Finite Group Databasebeta

← #77140

Subgroups — #77140

Order 384.

688 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#937C211yesmaximal, characteristic
2C2×#885C211yesmaximal, characteristic, complemented
3C3×#394C211yesmaximal, characteristic, complemented
4#5220C211yesmaximal, characteristic, complemented
5#4992C211yesmaximal, characteristic, complemented
6#4990C211yesmaximal, characteristic, complemented
7#4961C211yesmaximal, characteristic, complemented
8#4958C211yesmaximal, characteristic, complemented
9#4958C211yesmaximal, characteristic, complemented
10#4955C211yesmaximal, characteristic, complemented
11#4955C211yesmaximal, characteristic, complemented
12#4951C211yesmaximal, characteristic, complemented
13#4947C211yesmaximal, characteristic, complemented
14#4674C211yesmaximal, characteristic, complemented
15#4673C211yesmaximal, characteristic, complemented
16#2684—31nomaximal, Sylow
17C24×D3C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C23×Dic3C2211yescharacteristic, complemented
22C22×Dic6C2211yescharacteristic, complemented
23C2×C3×C4⋊C4C2211yescharacteristic, complemented
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic, complemented
27C2×C3×C22⋊C4C2211yescharacteristic, complemented
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic, complemented
32#937—24no—
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
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×C4⋊Dic3C2211yescharacteristic, complemented
44C2×C4×Dic3C2211yescharacteristic, complemented
45#885—24no—
46C3×#95C2211yescharacteristic, complemented
47C3⋊#95C2211yescharacteristic, complemented
48C3⋊#95C2211yescharacteristic, complemented
49C3⋊#95C2211yescharacteristic, complemented
50C3⋊#95C2211yescharacteristic, complemented
51C3⋊#95C2211yescharacteristic, complemented
52C3⋊#95C2211yescharacteristic, complemented
53C3⋊#95C2211yescharacteristic, complemented
54C2×#124—31no—
55C2×#120—31no—
56#397—31no—
57#397—31no—
58#394—31no—
59#394D311yescharacteristic, complemented
60#393—31no—
61#393—31no—
62#388—31no—
63#388—31no—
64#386—31no—
65#386—31no—
66#386—31no—
67#379—31no—
68#379—31no—
69C24×C3C2311yescharacteristic
70C24×C3C2311yescharacteristic
71C23×D3—22no—
72C23×D3—22no—
73C23×D3—22no—
74C23×D3—41no—
75C23×D3—41no—
76C23×D3—41no—
77C23×D3—41no—
78C23×D3C2311yescharacteristic
79C23×D3C2311yescharacteristic
80C2×C3×D4—22no—
81C2×C3×D4—22no—
82C2×C3×D4—41no—
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C22×C3×C4C2311yescharacteristic
87C22×C3×C4C2311yescharacteristic
88C2×#67—22no—
89C2×#67—22no—
90C2×#67—41no—
91C2×#67—41no—
92C2×#67—41no—
93C2×#67D414yes—
94C22×Dic3—41no—
95C22×Dic3—41no—
96C22×Dic3—41no—
97C22×Dic3C2311yescharacteristic
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100C22×Dic3C2311yescharacteristic
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C2×Dic6—22no—
104C2×Dic6—41no—
105C2×Dic6—41no—
106C3×C4⋊C4—41no—
107C3×C22⋊C4—22no—
108C3×C22⋊C4—22no—
109C3×C22⋊C4—41no—
110C3×C22⋊C4—41no—
111#217—22no—
112#217—22no—
113#217—41no—
114#217—41no—
115(C2×D3)⋊C4—22no—
116(C2×D3)⋊C4—22no—
117(C2×D3)⋊C4—22no—
118(C2×D3)⋊C4—22no—
119(C2×D3)⋊C4—22no—
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—41no—
123(C2×D3)⋊C4—41no—
124(C2×D3)⋊C4—41no—
125C4⋊Dic3—41no—
126C4×Dic3—22no—
127C25—31no—
128C22×Q8—31no—
129C22×D4—31no—
130C22×D4—31no—
131C22×D4C2×D311yescharacteristic
132C23×C4—31no—
133#124—64no—
134#120—64no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4C2×D311yescharacteristic, complemented
137C2×C22⋊C4—31no—
138C2×C22⋊C4—31no—
139C2×C22⋊C4—31no—
140C2×C22⋊C4—31no—
141C2×C22⋊C4—31no—
142C2×C22⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4—31no—
145C2×C22⋊C4—31no—
146C2×C22⋊C4—31no—
147C2×C22⋊C4—31no—
148C2×C22⋊C4—31no—
149C2×C22⋊C4—31no—
150C2×C22⋊C4—31no—
151C2×C22⋊C4C2×D311yescharacteristic
152C2×C22⋊C4C2×D311yescharacteristic
153C2×C22⋊C4C2×D311yescharacteristic, complemented
154C2×C22⋊C4C2×D311yescharacteristic, complemented
155C2×C42—31no—
156#95—31no—
157#95—31no—
158#95—31no—
159#95—31no—
160#95—31no—
161#95—31no—
162#95—31no—
163#95C2×D311yescharacteristic, complemented
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3—21no—
171C23×C3—21no—
172C23×C3—21no—
173C23×C3—21no—
174C23×C3—21no—
175C23×C3—21no—
176C23×C3C2×D412yes—
177C23×C3#4112yes—
178C23×C3C2411yescharacteristic
179C22×D3—21no—
180C22×D3—21no—
181C22×D3—21no—
182C22×D3—21no—
183C22×D3—21no—
184C22×D3—21no—
185C22×D3—21no—
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—44no—
192C22×D3—44no—
193C22×D3—44no—
194C22×D3—44no—
195C22×D3—44no—
196C22×D3—44no—
197C22×D3—44no—
198C22×D3C2×D412yes—
199C22×D3#4112yes—
200C3×D4—44no—
201C2×C3×C4—21no—
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—21no—
215C2×C3×C4—21no—
216#67—44no—
217#67—44no—
218C2×Dic3—21no—
219C2×Dic3—21no—
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—44no—
236C2×Dic3C2×D412yes—
237Dic6—44no—
238C24—31no—
239C24—31no—
240C24—62no—
241C24—62no—
242C24—62no—
243C24—121no—
244C24—121no—
245C24—121no—
246C24—121no—
247C24C22×D311yescharacteristic
248C24C22×D311yescharacteristic
249C2×Q8—62no—
250C2×Q8—121no—
251C2×Q8—121no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4—34no—
255C2×D4—41no—
256C2×D4—62no—
257C2×D4—62no—
258C2×D4—121no—
259C2×D4—121no—
260C2×D4—121no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—121no—
268C22×C4—121no—
269C22×C4—121no—
270C22×C4C22×D311yescharacteristic
271C22×C4C22×D311yescharacteristic
272C22×C4C22×D311yescharacteristic
273C22×C4C22×D311yescharacteristic
274C22×C4C22×D311yescharacteristic
275C4⋊C4—41no—
276C4⋊C4—121no—
277C22⋊C4—22no—
278C22⋊C4—22no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—62no—
282C22⋊C4—62no—
283C22⋊C4—62no—
284C22⋊C4—62no—
285C22⋊C4—62no—
286C22⋊C4—62no—
287C22⋊C4—62no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C22⋊C4—121no—
294C22⋊C4—121no—
295C42—62no—
296C22×C3—22no—
297C22×C3—22no—
298C22×C3—22no—
299C22×C3—22no—
300C22×C3—22no—
301C22×C3—22no—
302C22×C3—41no—
303C22×C3—41no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3C22×D411yescharacteristic
311C22×C3C2×#4111yescharacteristic
312C22×C3C2×#4111yescharacteristic
313C22×C3H(5,𝔽2)11yescharacteristic
314C22×C3H(5,𝔽2)11yescharacteristic
315C22×C3H(5,𝔽2)11yescharacteristic
316C22×C3H(5,𝔽2)11yescharacteristic
317C2×D3—22no—
318C2×D3—22no—
319C2×D3—22no—
320C2×D3—22no—
321C2×D3—22no—
322C2×D3—22no—
323C2×D3—416no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—41no—
329C2×D3—41no—
330C2×D3—41no—
331C2×D3—41no—
332C3×C4—41no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—41no—
337Dic3—22no—
338Dic3—41no—
339Dic3—41no—
340Dic3—41no—
341Dic3—41no—
342Dic3—41no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—32no—
356C23—32no—
357C23—61no—
358C23—61no—
359C23—61no—
360C23—61no—
361C23—61no—
362C23—61no—
363C23—61no—
364C23—61no—
365C23—61no—
366C23—61no—
367C23—61no—
368C23—61no—
369C23—124no—
370C23—124no—
371C23—124no—
372C23—124no—
373C23—124no—
374C23—124no—
375C23—124no—
376C23D3×D412yes—
377C23#23712yes—
378C23C23×D311yescharacteristic
379Q8—124no—
380D4—44no—
381D4—124no—
382D4—124no—
383C2×C4—21no—
384C2×C4—21no—
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—21no—
398C2×C4—32no—
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—124no—
417C2×C3—42no—
418C2×C3—42no—
419C2×C3#53411yescharacteristic
420C2×C3#53811yescharacteristic
421C2×C3#53811yescharacteristic
422C2×C3#54611yescharacteristic
423C2×C3#54611yescharacteristic
424C2×C3#55111yescharacteristic
425C2×C3H(3,𝔽4)11yescharacteristic
426D3—42no—
427D3—42no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—22no—
432C22—22no—
433C22—22no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—62no—
443C22—62no—
444C22—62no—
445C22—62no—
446C22—62no—
447C22—62no—
448C22—1216no—
449C22—121no—
450C22—121no—
451C22—121no—
452C22—121no—
453C22—121no—
454C22—121no—
455C22—121no—
456C22—121no—
457C22C2×D3×D411yescharacteristic
458C22C2×#23711yescharacteristic
459C22#100411yescharacteristic
460C22#100411yescharacteristic
461C22D3×#4111yescharacteristic
462C22#100911yescharacteristic
463C22#100911yescharacteristic
464C4—41no—
465C4—41no—
466C4—41no—
467C4—41no—
468C4—41no—
469C4—62no—
470C4—121no—
471C4—121no—
472C4—121no—
473C4—121no—
474C4—121no—
475C3#268411yescharacteristic, complemented, Sylow
476C2—42no—
477C2—42no—
478C2—122no—
479C2—122no—
480C2#559111yescharacteristic
481C2#560511yescharacteristic
482C2#561311yescharacteristic
483C2#561711yescharacteristic
484C2#565411yescharacteristic
485C2#565811yescharacteristic
486C2#568311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#394C2
prime radicalC2×#937C2
hypercenter#394D3
hypocenterC3#2684
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#885C2semidirect
3. C3×#394C2semidirect
4. #5220C2semidirect
5. #4992C2semidirect
6. #4990C2semidirect
7. #4961C2semidirect
8. #4958C2semidirect
9. #4958C2semidirect
10. #4955C2semidirect
11. #4955C2semidirect
12. #4951C2semidirect
13. #4947C2semidirect
14. #4674C2semidirect
15. #4673C2semidirect
21. C23×Dic3C22semidirect
22. C22×Dic6C22semidirect
23. C2×C3×C4⋊C4C22semidirect
26. C2×C3×C22⋊C4C22semidirect
27. C2×C3×C22⋊C4C22semidirect
31. C2×#217C22semidirect
43. C2×C4⋊Dic3C22semidirect
44. C2×C4×Dic3C22semidirect
46. C3×#95C22semidirect
47. C3⋊#95C22semidirect
48. C3⋊#95C22semidirect
49. C3⋊#95C22semidirect
50. C3⋊#95C22semidirect
51. C3⋊#95C22semidirect
52. C3⋊#95C22semidirect
53. C3⋊#95C22semidirect
59. #394D3semidirect
136. C2×C4⋊C4C2×D3semidirect
153. C2×C22⋊C4C2×D3semidirect
154. C2×C22⋊C4C2×D3semidirect
163. #95C2×D3semidirect
475. C3#2684semidirect

Maximal subgroups

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

subgroupindex
C2×#9372
C2×#8852
C3×#3942
#52202
#49922
#49902
#49612
#49582
#49552
#49512
#49472
#46742
#46732
#26843