Finite Group Databasebeta

← #74676

Subgroups — #74676

Order 384.

788 conjugacy classes of subgroups.

The subgroup lattice has 466 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#939C211yesmaximal, characteristic
2C2×D4×Dic3C211yesmaximal, characteristic, complemented
3C2×C4×#67C211yesmaximal, characteristic, complemented
4C22×Dic3⋊C4C211yesmaximal, characteristic, complemented
5C2×#67⋊C4C211yesmaximal, characteristic, complemented
6C3×#386C211yesmaximal, characteristic, complemented
7#4958C211yesmaximal, characteristic, complemented
8#4957C211yesmaximal, characteristic, complemented
9#4946C211yesmaximal, characteristic, complemented
10#4945C211yesmaximal, characteristic, complemented
11Dic3×C22⋊C4C211yesmaximal, characteristic, complemented
12#4669C211yesmaximal, characteristic, complemented
13#4668C211yesmaximal, characteristic, complemented
14#4649C211yesmaximal, characteristic, complemented
15#4647C211yesmaximal, characteristic, complemented
16#2361—31nomaximal, Sylow
17C22×C3×D4C2211yescharacteristic
18C23×C3×C4C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×#67C2211yescharacteristic
21C23×Dic3C2211yescharacteristic
22C23×Dic3C2211yescharacteristic
23C23×Dic3C2211yescharacteristic
24C22×C4×D3C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×C3×C22⋊C4C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C2×#217C2211yescharacteristic
30C2×#217C2211yescharacteristic
31#939C418yescomplemented
32D4×Dic3—24no—
33C4×#67—24no—
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×C4⋊Dic3C2211yescharacteristic
39C2×Dic3⋊C4—24no—
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44C2×Dic3⋊C4C2211yescharacteristic
45C2×Dic3⋊C4C2211yescharacteristic
46C2×C4×Dic3C2211yescharacteristic
47C2×C4×Dic3C2211yescharacteristic
48C2×C4×Dic3C2211yescharacteristic
49C2×C4×Dic3C2211yescharacteristic
50#67⋊C4—24no—
51C3×#95C2211yescharacteristic
52C3×#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C3⋊#95C2211yescharacteristic
57C2×#121—31no—
58C2×C4×D4—31no—
59C2×C4×D4—31no—
60C2×C4×D4—31no—
61C22×C4⋊C4—31no—
62#390—31no—
63#389—31no—
64#388—31no—
65#388—31no—
66#386—31no—
67#386D311yescharacteristic, complemented
68#385—31no—
69#385—31no—
70C4×C4⋊C4—31no—
71C4×C22⋊C4—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×#67—22no—
89C2×#67—22no—
90C2×#67C2×C414yescomplemented
91C2×#67C2×C414yescomplemented
92C22×Dic3—22no—
93C22×Dic3—22no—
94C22×Dic3—22no—
95C22×Dic3—22no—
96C22×Dic3—22no—
97C22×Dic3—22no—
98C22×Dic3—41no—
99C22×Dic3—41no—
100C22×Dic3C2×C414yescomplemented
101C22×Dic3C2311yescharacteristic
102C22×Dic3C2311yescharacteristic
103C22×Dic3C2311yescharacteristic
104C22×Dic3C2311yescharacteristic
105C22×Dic3C2311yescharacteristic
106C22×Dic3C2311yescharacteristic
107C22×Dic3C2311yescharacteristic
108C2×C4×D3—22no—
109C2×C4×D3—22no—
110C2×C4×D3—41no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C22⋊C4—41no—
114#217—22no—
115#217C2×C414yescomplemented
116#217D414yescomplemented
117(C2×D3)⋊C4—22no—
118(C2×D3)⋊C4—22no—
119(C2×D3)⋊C4—41no—
120(C2×D3)⋊C4C2×C414yescomplemented
121C4⋊Dic3—22no—
122Dic3⋊C4—22no—
123Dic3⋊C4—22no—
124Dic3⋊C4—22no—
125Dic3⋊C4—22no—
126Dic3⋊C4C2×C414yescomplemented
127Dic3⋊C4D414yescomplemented
128C4×Dic3—22no—
129C4×Dic3—22no—
130C4×Dic3—22no—
131C4×Dic3—22no—
132C22×D4—31no—
133C22×D4—31no—
134C22×D4C2×D311yescharacteristic
135C23×C4—31no—
136C23×C4—31no—
137C23×C4—31no—
138C23×C4—31no—
139C23×C4C2×D311yescharacteristic
140#121—38no—
141C4×D4—64no—
142C4×D4—64no—
143C4×D4—64no—
144C2×C4⋊C4—31no—
145C2×C4⋊C4—31no—
146C2×C4⋊C4—31no—
147C2×C4⋊C4—31no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—64no—
152C2×C22⋊C4—31no—
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⋊C4C2×D311yescharacteristic
160C2×C22⋊C4C2×D311yescharacteristic
161C2×C22⋊C4C2×D311yescharacteristic
162C2×C42—31no—
163C2×C42—31no—
164C2×C42—31no—
165C2×C42—31no—
166#95—31no—
167#95—31no—
168#95—31no—
169#95—31no—
170#95C2×D311yescharacteristic
171#95C2×D311yescharacteristic
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×C3C22×C412yes—
183C23×C3C22×C412yes—
184C23×C3C2×D412yes—
185C23×C3#4112yes—
186C23×C3C2411yescharacteristic
187C22×D3—21no—
188C22×D3—21no—
189C22×D3—21no—
190C22×D3—21no—
191C22×D3—21no—
192C22×D3—21no—
193C22×D3C22×C412yes—
194C3×D4—28no—
195C2×C3×C4—21no—
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—44no—
209C2×C3×C4C22×C412yes—
210C2×C3×C4C2×D412yes—
211#67—28no—
212#67—28no—
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—28no—
224C2×Dic3—21no—
225C2×Dic3—21no—
226C2×Dic3—21no—
227C2×Dic3—44no—
228C2×Dic3—44no—
229C2×Dic3C22×C412yes—
230C2×Dic3C22×C412yes—
231C2×Dic3C22×C412yes—
232C2×Dic3C2×D412yes—
233C2×Dic3C2×D412yes—
234C2×Dic3C2×D412yes—
235C2×Dic3C2×D412yes—
236C2×Dic3#4112yes—
237C4×D3—44no—
238C24—31no—
239C24C22×D311yescharacteristic
240C24C22×D311yescharacteristic
241C2×D4—22no—
242C2×D4—22no—
243C2×D4—34no—
244C2×D4—34no—
245C2×D4—62no—
246C2×D4—62no—
247C2×D4—62no—
248C2×D4—62no—
249C2×D4C4×D314yescomplemented
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—31no—
253C22×C4—34no—
254C22×C4—31no—
255C22×C4—31no—
256C22×C4—31no—
257C22×C4—31no—
258C22×C4—31no—
259C22×C4—31no—
260C22×C4—41no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4—62no—
264C22×C4—62no—
265C22×C4—62no—
266C22×C4—62no—
267C22×C4—62no—
268C22×C4—62no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4C22×D311yescharacteristic
273C22×C4C22×D311yescharacteristic
274C22×C4C22×D311yescharacteristic
275C22×C4C22×D311yescharacteristic
276C22×C4C22×D311yescharacteristic
277C4⋊C4—34no—
278C4⋊C4—34no—
279C4⋊C4—62no—
280C4⋊C4—62no—
281C4⋊C4—62no—
282C4⋊C4—62no—
283C4⋊C4—62no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—34no—
287C22⋊C4—34no—
288C22⋊C4—34no—
289C22⋊C4—41no—
290C22⋊C4—62no—
291C22⋊C4—62no—
292C22⋊C4—62no—
293C22⋊C4—121no—
294C42—62no—
295C42—62no—
296C42—62no—
297C42—62no—
298C22×C3—22no—
299C22×C3—22no—
300C22×C3—22no—
301C22×C3—22no—
302C22×C3—22no—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3—22no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3—41no—
311C22×C3C4×D414yescomplemented
312C22×C3C23×C411yescharacteristic
313C22×C3C22×D411yescharacteristic
314C22×C3C22×D411yescharacteristic
315C22×C3C2×#4111yescharacteristic
316C22×C3C2×#4111yescharacteristic
317C22×C3H(5,𝔽2)11yescharacteristic
318C22×C32−1+411yescharacteristic
319C2×D3—22no—
320C2×D3—22no—
321C2×D3—22no—
322C2×D3—41no—
323C2×D3—41no—
324C2×D3—41no—
325C2×D3—41no—
326C3×C4—22no—
327C3×C4—22no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4—41no—
331Dic3—22no—
332Dic3—22no—
333Dic3—22no—
334Dic3—22no—
335Dic3—22no—
336Dic3—41no—
337Dic3C4×D414yescomplemented
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—21no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—32no—
349C23—61no—
350C23—61no—
351C23—61no—
352C23—61no—
353C23—61no—
354C23—61no—
355C23C2×C4×D312yes—
356C23C2×C4×D312yes—
357C23D3×D412yes—
358C23#23712yes—
359C23C23×D311yescharacteristic
360D4—28no—
361D4—68no—
362D4—68no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—32no—
377C2×C4—32no—
378C2×C4—32no—
379C2×C4—32no—
380C2×C4—32no—
381C2×C4—32no—
382C2×C4—32no—
383C2×C4—32no—
384C2×C4—44no—
385C2×C4—68no—
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—124no—
400C2×C4—124no—
401C2×C4—124no—
402C2×C4C2×C4×D312yes—
403C2×C4D3×D412yes—
404C2×C3—24no—
405C2×C3—42no—
406C2×C3C2×C4×D411yescharacteristic
407C2×C3C2×C4×D411yescharacteristic
408C2×C3#52011yescharacteristic
409C2×C3D4211yescharacteristic
410C2×C3#54711yescharacteristic
411C2×C3#54811yescharacteristic
412C2×C3#55311yescharacteristic
413D3—42no—
414C22—22no—
415C22—22no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—41no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—62no—
428C22—62no—
429C22—62no—
430C22—121no—
431C22—121no—
432C22—121no—
433C22—121no—
434C22#67⋊C414yescomplemented
435C22C22×C4×D311yescharacteristic
436C22C2×D3×D411yescharacteristic
437C22C2×D3×D411yescharacteristic
438C22C2×#23711yescharacteristic
439C22#100411yescharacteristic
440C22D3×#4111yescharacteristic
441C22#101011yescharacteristic
442C4—22no—
443C4—22no—
444C4—34no—
445C4—41no—
446C4—41no—
447C4—41no—
448C4—62no—
449C4—62no—
450C4—62no—
451C4—62no—
452C4—62no—
453C4—121no—
454C3#236111yescharacteristic, complemented, Sylow
455C2—24no—
456C2—42no—
457C2—122no—
458C2C2×#67⋊C411yescharacteristic
459C2C4×D3×D411yescharacteristic
460C2#554611yescharacteristic
461C2#559011yescharacteristic
462C2#559811yescharacteristic
463C2#560111yescharacteristic
464C2#565511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#386C2
socle, cosocleC23×C3C24
prime radicalC2×#939C2
hypercenter#386D3
hypocenterC3#2361
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic3C2semidirect
3. C2×C4×#67C2semidirect
4. C22×Dic3⋊C4C2semidirect
5. C2×#67⋊C4C2semidirect
6. C3×#386C2semidirect
7. #4958C2semidirect
8. #4957C2semidirect
9. #4946C2semidirect
10. #4945C2semidirect
11. Dic3×C22⋊C4C2semidirect
12. #4669C2semidirect
13. #4668C2semidirect
14. #4649C2semidirect
15. #4647C2semidirect
31. #939C4semidirect
67. #386D3semidirect
77. C2×C3×D4C2×C4semidirect
90. C2×#67C2×C4semidirect
91. C2×#67C2×C4semidirect
100. C22×Dic3C2×C4semidirect
115. #217C2×C4semidirect
116. #217D4semidirect
120. (C2×D3)⋊C4C2×C4semidirect
126. Dic3⋊C4C2×C4semidirect
127. Dic3⋊C4D4semidirect
249. C2×D4C4×D3semidirect
311. C22×C3C4×D4semidirect
337. Dic3C4×D4semidirect
434. C22#67⋊C4semidirect
454. C3#2361semidirect

Maximal subgroups

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

subgroupindex
C2×#9392
C2×D4×Dic32
C2×C4×#672
C22×Dic3⋊C42
C2×#67⋊C42
C3×#3862
#49582
#49572
#49462
#49452
Dic3×C22⋊C42
#46692
#46682
#46492
#46472
#23613