Finite Group Databasebeta

← #74401

Subgroups — #74401

Order 384.

576 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#884C211yesmaximal, characteristic
2C3×#385C211yesmaximal, characteristic, complemented
3#5213C211yesmaximal, characteristic, complemented
4#5213C211yesmaximal, characteristic, complemented
5#4990C211yesmaximal, characteristic, complemented
6#4988C211yesmaximal, characteristic, complemented
7#4987C211yesmaximal, characteristic, complemented
8#4978C211yesmaximal, characteristic, complemented
9#4957C211yesmaximal, characteristic, complemented
10#4956C211yesmaximal, characteristic, complemented
11#4951C211yesmaximal, characteristic, complemented
12#4668C211yesmaximal, characteristic, complemented
13#4665C211yesmaximal, characteristic, complemented
14#4664C211yesmaximal, characteristic, complemented
15#4652C211yesmaximal, characteristic, complemented
16#2377—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
29C2×(C2×D3)⋊C4C2211yescharacteristic
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×C4⋊Dic3C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×C4×Dic3C2211yescharacteristic
41C2×C4×Dic3C2211yescharacteristic
42#884C418yescomplemented
43C3×#95C2211yescharacteristic
44C3×#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C2×#121—31no—
54#390—31no—
55#390—31no—
56#388—31no—
57#388—31no—
58#387—31no—
59#387—31no—
60#386—31no—
61#386—31no—
62#386—31no—
63#386—31no—
64#385—31no—
65#385D311yescharacteristic, complemented
66#381—31no—
67#380—31no—
68C24×C3C2311yescharacteristic
69C23×D3C2311yescharacteristic
70C23×D3C2311yescharacteristic
71C22×C3×C4—41no—
72C22×C3×C4—41no—
73C22×C3×C4—41no—
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—41no—
81C2×#67—41no—
82C2×#67—41no—
83C2×#67—41no—
84C2×#67C2×C414yes—
85C2×#67C2×C414yescomplemented
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C2×D12—41no—
93C2×D12—41no—
94C2×D12C2×C414yescomplemented
95C2×C4×D3—41no—
96C2×C4×D3—41no—
97C2×C4×D3—41no—
98C2×C4×D3—41no—
99C2×C4×D3—41no—
100C2×C4×D3C2×C414yescomplemented
101C3×C4⋊C4—41no—
102C3×C4⋊C4—41no—
103C3×C22⋊C4—41no—
104C3×C22⋊C4C2×C414yescomplemented
105#217—41no—
106#217—41no—
107(C2×D3)⋊C4—41no—
108(C2×D3)⋊C4—41no—
109(C2×D3)⋊C4—41no—
110(C2×D3)⋊C4—41no—
111(C2×D3)⋊C4—41no—
112(C2×D3)⋊C4—41no—
113(C2×D3)⋊C4—41no—
114(C2×D3)⋊C4C2×C414yescomplemented
115C4⋊Dic3—41no—
116Dic3⋊C4—41no—
117Dic3⋊C4C2×C414yescomplemented
118C4×Dic3—41no—
119C4×Dic3—41no—
120C22×D4—31no—
121C22×D4—31no—
122C22×D4—31no—
123C23×C4—31no—
124C23×C4—31no—
125C23×C4C2×D311yescharacteristic
126#121—38no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4C2×D311yescharacteristic
131C2×C4⋊C4C2×D311yescharacteristic
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⋊C4C2×D311yescharacteristic
143C2×C22⋊C4C2×D311yescharacteristic
144C2×C42—31no—
145C2×C42—31no—
146#95—31no—
147#95—31no—
148#95—31no—
149#95—31no—
150#95—31no—
151#95—31no—
152#95—31no—
153#95—31no—
154#95C2×D311yescharacteristic
155#95C2×D311yescharacteristic
156C23×C3—21no—
157C23×C3—21no—
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3C22×C412yes—
163C23×C3C2411yescharacteristic
164C22×D3—21no—
165C22×D3—21no—
166C22×D3—21no—
167C22×D3—21no—
168C22×D3—21no—
169C22×D3—21no—
170C22×D3—21no—
171C22×D3—21no—
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3C22×C412yes—
177C22×D3C22×C412yes—
178C2×C3×C4—21no—
179C2×C3×C4—21no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—21no—
183C2×C3×C4—21no—
184C2×C3×C4—21no—
185C2×C3×C4—21no—
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—44no—
195C2×C3×C4C22×C412yes—
196C2×C3×C4C22×C412yes—
197#67—44no—
198#67—44no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
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×Dic3C22×C412yes—
216C2×Dic3C22×C412yes—
217D12—44no—
218C4×D3—44no—
219C4×D3—44no—
220C24—31no—
221C24—31no—
222C24C22×D311yescharacteristic
223C2×D4—34no—
224C2×D4—34no—
225C2×D4—34no—
226C2×D4—121no—
227C2×D4—121no—
228C2×D4—121no—
229C2×D4—121no—
230C2×D4—121no—
231C2×D4—121no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—34no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—41no—
240C22×C4—41no—
241C22×C4—41no—
242C22×C4—121no—
243C22×C4—121no—
244C22×C4—121no—
245C22×C4—121no—
246C22×C4—121no—
247C22×C4C22×D311yescharacteristic
248C22×C4C22×D311yescharacteristic
249C22×C4C22×D311yescharacteristic
250C22×C4C22×D311yescharacteristic
251C22×C4C22×D311yescharacteristic
252C22×C4C22×D311yescharacteristic
253C4⋊C4—34no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—121no—
257C4⋊C4—121no—
258C22⋊C4—34no—
259C22⋊C4—41no—
260C22⋊C4—121no—
261C22⋊C4—121no—
262C22⋊C4—121no—
263C22⋊C4—121no—
264C22⋊C4—121no—
265C22⋊C4—121no—
266C22⋊C4—121no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C22⋊C4C4×D314yescomplemented
270C42—121no—
271C42—121no—
272C22×C3—22no—
273C22×C3—22no—
274C22×C3—22no—
275C22×C3—41no—
276C22×C3—41no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3C23×C411yescharacteristic
280C22×C3H(5,𝔽2)11yescharacteristic
281C22×C3H(5,𝔽2)11yescharacteristic
282C22×C3H(5,𝔽2)11yescharacteristic
283C22×C3H(5,𝔽2)11yescharacteristic
284C22×C3H(5,𝔽2)11yescharacteristic
285C22×C32−1+411yescharacteristic
286C2×D3—22no—
287C2×D3—22no—
288C2×D3—22no—
289C2×D3—22no—
290C2×D3—22no—
291C2×D3—22no—
292C2×D3—41no—
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—41no—
298C2×D3—41no—
299C2×D3—41no—
300C3×C4—22no—
301C3×C4—22no—
302C3×C4—41no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—41no—
306Dic3—22no—
307Dic3—22no—
308Dic3—41no—
309Dic3—41no—
310Dic3—41no—
311Dic3—41no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—32no—
319C23—32no—
320C23—61no—
321C23—61no—
322C23—61no—
323C23—61no—
324C23—61no—
325C23—61no—
326C23—61no—
327C23—61no—
328C23—61no—
329C23—61no—
330C23—61no—
331C23—61no—
332C23C2×C4×D312yes—
333C23C23×D311yescharacteristic
334D4—124no—
335D4—124no—
336D4—124no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—32no—
354C2×C4—32no—
355C2×C4—44no—
356C2×C4—61no—
357C2×C4—61no—
358C2×C4—61no—
359C2×C4—61no—
360C2×C4—61no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—61no—
365C2×C4—61no—
366C2×C4—61no—
367C2×C4—61no—
368C2×C4—61no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—61no—
372C2×C4—124no—
373C2×C4—124no—
374C2×C4C2×C4×D312yes—
375C2×C4C2×C4×D312yes—
376C2×C3—42no—
377C2×C3#51811yescharacteristic
378C2×C3#51811yescharacteristic
379C2×C3#52011yescharacteristic
380C2×C3#56011yescharacteristic
381C2×C3#56011yescharacteristic
382C2×C3#56211yescharacteristic
383C2×C3#56211yescharacteristic
384D3—42no—
385D3—42no—
386C22—22no—
387C22—22no—
388C22—22no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—62no—
394C22—62no—
395C22—62no—
396C22—62no—
397C22—62no—
398C22—62no—
399C22—121no—
400C22—121no—
401C22—121no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22C22×C4×D311yescharacteristic
408C22#100411yescharacteristic
409C22#100411yescharacteristic
410C22#100411yescharacteristic
411C22#100711yescharacteristic
412C22#100911yescharacteristic
413C22#100911yescharacteristic
414C4—22no—
415C4—22no—
416C4—41no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—62no—
421C4—62no—
422C4—121no—
423C4—121no—
424C4—121no—
425C4—121no—
426C3#237711yescharacteristic, complemented, Sylow
427C2—42no—
428C2—122no—
429C2—122no—
430C2#550411yescharacteristic
431C2#554511yescharacteristic
432C2#554511yescharacteristic
433C2#559511yescharacteristic
434C2#564711yescharacteristic
435C2#564711yescharacteristic
436C2#566611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutatorC22×C3C23×C4
FittingC3×#385C2
socle, cosocleC23×C3C24
prime radicalC2×#884C2
hypercenter#385D3
hypocenterC3#2377
LCSG ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C3×#385C2semidirect
3. #5213C2semidirect
4. #5213C2semidirect
5. #4990C2semidirect
6. #4988C2semidirect
7. #4987C2semidirect
8. #4978C2semidirect
9. #4957C2semidirect
10. #4956C2semidirect
11. #4951C2semidirect
12. #4668C2semidirect
13. #4665C2semidirect
14. #4664C2semidirect
15. #4652C2semidirect
42. #884C4semidirect
65. #385D3semidirect
85. C2×#67C2×C4semidirect
94. C2×D12C2×C4semidirect
100. C2×C4×D3C2×C4semidirect
104. C3×C22⋊C4C2×C4semidirect
114. (C2×D3)⋊C4C2×C4semidirect
117. Dic3⋊C4C2×C4semidirect
269. C22⋊C4C4×D3semidirect
426. C3#2377semidirect

Maximal subgroups

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

subgroupindex
C2×#8842
C3×#3852
#52132
#49902
#49882
#49872
#49782
#49572
#49562
#49512
#46682
#46652
#46642
#46522
#23773