Finite Group Databasebeta

← #74639

Subgroups — #74639

Order 384.

568 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#930C211yesmaximal, characteristic
2C2×#894C211yesmaximal, characteristic, complemented
3C3×#385C211yesmaximal, characteristic, complemented
4#5210C211yesmaximal, characteristic, complemented
5#4992C211yesmaximal, characteristic, complemented
6#4989C211yesmaximal, characteristic, complemented
7#4981C211yesmaximal, characteristic, complemented
8#4959C211yesmaximal, characteristic, complemented
9#4959C211yesmaximal, characteristic, complemented
10#4956C211yesmaximal, characteristic, complemented
11#4956C211yesmaximal, characteristic, complemented
12#4675C211yesmaximal, characteristic, complemented
13#4667C211yesmaximal, characteristic, complemented
14#4665C211yesmaximal, characteristic, complemented
15#4659C211yesmaximal, characteristic, complemented
16#2708—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
29#930—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
43#894—24no—
44C3×#95C2211yescharacteristic
45C3×#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#123—31no—
55C2×#121—31no—
56#400—31no—
57#398—31no—
58#397—31no—
59#397—31no—
60#394—31no—
61#394—31no—
62#386—31no—
63#386—31no—
64#386—31no—
65#385—31no—
66#385—31no—
67#385D311yescharacteristic, complemented
68#380—31no—
69C24×C3C2311yescharacteristic
70C23×D3C2311yescharacteristic
71C23×D3C2311yescharacteristic
72C22×C3×C4—22no—
73C22×C3×C4—41no—
74C22×C3×C4—41no—
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C22×C3×C4C2311yescharacteristic
81C2×#67—22no—
82C2×#67—22no—
83C2×#67—22no—
84C2×#67—22no—
85C2×#67—41no—
86C2×#67—41no—
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C2×D12—41no—
94C2×D12—41no—
95C2×D12D414yes—
96C2×C4×D3—22no—
97C2×C4×D3—22no—
98C2×C4×D3—41no—
99C2×C4×D3—41no—
100C2×C4×D3—41no—
101C2×C4×D3—41no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—41no—
104C3×C22⋊C4—22no—
105C3×C22⋊C4—41no—
106#217—22no—
107#217—41no—
108(C2×D3)⋊C4—22no—
109(C2×D3)⋊C4—22no—
110(C2×D3)⋊C4—22no—
111(C2×D3)⋊C4—22no—
112(C2×D3)⋊C4—22no—
113(C2×D3)⋊C4—22no—
114(C2×D3)⋊C4—41no—
115(C2×D3)⋊C4—41no—
116C4⋊Dic3—22no—
117Dic3⋊C4—22no—
118Dic3⋊C4—41no—
119Dic3⋊C4—41no—
120Dic3⋊C4—41no—
121C22×D4—31no—
122C22×D4—31no—
123C22×D4—31no—
124C23×C4—31no—
125C23×C4—31no—
126C23×C4C2×D311yescharacteristic
127#123—64no—
128#121—64no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4C2×D311yescharacteristic
135C2×C4⋊C4C2×D311yescharacteristic
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⋊C4—31no—
143C2×C22⋊C4—31no—
144C2×C22⋊C4—31no—
145C2×C22⋊C4—31no—
146C2×C22⋊C4C2×D311yescharacteristic
147C2×C22⋊C4C2×D311yescharacteristic
148#95—31no—
149#95—31no—
150#95—31no—
151#95—31no—
152#95—31no—
153#95—31no—
154#95—31no—
155#95—31no—
156#95C2×D311yescharacteristic
157#95C2×D311yescharacteristic
158C23×C3—21no—
159C23×C3—21no—
160C23×C3—21no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—21no—
164C23×C3#4112yes—
165C23×C3C2411yescharacteristic
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×D3—21no—
177C22×D3—21no—
178C22×D3C2×D412yes—
179C22×D3C2×D412yes—
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—21no—
195C2×C3×C4—21no—
196C2×C3×C4—21no—
197C2×C3×C4—44no—
198C2×C3×C4C2×D412yes—
199#67—44no—
200#67—44no—
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×Dic3—21no—
216C2×Dic3—21no—
217C2×Dic3—21no—
218C2×Dic3#4112yes—
219D12—44no—
220C4×D3—44no—
221C4×D3—44no—
222C24—31no—
223C24—31no—
224C24C22×D311yescharacteristic
225C2×D4—34no—
226C2×D4—62no—
227C2×D4—62no—
228C2×D4—62no—
229C2×D4—62no—
230C2×D4—121no—
231C2×D4—121no—
232C2×D4—121no—
233C2×D4—121no—
234C22×C4—22no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—31no—
241C22×C4—41no—
242C22×C4—41no—
243C22×C4—62no—
244C22×C4—62no—
245C22×C4—121no—
246C22×C4—121no—
247C22×C4—121no—
248C22×C4—121no—
249C22×C4C22×D311yescharacteristic
250C22×C4C22×D311yescharacteristic
251C22×C4C22×D311yescharacteristic
252C22×C4C22×D311yescharacteristic
253C22×C4C22×D311yescharacteristic
254C22×C4C22×D311yescharacteristic
255C4⋊C4—22no—
256C4⋊C4—41no—
257C4⋊C4—62no—
258C4⋊C4—62no—
259C4⋊C4—121no—
260C4⋊C4—121no—
261C4⋊C4—121no—
262C22⋊C4—22no—
263C22⋊C4—41no—
264C22⋊C4—62no—
265C22⋊C4—62no—
266C22⋊C4—62no—
267C22⋊C4—62no—
268C22⋊C4—62no—
269C22⋊C4—62no—
270C22⋊C4—62no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C22⋊C4—121no—
274C22×C3—22no—
275C22×C3—22no—
276C22×C3—22no—
277C22×C3—41no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3C22×D411yescharacteristic
282C22×C3C2×#4111yescharacteristic
283C22×C3C2×#4111yescharacteristic
284C22×C3H(5,𝔽2)11yescharacteristic
285C22×C3H(5,𝔽2)11yescharacteristic
286C22×C3H(5,𝔽2)11yescharacteristic
287C22×C32−1+411yescharacteristic
288C2×D3—22no—
289C2×D3—22no—
290C2×D3—22no—
291C2×D3—22no—
292C2×D3—22no—
293C2×D3—22no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—41no—
297C2×D3—41no—
298C2×D3—41no—
299C2×D3—41no—
300C2×D3—41no—
301C2×D3—41no—
302C3×C4—22no—
303C3×C4—41no—
304C3×C4—41no—
305C3×C4—41no—
306C3×C4—41no—
307C3×C4—41no—
308Dic3—22no—
309Dic3—41no—
310Dic3—41no—
311Dic3—41no—
312Dic3—41no—
313Dic3—41no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23—32no—
321C23—32no—
322C23—61no—
323C23—61no—
324C23—61no—
325C23—61no—
326C23—61no—
327C23—61no—
328C23—61no—
329C23—61no—
330C23—61no—
331C23—61no—
332C23—61no—
333C23—61no—
334C23#23512yes—
335C23C23×D311yescharacteristic
336D4—124no—
337D4—124no—
338D4—124no—
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—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—32no—
357C2×C4—44no—
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—61no—
373C2×C4—61no—
374C2×C4—61no—
375C2×C4—124no—
376C2×C4—124no—
377C2×C4D3×D412yes—
378C2×C3—42no—
379C2×C3#53411yescharacteristic
380C2×C3#53811yescharacteristic
381C2×C3#53911yescharacteristic
382C2×C3#54611yescharacteristic
383C2×C3#54711yescharacteristic
384C2×C3#55311yescharacteristic
385C2×C3#56211yescharacteristic
386D3—42no—
387D3—42no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—62no—
396C22—62no—
397C22—62no—
398C22—62no—
399C22—62no—
400C22—62no—
401C22—121no—
402C22—121no—
403C22—121no—
404C22—121no—
405C22—121no—
406C22—121no—
407C22—121no—
408C22—121no—
409C22C2×#23511yescharacteristic
410C22C2×D3×D411yescharacteristic
411C22#100411yescharacteristic
412C22#100411yescharacteristic
413C22#100711yescharacteristic
414C22D3×#4111yescharacteristic
415C22#100911yescharacteristic
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—62no—
423C4—121no—
424C4—121no—
425C4—121no—
426C4—121no—
427C4—121no—
428C3#270811yescharacteristic, complemented, Sylow
429C2—42no—
430C2—122no—
431C2—122no—
432C2#551511yescharacteristic
433C2#554011yescharacteristic
434C2#555011yescharacteristic
435C2#558911yescharacteristic
436C2#563111yescharacteristic
437C2#564711yescharacteristic
438C2#565711yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC2×#930C2
hypercenter#385D3
hypocenterC3#2708
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. C3×#385C2semidirect
4. #5210C2semidirect
5. #4992C2semidirect
6. #4989C2semidirect
7. #4981C2semidirect
8. #4959C2semidirect
9. #4959C2semidirect
10. #4956C2semidirect
11. #4956C2semidirect
12. #4675C2semidirect
13. #4667C2semidirect
14. #4665C2semidirect
15. #4659C2semidirect
67. #385D3semidirect
428. C3#2708semidirect

Maximal subgroups

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

subgroupindex
C2×#9302
C2×#8942
C3×#3852
#52102
#49922
#49892
#49812
#49592
#49562
#46752
#46672
#46652
#46592
#27083