Finite Group Databasebeta

← #76721

Subgroups — #76721

Order 384.

612 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#939C211yesmaximal, characteristic, complemented
2C2×#883C211yesmaximal, characteristic, complemented
3C3×#392C211yesmaximal, characteristic, complemented
4#5223C211yesmaximal, characteristic, complemented
5#5223C211yesmaximal, characteristic, complemented
6#5222C211yesmaximal, characteristic, complemented
7#5221C211yesmaximal, characteristic, complemented
8#5220C211yesmaximal, characteristic, complemented
9#4956C211yesmaximal, characteristic, complemented
10#4956C211yesmaximal, characteristic, complemented
11#4952C211yesmaximal, characteristic, complemented
12#4945C211yesmaximal, characteristic, complemented
13#4942C211yesmaximal, characteristic, complemented
14#4669C211yesmaximal, characteristic, complemented
15#4659C211yesmaximal, characteristic, complemented
16#2703—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C23×Dic3C2211yescharacteristic
24C22×C4×D3C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36#939—24no—
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×C4⋊Dic3C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44C2×C4×Dic3C2211yescharacteristic
45#883—24no—
46C3×#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#123—31no—
55C2×#121—31no—
56#400—31no—
57#397—31no—
58#394—31no—
59#392—31no—
60#392—31no—
61#392D311yescharacteristic, complemented
62#390—31no—
63#388—31no—
64#388—31no—
65#386—31no—
66#386—31no—
67#386—31no—
68#380—31no—
69C24×C3C2311yescharacteristic
70C24×C3C2311yescharacteristic
71C24×C3C2311yescharacteristic
72C23×D3C2311yescharacteristic
73C2×C3×D4—22no—
74C2×C3×D4—22no—
75C2×C3×D4—22no—
76C2×C3×D4—41no—
77C2×C3×D4—41no—
78C2×C3×D4—41no—
79C2×C3×D4—41no—
80C2×C3×D4—41no—
81C2×C3×D4—41no—
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C2×#67—22no—
87C2×#67—22no—
88C2×#67—22no—
89C2×#67—22no—
90C2×#67—22no—
91C2×#67—41no—
92C2×#67—41no—
93C2×#67—41no—
94C2×#67—41no—
95C22×Dic3—22no—
96C22×Dic3—22no—
97C22×Dic3—41no—
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100C22×Dic3C2311yescharacteristic
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C2×C4×D3—22no—
106C2×C4×D3—41no—
107C2×C4×D3—41no—
108C3×C22⋊C4—22no—
109C3×C22⋊C4—41no—
110C3×C22⋊C4—41no—
111#217—22no—
112#217—22no—
113#217—22no—
114#217—41no—
115#217—41no—
116#217—41no—
117#217—41no—
118#217—41no—
119(C2×D3)⋊C4—41no—
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4D414yescomplemented
122C4⋊Dic3—22no—
123Dic3⋊C4—22no—
124Dic3⋊C4—22no—
125Dic3⋊C4—41no—
126C4×Dic3—41no—
127C22×D4—31no—
128C22×D4—31no—
129C22×D4—31no—
130C22×D4C2×D311yescharacteristic
131C22×D4C2×D311yescharacteristic
132C22×D4C2×D311yescharacteristic
133C23×C4—31no—
134C23×C4—31no—
135#123—64no—
136#121—64no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊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⋊C4—31no—
152C2×C22⋊C4C2×D311yescharacteristic
153C2×C22⋊C4C2×D311yescharacteristic
154C2×C22⋊C4C2×D311yescharacteristic
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
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×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#4112yes—
185C23×C3C2411yescharacteristic
186C22×D3—21no—
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—21no—
192C22×D3C2×D412yes—
193C3×D4—44no—
194C3×D4—44no—
195C3×D4—44no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4—21no—
200C2×C3×C4—21no—
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×C4C2×D412yes—
208#67—44no—
209#67—44no—
210#67—44no—
211C2×Dic3—21no—
212C2×Dic3—21no—
213C2×Dic3—21no—
214C2×Dic3—21no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
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—44no—
231C2×Dic3C2×D412yes—
232C2×Dic3#4112yes—
233C4×D3—44no—
234C24—31no—
235C24C22×D311yescharacteristic
236C24C22×D311yescharacteristic
237C24C22×D311yescharacteristic
238C2×D4—22no—
239C2×D4—22no—
240C2×D4—22no—
241C2×D4—41no—
242C2×D4—41no—
243C2×D4—41no—
244C2×D4—41no—
245C2×D4—41no—
246C2×D4—41no—
247C2×D4—62no—
248C2×D4—62no—
249C2×D4—62no—
250C2×D4—62no—
251C2×D4—62no—
252C2×D4—121no—
253C2×D4—121no—
254C2×D4—121no—
255C2×D4—121no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—31no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—62no—
264C22×C4—62no—
265C22×C4—62no—
266C22×C4—121no—
267C22×C4—121no—
268C22×C4—121no—
269C22×C4C22×D311yescharacteristic
270C22×C4C22×D311yescharacteristic
271C22×C4C22×D311yescharacteristic
272C22×C4C22×D311yescharacteristic
273C4⋊C4—62no—
274C4⋊C4—62no—
275C4⋊C4—62no—
276C4⋊C4—121no—
277C22⋊C4—22no—
278C22⋊C4—34no—
279C22⋊C4—41no—
280C22⋊C4—41no—
281C22⋊C4—62no—
282C22⋊C4—62no—
283C22⋊C4—62no—
284C22⋊C4—121no—
285C22⋊C4—121no—
286C22⋊C4—121no—
287C22⋊C4—121no—
288C22⋊C4—121no—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C42—121no—
292C22×C3—22no—
293C22×C3—22no—
294C22×C3—22no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—41no—
301C22×C3—41no—
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×C3—41no—
311C22×C3—41no—
312C22×C3—41no—
313C22×C3C22×D411yescharacteristic
314C22×C3C2×#4111yescharacteristic
315C22×C3C2×#4111yescharacteristic
316C22×C3H(5,𝔽2)11yescharacteristic
317C22×C3H(5,𝔽2)11yescharacteristic
318C22×C3H(5,𝔽2)11yescharacteristic
319C22×C3H(5,𝔽2)11yescharacteristic
320C2×D3—22no—
321C2×D3—22no—
322C2×D3—22no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C2×D3—41no—
327C3×C4—22no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—41no—
331Dic3—22no—
332Dic3—22no—
333Dic3—41no—
334Dic3—41no—
335Dic3—41no—
336Dic3—41no—
337Dic3—41no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
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—21no—
356C23—21no—
357C23—21no—
358C23—32no—
359C23—61no—
360C23—61no—
361C23—61no—
362C23—61no—
363C23—61no—
364C23—61no—
365C23#23712yes—
366C23C23×D311yescharacteristic
367D4—44no—
368D4—44no—
369D4—44no—
370D4—124no—
371D4—124no—
372D4—124no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—21no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—32no—
385C2×C4—32no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—61no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—61no—
396C2×C4—61no—
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—124no—
406C2×C4—124no—
407C2×C4D3×D412yes—
408C2×C3—42no—
409C2×C3—81no—
410C2×C3—81no—
411C2×C3#53511yescharacteristic
412C2×C3#53811yescharacteristic
413C2×C3#54011yescharacteristic
414C2×C3#54611yescharacteristic
415C2×C3#54611yescharacteristic
416C2×C3#55311yescharacteristic
417C2×C3#56011yescharacteristic
418D3—42no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—41no—
429C22—41no—
430C22—41no—
431C22—41no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—62no—
441C22—62no—
442C22—62no—
443C22—121no—
444C22—121no—
445C22—121no—
446C22—121no—
447C22C2×D3×D411yescharacteristic
448C22C2×#23711yescharacteristic
449C22#100411yescharacteristic
450C22#100411yescharacteristic
451C22#100411yescharacteristic
452C22#100411yescharacteristic
453C22D3×#4111yescharacteristic
454C4—22no—
455C4—41no—
456C4—41no—
457C4—41no—
458C4—62no—
459C4—62no—
460C4—121no—
461C4—121no—
462C4—121no—
463C4—121no—
464C4—121no—
465C3#270311yescharacteristic, complemented, Sylow
466C2—42no—
467C2—81no—
468C2—81no—
469C2—122no—
470C2#559111yescharacteristic
471C2#559311yescharacteristic
472C2#559511yescharacteristic
473C2#560111yescharacteristic
474C2#561211yescharacteristic
475C2#561511yescharacteristic
476C2#571511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2703
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C2×#939C2semidirect
2. C2×#883C2semidirect
3. C3×#392C2semidirect
4. #5223C2semidirect
5. #5223C2semidirect
6. #5222C2semidirect
7. #5221C2semidirect
8. #5220C2semidirect
9. #4956C2semidirect
10. #4956C2semidirect
11. #4952C2semidirect
12. #4945C2semidirect
13. #4942C2semidirect
14. #4669C2semidirect
15. #4659C2semidirect
61. #392D3semidirect
121. (C2×D3)⋊C4D4semidirect
465. C3#2703semidirect

Maximal subgroups

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

subgroupindex
C2×#9392
C2×#8832
C3×#3922
#52232
#52222
#52212
#52202
#49562
#49522
#49452
#49422
#46692
#46592
#27033