Finite Group Databasebeta

← #74385

Subgroups — #74385

Order 384.

668 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#67C211yesmaximal, characteristic, complemented
2C2×#892C211yesmaximal, characteristic, complemented
3C2×#884C211yesmaximal, characteristic
4C3×#385C211yesmaximal, characteristic, complemented
5#5213C211yesmaximal, characteristic, complemented
6#4989C211yesmaximal, characteristic, complemented
7#4988C211yesmaximal, characteristic, complemented
8#4976C211yesmaximal, characteristic, complemented
9#4957C211yesmaximal, characteristic, complemented
10#4957C211yesmaximal, characteristic, complemented
11#4951C211yesmaximal, characteristic, complemented
12#4670C211yesmaximal, characteristic, complemented
13#4667C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4649C211yesmaximal, characteristic, complemented
16#2324—31nomaximal, Sylow
17C23×C3×C4C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C22×#67C2211yescharacteristic
20C22×D12C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C22×C4×D3C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2211yescharacteristic
27C2×#217C2211yescharacteristic
28C2×#217C2211yescharacteristic
29C4×#67—24no—
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×(C2×D3)⋊C4C2211yescharacteristic
34C2×(C2×D3)⋊C4C2211yescharacteristic
35C2×(C2×D3)⋊C4C2211yescharacteristic
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×(C2×D3)⋊C4C2211yescharacteristic
38C2×C4⋊Dic3C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×C4×Dic3C2211yescharacteristic
44C2×C4×Dic3C2211yescharacteristic
45C2×C4×Dic3C2211yescharacteristic
46C2×C4×Dic3C2211yescharacteristic
47#892—24no—
48#884C418yescomplemented
49C3×#95C2211yescharacteristic
50C3×#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C2×#121—31no—
56C2×C4×D4—31no—
57C2×#117—31no—
58#390—31no—
59#390—31no—
60#390—31no—
61#389—31no—
62#388—31no—
63#388—31no—
64#386—31no—
65#386—31no—
66#385—31no—
67#385D311yescharacteristic, complemented
68#380—31no—
69C4×C4⋊C4—31no—
70C24×C3C2311yescharacteristic
71C23×D3C2311yescharacteristic
72C23×D3C2311yescharacteristic
73C22×C3×C4—22no—
74C22×C3×C4—41no—
75C22×C3×C4—41no—
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C22×C3×C4C2311yescharacteristic
82C2×#67—22no—
83C2×#67—22no—
84C2×#67—41no—
85C2×#67—41no—
86C2×#67C2×C414yes—
87C2×#67C2×C414yescomplemented
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C2×D12—41no—
95C2×D12—41no—
96C2×D12C2×C414yescomplemented
97C2×C4×D3—22no—
98C2×C4×D3—22no—
99C2×C4×D3—22no—
100C2×C4×D3—41no—
101C2×C4×D3—41no—
102C2×C4×D3C2×C414yescomplemented
103C3×C4⋊C4—22no—
104C3×C4⋊C4—41no—
105C3×C22⋊C4—41no—
106C3×C22⋊C4C2×C414yescomplemented
107#217—22no—
108#217—41no—
109(C2×D3)⋊C4—22no—
110(C2×D3)⋊C4—22no—
111(C2×D3)⋊C4—22no—
112(C2×D3)⋊C4—41no—
113(C2×D3)⋊C4—41no—
114(C2×D3)⋊C4—41no—
115(C2×D3)⋊C4—41no—
116(C2×D3)⋊C4C2×C414yescomplemented
117C4⋊Dic3—22no—
118Dic3⋊C4—22no—
119Dic3⋊C4—22no—
120Dic3⋊C4—41no—
121Dic3⋊C4C2×C414yescomplemented
122C4×Dic3—22no—
123C4×Dic3—41no—
124C4×Dic3—41no—
125C4×Dic3D414yescomplemented
126C22×D4—31no—
127C22×D4—31no—
128C22×D4—31no—
129C23×C4—31no—
130C23×C4—31no—
131C23×C4C2×D311yescharacteristic
132#121—38no—
133C4×D4—64no—
134#117—64no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4C2×D311yescharacteristic
141C2×C4⋊C4C2×D311yescharacteristic
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×C42—31no—
155C2×C42—31no—
156C2×C42—31no—
157C2×C42—31no—
158#95—31no—
159#95—31no—
160#95—31no—
161#95—31no—
162#95C2×D311yescharacteristic
163#95C2×D311yescharacteristic
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3C22×C412yes—
171C23×C3C2411yescharacteristic
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3—21no—
179C22×D3—21no—
180C22×D3—21no—
181C22×D3—21no—
182C22×D3—21no—
183C22×D3—21no—
184C22×D3C22×C412yes—
185C22×D3C22×C412yes—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
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—21no—
200C2×C3×C4—21no—
201C2×C3×C4—44no—
202C2×C3×C4C22×C412yes—
203C2×C3×C4C22×C412yes—
204C2×C3×C4C2×D412yes—
205#67—28no—
206#67—44no—
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×Dic3C22×C412yes—
221C2×Dic3C22×C412yes—
222C2×Dic3C2×D412yes—
223C2×Dic3C2×D412yes—
224C2×Dic3#4112yes—
225D12—44no—
226C4×D3—28no—
227C4×D3—44no—
228C24—31no—
229C24—31no—
230C24C22×D311yescharacteristic
231C2×D4—34no—
232C2×D4—34no—
233C2×D4—34no—
234C2×D4—62no—
235C2×D4—62no—
236C2×D4—121no—
237C2×D4—121no—
238C2×D4—121no—
239C2×D4—121no—
240C22×C4—22no—
241C22×C4—31no—
242C22×C4—31no—
243C22×C4—31no—
244C22×C4—34no—
245C22×C4—31no—
246C22×C4—31no—
247C22×C4—31no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—62no—
251C22×C4—62no—
252C22×C4—62no—
253C22×C4—121no—
254C22×C4—121no—
255C22×C4C22×D311yescharacteristic
256C22×C4C22×D311yescharacteristic
257C22×C4C22×D311yescharacteristic
258C22×C4C22×D311yescharacteristic
259C22×C4C22×D311yescharacteristic
260C22×C4C22×D311yescharacteristic
261C4⋊C4—22no—
262C4⋊C4—34no—
263C4⋊C4—41no—
264C4⋊C4—62no—
265C4⋊C4—62no—
266C4⋊C4—62no—
267C4⋊C4—121no—
268C22⋊C4—34no—
269C22⋊C4—41no—
270C22⋊C4—62no—
271C22⋊C4—62no—
272C22⋊C4—62no—
273C22⋊C4—62no—
274C22⋊C4—121no—
275C22⋊C4—121no—
276C22⋊C4—121no—
277C22⋊C4—121no—
278C22⋊C4—121no—
279C22⋊C4C4×D314yescomplemented
280C42—34no—
281C42—62no—
282C42—121no—
283C42—121no—
284C22×C3—22no—
285C22×C3—22no—
286C22×C3—22no—
287C22×C3—41no—
288C22×C3—41no—
289C22×C3—41no—
290C22×C3—41no—
291C22×C3C23×C411yescharacteristic
292C22×C3C22×D411yescharacteristic
293C22×C3C2×#4111yescharacteristic
294C22×C3H(5,𝔽2)11yescharacteristic
295C22×C3H(5,𝔽2)11yescharacteristic
296C22×C3H(5,𝔽2)11yescharacteristic
297C22×C32−1+411yescharacteristic
298C2×D3—22no—
299C2×D3—22no—
300C2×D3—22no—
301C2×D3—22no—
302C2×D3—22no—
303C2×D3—22no—
304C2×D3—41no—
305C2×D3—41no—
306C2×D3—41no—
307C2×D3—41no—
308C2×D3—41no—
309C2×D3—41no—
310C2×D3—41no—
311C2×D3—41no—
312C3×C4—22no—
313C3×C4—22no—
314C3×C4—22no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4—41no—
318Dic3—22no—
319Dic3—22no—
320Dic3—41no—
321Dic3—41no—
322Dic3—41no—
323Dic3C4×D414yes—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—32no—
331C23—32no—
332C23—61no—
333C23—61no—
334C23—61no—
335C23—61no—
336C23—61no—
337C23—61no—
338C23—61no—
339C23—61no—
340C23—61no—
341C23—61no—
342C23—61no—
343C23—61no—
344C23C2×C4×D312yes—
345C23C23×D311yescharacteristic
346D4—68no—
347D4—124no—
348D4—124no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—32no—
365C2×C4—32no—
366C2×C4—32no—
367C2×C4—32no—
368C2×C4—32no—
369C2×C4—44no—
370C2×C4—68no—
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—124no—
385C2×C4C2×C4×D312yes—
386C2×C4C2×C4×D312yes—
387C2×C4D3×D412yes—
388C2×C3—42no—
389C2×C3C2×C4×D411yescharacteristic
390C2×C3#51811yescharacteristic
391C2×C3#52011yescharacteristic
392C2×C3#53511yescharacteristic
393C2×C3#53711yescharacteristic
394C2×C3#54011yescharacteristic
395C2×C3#54211yescharacteristic
396D3—42no—
397D3—42no—
398C22—22no—
399C22—22no—
400C22—22no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—62no—
406C22—62no—
407C22—62no—
408C22—62no—
409C22—62no—
410C22—62no—
411C22—121no—
412C22—121no—
413C22—121no—
414C22—121no—
415C22—121no—
416C22—121no—
417C22—121no—
418C22—121no—
419C22C22×C4×D311yescharacteristic
420C22C2×D3×D411yescharacteristic
421C22#100411yescharacteristic
422C22#100411yescharacteristic
423C22#100711yescharacteristic
424C22D3×#4111yescharacteristic
425C22#100911yescharacteristic
426C4—22no—
427C4—22no—
428C4—22no—
429C4—34no—
430C4—41no—
431C4—41no—
432C4—41no—
433C4—62no—
434C4—62no—
435C4—121no—
436C4—121no—
437C4—121no—
438C3#232411yescharacteristic, complemented, Sylow
439C2—42no—
440C2—122no—
441C2—122no—
442C2#550411yescharacteristic
443C2C4×D3×D411yescharacteristic
444C2#554511yescharacteristic
445C2#559211yescharacteristic
446C2#562911yescharacteristic
447C2#564011yescharacteristic
448C2#566311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#385C2
socle, cosocleC23×C3C24
prime radicalC2×#884C2
hypercenter#385D3
hypocenterC3#2324
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
1. C2×C4×#67C2semidirect
2. C2×#892C2semidirect
4. C3×#385C2semidirect
5. #5213C2semidirect
6. #4989C2semidirect
7. #4988C2semidirect
8. #4976C2semidirect
9. #4957C2semidirect
10. #4957C2semidirect
11. #4951C2semidirect
12. #4670C2semidirect
13. #4667C2semidirect
14. #4665C2semidirect
15. #4649C2semidirect
48. #884C4semidirect
67. #385D3semidirect
87. C2×#67C2×C4semidirect
96. C2×D12C2×C4semidirect
102. C2×C4×D3C2×C4semidirect
106. C3×C22⋊C4C2×C4semidirect
116. (C2×D3)⋊C4C2×C4semidirect
121. Dic3⋊C4C2×C4semidirect
125. C4×Dic3D4semidirect
279. C22⋊C4C4×D3semidirect
438. C3#2324semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#672
C2×#8922
C2×#8842
C3×#3852
#52132
#49892
#49882
#49762
#49572
#49512
#46702
#46672
#46652
#46492
#23243