Finite Group Databasebeta

← #76749

Subgroups — #76749

Order 384.

508 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#392C211yesmaximal, characteristic, complemented
2#5223C211yesmaximal, characteristic, complemented
3#5223C211yesmaximal, characteristic, complemented
4#5223C211yesmaximal, characteristic, complemented
5#5222C211yesmaximal, characteristic, complemented
6#5222C211yesmaximal, characteristic, complemented
7#5221C211yesmaximal, characteristic, complemented
8#4961C211yesmaximal, characteristic, complemented
9#4961C211yesmaximal, characteristic, complemented
10#4959C211yesmaximal, characteristic, complemented
11#4953C211yesmaximal, characteristic, complemented
12#4952C211yesmaximal, characteristic, complemented
13#4951C211yesmaximal, characteristic, complemented
14#4669C211yesmaximal, characteristic, complemented
15#4660C211yesmaximal, characteristic, complemented
16#2828—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C22×C3×D4C2211yescharacteristic
19C22×C3×D4C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C22×#67C2211yescharacteristic
22C22×#67C2211yescharacteristic
23C2×C3×C22⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31C2×#217C2211yescharacteristic
32C2×#217C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×C4⋊Dic3C2211yescharacteristic
40C2×C4⋊Dic3C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×C4×Dic3C2211yescharacteristic
44C3×#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52#400—31no—
53#397—31no—
54#397—31no—
55#394—31no—
56#394—31no—
57#394—31no—
58#394—31no—
59#394—31no—
60#392—31no—
61#392—31no—
62#392—31no—
63#392D311yescharacteristic, complemented
64#390—31no—
65#388—31no—
66#388—31no—
67C24×C3C2311yescharacteristic
68C24×C3C2311yescharacteristic
69C24×C3C2311yescharacteristic
70C23×D3C2311yescharacteristic
71C2×C3×D4—22no—
72C2×C3×D4—41no—
73C2×C3×D4—41no—
74C2×C3×D4—41no—
75C2×C3×D4—41no—
76C2×C3×D4—41no—
77C2×C3×D4—41no—
78C2×C3×D4—41no—
79C2×C3×D4—41no—
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C2×#67—41no—
85C2×#67—41no—
86C2×#67—41no—
87C2×#67—41no—
88C2×#67—41no—
89C2×#67—41no—
90C2×#67—41no—
91C2×#67—41no—
92C2×#67—41no—
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97C22×Dic3C2311yescharacteristic
98C22×Dic3C2311yescharacteristic
99C22×Dic3C2311yescharacteristic
100C3×C22⋊C4—22no—
101C3×C22⋊C4—41no—
102C3×C22⋊C4—41no—
103#217—41no—
104#217—41no—
105#217—41no—
106#217—41no—
107#217—41no—
108#217—41no—
109#217—41no—
110#217—41no—
111#217—41no—
112(C2×D3)⋊C4—22no—
113(C2×D3)⋊C4—41no—
114(C2×D3)⋊C4—41no—
115(C2×D3)⋊C4—41no—
116C4⋊Dic3—41no—
117C4⋊Dic3—41no—
118Dic3⋊C4—41no—
119Dic3⋊C4—41no—
120C4×Dic3—41no—
121C22×D4—31no—
122C22×D4—31no—
123C22×D4—31no—
124C22×D4C2×D311yescharacteristic
125C22×D4C2×D311yescharacteristic
126C22×D4C2×D311yescharacteristic
127C2×C4⋊C4—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C22⋊C4—31no—
132C2×C22⋊C4—31no—
133C2×C22⋊C4—31no—
134C2×C22⋊C4—31no—
135C2×C22⋊C4—31no—
136C2×C22⋊C4—31no—
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⋊C4C2×D311yescharacteristic
145C2×C22⋊C4C2×D311yescharacteristic
146C2×C22⋊C4C2×D311yescharacteristic
147C2×C42—31no—
148#95—31no—
149#95—31no—
150#95—31no—
151#95—31no—
152#95—31no—
153#95—31no—
154#95—31no—
155#95C2×D311yescharacteristic
156C23×C3—21no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—21no—
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×C3C2411yescharacteristic
178C22×D3—21no—
179C22×D3—21no—
180C22×D3—21no—
181C22×D3—21no—
182C22×D3—21no—
183C22×D3—21no—
184C22×D3—21no—
185C3×D4—44no—
186C3×D4—44no—
187C3×D4—44no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—21no—
192C2×C3×C4—21no—
193C2×C3×C4—21no—
194C2×C3×C4—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—21no—
198C2×C3×C4—21no—
199C2×C3×C4#4112yes—
200#67—44no—
201#67—44no—
202#67—44no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
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—
224C24—31no—
225C24C22×D311yescharacteristic
226C24C22×D311yescharacteristic
227C24C22×D311yescharacteristic
228C2×D4—22no—
229C2×D4—41no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4—41no—
233C2×D4—41no—
234C2×D4—41no—
235C2×D4—41no—
236C2×D4—41no—
237C2×D4—121no—
238C2×D4—121no—
239C2×D4—121no—
240C2×D4—121no—
241C2×D4—121no—
242C2×D4—121no—
243C2×D4—121no—
244C2×D4—121no—
245C2×D4—121no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—31no—
249C22×C4—31no—
250C22×C4—31no—
251C22×C4—31no—
252C22×C4—31no—
253C22×C4C22×D311yescharacteristic
254C22×C4C22×D311yescharacteristic
255C22×C4C22×D311yescharacteristic
256C22×C4C22×D311yescharacteristic
257C4⋊C4—121no—
258C4⋊C4—121no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C22⋊C4—22no—
262C22⋊C4—41no—
263C22⋊C4—41no—
264C22⋊C4—62no—
265C22⋊C4—121no—
266C22⋊C4—121no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C22⋊C4—121no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C42—121no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3—41no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—41no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3—41no—
292C22×C3—41no—
293C22×C3—41no—
294C22×C3—41no—
295C22×C3—41no—
296C22×C3—41no—
297C22×C3—41no—
298C22×C3—41no—
299C22×C3C2×#4111yescharacteristic
300C22×C3H(5,𝔽2)11yescharacteristic
301C22×C3H(5,𝔽2)11yescharacteristic
302C22×C3H(5,𝔽2)11yescharacteristic
303C22×C3H(5,𝔽2)11yescharacteristic
304C22×C3H(5,𝔽2)11yescharacteristic
305C22×C3H(5,𝔽2)11yescharacteristic
306C2×D3—41no—
307C2×D3—41no—
308C2×D3—41no—
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—41no—
312C2×D3—41no—
313C3×C4—22no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4—41no—
317Dic3—41no—
318Dic3—41no—
319Dic3—41no—
320Dic3—41no—
321Dic3—41no—
322Dic3—41no—
323Dic3—41no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—21no—
335C23—21no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—61no—
346C23—61no—
347C23—61no—
348C23—61no—
349C23—61no—
350C23—61no—
351C23—61no—
352C23C23×D311yescharacteristic
353D4—44no—
354D4—44no—
355D4—44no—
356D4—124no—
357D4—124no—
358D4—124no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—61no—
376C2×C4—61no—
377C2×C4—61no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—61no—
391C2×C4#23712yes—
392C2×C3—81no—
393C2×C3—81no—
394C2×C3—81no—
395C2×C3#53811yescharacteristic
396C2×C3#54011yescharacteristic
397C2×C3#54011yescharacteristic
398C2×C3#56011yescharacteristic
399C2×C3#56011yescharacteristic
400C2×C3#56011yescharacteristic
401C2×C3H(3,𝔽4)11yescharacteristic
402D3—81no—
403C22—41no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—41no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—41no—
413C22—41no—
414C22—41no—
415C22—41no—
416C22—41no—
417C22—41no—
418C22—41no—
419C22—41no—
420C22—41no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—121no—
425C22—121no—
426C22—121no—
427C22—121no—
428C22—121no—
429C22—121no—
430C22—121no—
431C22C2×#23711yescharacteristic
432C22#100411yescharacteristic
433C22#100411yescharacteristic
434C22#100411yescharacteristic
435C22#100411yescharacteristic
436C22#100411yescharacteristic
437C22#100911yescharacteristic
438C4—22no—
439C4—41no—
440C4—41no—
441C4—41no—
442C4—121no—
443C4—121no—
444C4—121no—
445C4—121no—
446C4—121no—
447C4—121no—
448C4—121no—
449C3#282811yescharacteristic, complemented, Sylow
450C2—81no—
451C2—81no—
452C2—81no—
453C2—241no—
454C2#559311yescharacteristic
455C2#559411yescharacteristic
456C2#559511yescharacteristic
457C2#561911yescharacteristic
458C2#562011yescharacteristic
459C2#562011yescharacteristic
460C2#571911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#392C2
hypercenter#392D3
hypocenterC3#2828
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #392

Complements

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

NH ≅ G/Ntype
1. C3×#392C2semidirect
2. #5223C2semidirect
3. #5223C2semidirect
4. #5223C2semidirect
5. #5222C2semidirect
6. #5222C2semidirect
7. #5221C2semidirect
8. #4961C2semidirect
9. #4961C2semidirect
10. #4959C2semidirect
11. #4953C2semidirect
12. #4952C2semidirect
13. #4951C2semidirect
14. #4669C2semidirect
15. #4660C2semidirect
63. #392D3semidirect
449. C3#2828semidirect

Maximal subgroups

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

subgroupindex
C3×#3922
#52232
#52222
#52212
#49612
#49592
#49532
#49522
#49512
#46692
#46602
#28283