C2×#8832
#52232
#52222
#52202
#52132
#52122
#49892
#49822
#49612
#49502
#49492
#26713