Finite Group Databasebeta

← #74672

Subgroups — #74672

Order 384.

928 conjugacy classes of subgroups.

The subgroup lattice has 474 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#237C211yesmaximal, characteristic
2C2×C4×#67C211yesmaximal, characteristic, complemented
3C2×#67⋊C4C211yesmaximal, characteristic, complemented
4C3×#386C211yesmaximal, characteristic, complemented
5#5220C211yesmaximal, characteristic, complemented
6C2×C3⋊#95C211yesmaximal, characteristic, complemented
7#4956C211yesmaximal, characteristic, complemented
8#4956C211yesmaximal, characteristic, complemented
9#4943C211yesmaximal, characteristic, complemented
10#4942C211yesmaximal, characteristic, complemented
11Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
12#4664C211yesmaximal, characteristic, complemented
13D3×#95C211yesmaximal, characteristic, complemented
14#4646C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2339—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C23×Dic3C2211yescharacteristic
22C23×Dic3C2211yescharacteristic
23C23×Dic3C2211yescharacteristic
24C2×#237—24no—
25C2×#237—24no—
26C2×#237C418yescomplemented
27C22×C4×D3C2211yescharacteristic
28C22×C4×D3C2211yescharacteristic
29C22×C4×D3C2211yescharacteristic
30C22×Dic6C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×C3×C22⋊C4C2211yescharacteristic
34C2×#217C2211yescharacteristic
35C2×#217C2211yescharacteristic
36C2×#217C2211yescharacteristic
37C4×#67—24no—
38C2×(C2×D3)⋊C4C2211yescharacteristic
39C2×(C2×D3)⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×C4×Dic3C2211yescharacteristic
43C2×C4×Dic3C2211yescharacteristic
44#67⋊C4—24no—
45C3×#95C2211yescharacteristic
46C3×#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C3⋊#95C2211yescharacteristic
57C22×#41—31no—
58C2×C4×D4—31no—
59C2×C4×D4—31no—
60#391—31no—
61#391—31no—
62#386—31no—
63#386—31no—
64#386—31no—
65#386D311yescharacteristic, complemented
66#381—31no—
67#381—31no—
68#380—31no—
69C4×C22⋊C4—31no—
70C2×#95—31no—
71C2×#95—31no—
72C24×C3C2311yescharacteristic
73C24×C3C2311yescharacteristic
74C23×D3C2311yescharacteristic
75C2×C3×D4—22no—
76C2×C3×D4—22no—
77C2×C3×D4C2×C414yescomplemented
78C22×C3×C4—22no—
79C22×C3×C4—22no—
80C22×C3×C4—41no—
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C22×C3×C4C2311yescharacteristic
86C2×#67—22no—
87C2×#67—22no—
88C2×#67C2×C414yes—
89C2×#67C2×C414yescomplemented
90C2×#67D414yes—
91C2×#67D414yes—
92C22×Dic3—22no—
93C22×Dic3—22no—
94C22×Dic3—22no—
95C22×Dic3—22no—
96C22×Dic3—22no—
97C22×Dic3—22no—
98C22×Dic3—41no—
99C22×Dic3C2×C414yescomplemented
100C22×Dic3C2×C414yescomplemented
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C22×Dic3C2311yescharacteristic
106C22×Dic3C2311yescharacteristic
107C22×Dic3C2311yescharacteristic
108#237—232no—
109C2×C4×D3—22no—
110C2×C4×D3—22no—
111C2×C4×D3—22no—
112C2×C4×D3—22no—
113C2×C4×D3—22no—
114C2×C4×D3—22no—
115C2×C4×D3—41no—
116C2×C4×D3—41no—
117C2×C4×D3C2×C414yescomplemented
118C2×Dic6—22no—
119C2×Dic6—22no—
120C2×Dic6C2×C414yescomplemented
121C3×C22⋊C4—22no—
122C3×C22⋊C4—22no—
123C3×C22⋊C4—41no—
124#217—22no—
125#217—22no—
126#217—41no—
127(C2×D3)⋊C4—22no—
128(C2×D3)⋊C4—22no—
129Dic3⋊C4—22no—
130Dic3⋊C4—22no—
131C4×Dic3—22no—
132C4×Dic3—22no—
133C2×#41—38no—
134C2×#41—64no—
135C2×#41—64no—
136C22×Q8—31no—
137C22×D4—31no—
138C22×D4—31no—
139C22×D4C2×D311yescharacteristic
140C23×C4—31no—
141C23×C4—31no—
142C23×C4—31no—
143C23×C4—31no—
144C23×C4—31no—
145C23×C4—31no—
146C23×C4C2×D311yescharacteristic
147C4×D4—64no—
148C4×D4—64no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—31no—
151C2×C22⋊C4—31no—
152C2×C22⋊C4—31no—
153C2×C22⋊C4—31no—
154C2×C22⋊C4—31no—
155C2×C22⋊C4—31no—
156C2×C22⋊C4C2×D311yescharacteristic
157C2×C22⋊C4C2×D311yescharacteristic
158C2×C22⋊C4C2×D311yescharacteristic
159C2×C42—31no—
160C2×C42—31no—
161#95—31no—
162#95—31no—
163#95—31no—
164#95—31no—
165#95—31no—
166#95—31no—
167#95—31no—
168#95—31no—
169#95—31no—
170#95—31no—
171#95C2×D311yescharacteristic
172#95C2×D311yescharacteristic
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×C3C22×C412yes—
184C23×C3C22×C412yes—
185C23×C3C2×D412yes—
186C23×C3C2×D412yes—
187C23×C3C2411yescharacteristic
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—21no—
192C22×D3C22×C412yes—
193C22×D3C2×D412yes—
194C22×D3C2×D412yes—
195C3×D4—28no—
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×C4—21no—
208C2×C3×C4—21no—
209C2×C3×C4—44no—
210C2×C3×C4C22×C412yes—
211C2×C3×C4#4112yescomplemented
212#67—28no—
213#67C22⋊C4116yescomplemented
214C2×Dic3—28no—
215C2×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3—21no—
219C2×Dic3—21no—
220C2×Dic3—21no—
221C2×Dic3—28no—
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—44no—
232C2×Dic3C22×C412yes—
233C2×Dic3C22×C412yes—
234C2×Dic3C22×C412yes—
235C2×Dic3C2×D412yes—
236C2×Dic3C2×D412yes—
237C2×Dic3#4112yescomplemented
238C4×D3—28no—
239C4×D3—44no—
240C4×D3—44no—
241Dic6—28no—
242C24—31no—
243C24C22×D311yescharacteristic
244C24C22×D311yescharacteristic
245#41—632no—
246C2×Q8—34no—
247C2×Q8—62no—
248C2×Q8—62no—
249C2×D4—22no—
250C2×D4—22no—
251C2×D4—34no—
252C2×D4—34no—
253C2×D4—34no—
254C2×D4—34no—
255C2×D4—62no—
256C2×D4—62no—
257C2×D4C4×D314yescomplemented
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—31no—
261C22×C4—34no—
262C22×C4—34no—
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—34no—
267C22×C4—31no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—41no—
271C22×C4—62no—
272C22×C4—62no—
273C22×C4—62no—
274C22×C4—62no—
275C22×C4—62no—
276C22×C4—62no—
277C22×C4—62no—
278C22×C4—62no—
279C22×C4—62no—
280C22×C4—62no—
281C22×C4—62no—
282C22×C4—62no—
283C22×C4—121no—
284C22×C4—121no—
285C22×C4—121no—
286C22×C4C22×D311yescharacteristic
287C22×C4C22×D311yescharacteristic
288C22×C4C22×D311yescharacteristic
289C22×C4C22×D311yescharacteristic
290C22×C4C22×D311yescharacteristic
291C4⋊C4—62no—
292C4⋊C4—62no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—41no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—121no—
301C42—62no—
302C42—62no—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3—22no—
307C22×C3—22no—
308C22×C3—22no—
309C22×C3—22no—
310C22×C3—22no—
311C22×C3—22no—
312C22×C3—41no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3C2×C22⋊C414yes—
317C22×C3C23×C411yescharacteristic
318C22×C3C22×D411yescharacteristic
319C22×C3C22×D411yescharacteristic
320C22×C3C2×#4111yescharacteristic
321C22×C3C2×#4111yescharacteristic
322C22×C3H(5,𝔽2)11yescharacteristic
323C22×C32−1+411yescharacteristic
324C2×D3—22no—
325C2×D3—22no—
326C2×D3—22no—
327C2×D3—22no—
328C2×D3—22no—
329C2×D3—22no—
330C2×D3C2×C22⋊C414yes—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—41no—
334C3×C4—41no—
335C3×C4—41no—
336Dic3—22no—
337Dic3—22no—
338Dic3—22no—
339Dic3—41no—
340Dic3—41no—
341Dic3—41no—
342Dic3C2×C22⋊C414yes—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23—21no—
351C23—21no—
352C23—21no—
353C23—32no—
354C23—32no—
355C23—32no—
356C23—61no—
357C23—61no—
358C23—61no—
359C23—61no—
360C23C2×C4×D312yes—
361C23C2×C4×D312yes—
362C23D3×D412yes—
363C23D3×D412yes—
364C23C23×D311yescharacteristic
365Q8—68no—
366D4—28no—
367D4—316no—
368D4—68no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
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—32no—
383C2×C4—32no—
384C2×C4—32no—
385C2×C4—32no—
386C2×C4—32no—
387C2×C4—32no—
388C2×C4—44no—
389C2×C4—68no—
390C2×C4—68no—
391C2×C4—68no—
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—61no—
406C2×C4—61no—
407C2×C4—124no—
408C2×C4—124no—
409C2×C4—124no—
410C2×C4C2×C4×D312yes—
411C2×C4#23712yescomplemented
412C2×C3—24no—
413C2×C3—42no—
414C2×C3C22×C22⋊C411yescharacteristic
415C2×C3C4×#4111yescharacteristic
416C2×C3#52011yescharacteristic
417C2×C3#54611yescharacteristic
418C2×C3#54611yescharacteristic
419C2×C3#54711yescharacteristic
420C2×C3#54711yescharacteristic
421D3—24no—
422C22—22no—
423C22—22no—
424C22—22no—
425C22—22no—
426C22—22no—
427C22—22no—
428C22—22no—
429C22—22no—
430C22—22no—
431C22—34no—
432C22—41no—
433C22—41no—
434C22—41no—
435C22—41no—
436C22—62no—
437C22—62no—
438C22—62no—
439C22—62no—
440C22—62no—
441C22—62no—
442C22D3×C22⋊C414yes—
443C22C22×C4×D311yescharacteristic
444C22C2×D3×D411yescharacteristic
445C22C2×D3×D411yescharacteristic
446C22C2×#23711yescharacteristic
447C22#100411yescharacteristic
448C22D3×#4111yescharacteristic
449C22#101011yescharacteristic
450C4—22no—
451C4—22no—
452C4—34no—
453C4—41no—
454C4—41no—
455C4—41no—
456C4—62no—
457C4—62no—
458C4—62no—
459C4—121no—
460C4—121no—
461C4—121no—
462C3#233911yescharacteristic, complemented, Sylow
463C2—24no—
464C2—42no—
465C2—64no—
466C2C2×D3×C22⋊C411yescharacteristic
467C2C4×#23711yescharacteristic
468C2#554611yescharacteristic
469C2#559111yescharacteristic
470C2#561011yescharacteristic
471C2#561111yescharacteristic
472C2#565511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#386C2
socle, cosocleC23×C3C24
prime radicalC22×#237C2
hypercenter#386D3
hypocenterC3#2339
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×C4×#67C2semidirect
3. C2×#67⋊C4C2semidirect
4. C3×#386C2semidirect
5. #5220C2semidirect
6. C2×C3⋊#95C2semidirect
7. #4956C2semidirect
8. #4956C2semidirect
9. #4943C2semidirect
10. #4942C2semidirect
11. Dic3×C22⋊C4C2semidirect
12. #4664C2semidirect
13. D3×#95C2semidirect
14. #4646C2semidirect
15. #4646C2semidirect
26. C2×#237C4semidirect
65. #386D3semidirect
77. C2×C3×D4C2×C4semidirect
89. C2×#67C2×C4semidirect
99. C22×Dic3C2×C4semidirect
100. C22×Dic3C2×C4semidirect
117. C2×C4×D3C2×C4semidirect
120. C2×Dic6C2×C4semidirect
211. C2×C3×C4#41semidirect
213. #67C22⋊C4semidirect
237. C2×Dic3#41semidirect
257. C2×D4C4×D3semidirect
411. C2×C4#237semidirect
462. C3#2339semidirect

Maximal subgroups

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

subgroupindex
C22×#2372
C2×C4×#672
C2×#67⋊C42
C3×#3862
#52202
C2×C3⋊#952
#49562
#49432
#49422
Dic3×C22⋊C42
#46642
D3×#952
#46462
#23393