Finite Group Databasebeta

← #74600

Subgroups — #74600

Order 384.

864 conjugacy classes of subgroups.

The subgroup lattice has 472 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#235C211yesmaximal, characteristic
2C2×#894C211yesmaximal, characteristic, complemented
3C2×D3×C4⋊C4C211yesmaximal, characteristic, complemented
4C2×#883C211yesmaximal, characteristic, complemented
5C3×#385C211yesmaximal, characteristic, complemented
6#5210C211yesmaximal, characteristic, complemented
7#4989C211yesmaximal, characteristic, complemented
8#4979C211yesmaximal, characteristic, complemented
9#4961C211yesmaximal, characteristic, complemented
10#4959C211yesmaximal, characteristic, complemented
11#4956C211yesmaximal, characteristic, complemented
12#4668C211yesmaximal, characteristic, complemented
13#4665C211yesmaximal, characteristic, complemented
14#4664C211yesmaximal, characteristic, complemented
15#4656C211yesmaximal, characteristic, complemented
16#2406—31nomaximal, Sylow
17C23×C3×C4C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C2×#235—24no—
21C2×#235—24no—
22C2×#235—24no—
23C22×D12C2211yescharacteristic
24C22×C4×D3C2211yescharacteristic
25C22×C4×D3C2211yescharacteristic
26C22×C4×D3C2211yescharacteristic
27C22×C4×D3C2211yescharacteristic
28C22×Dic6C2211yescharacteristic
29C2×C3×C4⋊C4C2211yescharacteristic
30C2×C3×C4⋊C4C2211yescharacteristic
31C2×C3×C22⋊C4C2211yescharacteristic
32C2×C3×C22⋊C4C2211yescharacteristic
33C2×#217C2211yescharacteristic
34C2×#217C2211yescharacteristic
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×C4⋊Dic3C2211yescharacteristic
42C2×C4⋊Dic3C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44C2×Dic3⋊C4C2211yescharacteristic
45C2×Dic3⋊C4C2211yescharacteristic
46C2×Dic3⋊C4C2211yescharacteristic
47#894—24no—
48D3×C4⋊C4—24no—
49#883—24no—
50C3×#95C2211yescharacteristic
51C3×#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C3⋊#95C2211yescharacteristic
57C3⋊#95C2211yescharacteristic
58C22×#41—31no—
59C2×#123—31no—
60C2×#123—31no—
61C22×C4⋊C4—31no—
62#395—31no—
63#395—31no—
64#394—31no—
65#394—31no—
66#386—31no—
67#386—31no—
68#385—31no—
69#385—31no—
70#385—31no—
71#385D311yescharacteristic, complemented
72#381—31no—
73C24×C3C2311yescharacteristic
74C23×D3C2311yescharacteristic
75C23×D3C2311yescharacteristic
76C22×C3×C4—22no—
77C22×C3×C4—22no—
78C22×C3×C4—22no—
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C22×C3×C4C2311yescharacteristic
83C22×C3×C4C2311yescharacteristic
84C22×C3×C4C2311yescharacteristic
85C2×#67—22no—
86C2×#67—22no—
87C2×#67—22no—
88C2×#67—22no—
89C2×#67—22no—
90C2×#67D414yes—
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C22×Dic3C2311yescharacteristic
97#235—232no—
98C2×D12—22no—
99C2×D12—22no—
100C2×D12D414yes—
101C2×C4×D3—22no—
102C2×C4×D3—22no—
103C2×C4×D3—22no—
104C2×C4×D3—22no—
105C2×C4×D3—22no—
106C2×C4×D3—22no—
107C2×C4×D3—22no—
108C2×C4×D3—22no—
109C2×C4×D3—22no—
110C2×C4×D3—22no—
111C2×C4×D3—22no—
112C2×C4×D3D414yes—
113C2×Dic6—22no—
114C2×Dic6—22no—
115C2×Dic6—22no—
116C3×C4⋊C4—22no—
117C3×C4⋊C4—22no—
118C3×C22⋊C4—22no—
119C3×C22⋊C4—22no—
120#217—22no—
121#217—22no—
122(C2×D3)⋊C4—22no—
123(C2×D3)⋊C4—22no—
124(C2×D3)⋊C4—22no—
125(C2×D3)⋊C4—22no—
126(C2×D3)⋊C4—22no—
127(C2×D3)⋊C4—22no—
128C4⋊Dic3—22no—
129C4⋊Dic3—22no—
130Dic3⋊C4—22no—
131Dic3⋊C4—22no—
132Dic3⋊C4—22no—
133Dic3⋊C4—22no—
134C2×#41—64no—
135C2×#41—64no—
136C2×#41—64no—
137C22×Q8—31no—
138C22×D4—31no—
139C22×D4—31no—
140C22×D4—31no—
141C23×C4—31no—
142C23×C4—31no—
143C23×C4—31no—
144C23×C4—31no—
145C23×C4C2×D311yescharacteristic
146#123—64no—
147#123—64no—
148C2×C4⋊C4—31no—
149C2×C4⋊C4—31no—
150C2×C4⋊C4—31no—
151C2×C4⋊C4—31no—
152C2×C4⋊C4—31no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—64no—
155C2×C4⋊C4C2×D311yescharacteristic
156C2×C4⋊C4C2×D311yescharacteristic
157C2×C22⋊C4—31no—
158C2×C22⋊C4—31no—
159C2×C22⋊C4—31no—
160C2×C22⋊C4—31no—
161C2×C22⋊C4—31no—
162C2×C22⋊C4—31no—
163C2×C22⋊C4—31no—
164C2×C22⋊C4—31no—
165C2×C22⋊C4C2×D311yescharacteristic
166C2×C22⋊C4C2×D311yescharacteristic
167#95—31no—
168#95—31no—
169#95—31no—
170#95—31no—
171#95—31no—
172#95—31no—
173#95C2×D311yescharacteristic
174#95C2×D311yescharacteristic
175C23×C3—21no—
176C23×C3—21no—
177C23×C3—21no—
178C23×C3—21no—
179C23×C3—21no—
180C23×C3—21no—
181C23×C3C2×D412yes—
182C23×C3C2411yescharacteristic
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—21no—
192C22×D3—21no—
193C22×D3C2×D412yes—
194C22×D3C2×D412yes—
195C22×D3C2×D412yes—
196C22×D3C2×D412yes—
197C2×C3×C4—28no—
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—21no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4C2×D412yes—
214C2×C3×C4C2×D412yes—
215C2×C3×C4#4112yescomplemented
216#67—28no—
217#67—28no—
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×Dic3C2×D412yes—
234C2×Dic3C2×D412yes—
235C2×Dic3#4112yescomplemented
236D12—28no—
237C4×D3—28no—
238C4×D3—28no—
239C4×D3—28no—
240C4×D3—44no—
241Dic6—28no—
242C24—31no—
243C24—31no—
244C24C22×D311yescharacteristic
245#41—632no—
246C2×Q8—62no—
247C2×Q8—62no—
248C2×Q8—62no—
249C2×D4—34no—
250C2×D4—34no—
251C2×D4—62no—
252C2×D4—62no—
253C2×D4—62no—
254C2×D4—62no—
255C2×D4—62no—
256C2×D4—62no—
257C2×D4—62no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—31no—
262C22×C4—31no—
263C22×C4—31no—
264C22×C4—34no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—31no—
268C22×C4—62no—
269C22×C4—62no—
270C22×C4—62no—
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×C4C22×D311yescharacteristic
280C22×C4C22×D311yescharacteristic
281C22×C4C22×D311yescharacteristic
282C22×C4C22×D311yescharacteristic
283C22×C4C22×D311yescharacteristic
284C22×C4C22×D311yescharacteristic
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—62no—
288C4⋊C4—62no—
289C4⋊C4—62no—
290C4⋊C4—62no—
291C4⋊C4—62no—
292C4⋊C4—62no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—62no—
296C22⋊C4—62no—
297C22⋊C4—62no—
298C22⋊C4—62no—
299C22⋊C4—62no—
300C22⋊C4—62no—
301C22⋊C4—62no—
302C22⋊C4—62no—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3—41no—
307C22×C3—41no—
308C22×C3—41no—
309C22×C3—41no—
310C22×C3C22×D411yescharacteristic
311C22×C3C22×D411yescharacteristic
312C22×C3C22×D411yescharacteristic
313C22×C3C2×#4111yescharacteristic
314C22×C3C2×#4111yescharacteristic
315C22×C32−1+411yescharacteristic
316C22×C32−1+411yescharacteristic
317C2×D3—22no—
318C2×D3—22no—
319C2×D3—22no—
320C2×D3—22no—
321C2×D3—22no—
322C2×D3—22no—
323C2×D3—22no—
324C2×D3—22no—
325C2×D3—22no—
326C2×D3—41no—
327C2×D3—41no—
328C2×D3—41no—
329C2×D3—41no—
330C2×D3#12014yes—
331C3×C4—22no—
332C3×C4—22no—
333C3×C4—22no—
334C3×C4—41no—
335C3×C4—41no—
336C3×C4—41no—
337Dic3—22no—
338Dic3—22no—
339Dic3—22no—
340Dic3—41no—
341Dic3—41no—
342Dic3—41no—
343C23—21no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—32no—
350C23—32no—
351C23—32no—
352C23—32no—
353C23—61no—
354C23—61no—
355C23—61no—
356C23—61no—
357C23—61no—
358C23—61no—
359C23—61no—
360C23—61no—
361C23—61no—
362C23—61no—
363C23D3×D412yes—
364C23C23×D311yescharacteristic
365Q8—68no—
366D4—68no—
367D4—68no—
368D4—68no—
369C2×C4—28no—
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—21no—
383C2×C4—21no—
384C2×C4—21no—
385C2×C4—32no—
386C2×C4—32no—
387C2×C4—32no—
388C2×C4—68no—
389C2×C4—68no—
390C2×C4—68no—
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—61no—
406C2×C4—124no—
407C2×C4#23512yescomplemented
408C2×C4D3×D412yes—
409C2×C4D3×D412yes—
410C2×C3—42no—
411C2×C3C2×#12011yescharacteristic
412C2×C3#52511yescharacteristic
413C2×C3#53611yescharacteristic
414C2×C3#54711yescharacteristic
415C2×C3#54711yescharacteristic
416C2×C3#54711yescharacteristic
417C2×C3#54711yescharacteristic
418D3—24no—
419D3—42no—
420C22—22no—
421C22—22no—
422C22—22no—
423C22—34no—
424C22—41no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22—62no—
429C22—62no—
430C22—62no—
431C22—62no—
432C22—62no—
433C22—62no—
434C22—62no—
435C22—62no—
436C22—62no—
437C22—121no—
438C22—121no—
439C22—121no—
440C22—121no—
441C22C2×#23511yescharacteristic
442C22C2×D3×D411yescharacteristic
443C22C2×D3×D411yescharacteristic
444C22C2×D3×D411yescharacteristic
445C22#100711yescharacteristic
446C22D3×#4111yescharacteristic
447C22#101011yescharacteristic
448C4—22no—
449C4—22no—
450C4—22no—
451C4—41no—
452C4—41no—
453C4—41no—
454C4—62no—
455C4—62no—
456C4—62no—
457C4—121no—
458C4—121no—
459C4—121no—
460C3#240611yescharacteristic, complemented, Sylow
461C2—42no—
462C2—64no—
463C2—122no—
464C2#550711yescharacteristic
465C2#554711yescharacteristic
466C2#555111yescharacteristic
467C2D3×#12011yescharacteristic
468C2#562811yescharacteristic
469C2#563111yescharacteristic
470C2#565511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC22×#235C2
hypercenter#385D3
hypocenterC3#2406
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#894C2semidirect
3. C2×D3×C4⋊C4C2semidirect
4. C2×#883C2semidirect
5. C3×#385C2semidirect
6. #5210C2semidirect
7. #4989C2semidirect
8. #4979C2semidirect
9. #4961C2semidirect
10. #4959C2semidirect
11. #4956C2semidirect
12. #4668C2semidirect
13. #4665C2semidirect
14. #4664C2semidirect
15. #4656C2semidirect
71. #385D3semidirect
215. C2×C3×C4#41semidirect
235. C2×Dic3#41semidirect
407. C2×C4#235semidirect
460. C3#2406semidirect

Maximal subgroups

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

subgroupindex
C22×#2352
C2×#8942
C2×D3×C4⋊C42
C2×#8832
C3×#3852
#52102
#49892
#49792
#49612
#49592
#49562
#46682
#46652
#46642
#46562
#24063