Finite Group Databasebeta

← #74408

Subgroups — #74408

Order 384.

560 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#67C211yesmaximal, characteristic
2C2×#886C211yesmaximal, characteristic
3C3×#385C211yesmaximal, characteristic, complemented
4C2×C3⋊#95C211yesmaximal, characteristic, complemented
5#4992C211yesmaximal, characteristic, complemented
6#4989C211yesmaximal, characteristic, complemented
7#4982C211yesmaximal, characteristic, complemented
8#4980C211yesmaximal, characteristic, complemented
9#4958C211yesmaximal, characteristic
10#4943C211yesmaximal, characteristic, complemented
11#4942C211yesmaximal, characteristic, complemented
12#4675C211yesmaximal, characteristic, complemented
13#4664C211yesmaximal, characteristic, complemented
14#4660C211yesmaximal, characteristic, complemented
15#4655C211yesmaximal, characteristic, complemented
16#2419—31nomaximal, Sylow
17C23×C3×C4C2211yescharacteristic
18C22×#67C2211yescharacteristic
19C23×Dic3C2211yescharacteristic
20C23×Dic3C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C22⋊C4C2211yescharacteristic
25C2×C3×C22⋊C4C2211yescharacteristic
26C2×#217C2211yescharacteristic
27C4×#67—24no—
28C2×(C2×D3)⋊C4C2211yescharacteristic
29C2×(C2×D3)⋊C4C2211yescharacteristic
30C2×(C2×D3)⋊C4C2211yescharacteristic
31C2×(C2×D3)⋊C4C2211yescharacteristic
32C2×(C2×D3)⋊C4C2211yescharacteristic
33C2×C4⋊Dic3C2211yescharacteristic
34C2×C4⋊Dic3C2211yescharacteristic
35C2×Dic3⋊C4C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×C4×Dic3C2211yescharacteristic
39#886—24no—
40C3×#95C2211yescharacteristic
41C3×#95C2211yescharacteristic
42C3⋊#95C2211yescharacteristic
43C3⋊#95C2211yescharacteristic
44C3⋊#95C2211yescharacteristic
45C3⋊#95C2211yescharacteristic
46C3⋊#95C2211yescharacteristic
47C3⋊#95C2211yescharacteristic
48C3⋊#95C2211yescharacteristic
49C3⋊#95C2211yescharacteristic
50C3⋊#95C2211yescharacteristic
51C3⋊#95C2211yescharacteristic
52C3⋊#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C2×#123—31no—
55C2×C4×D4—31no—
56#400—31no—
57#398—31no—
58#397—31no—
59#397—31no—
60#387—31no—
61#387—31no—
62#386—31no—
63#385—31no—
64#385D311yescharacteristic, complemented
65#381—31no—
66#381—31no—
67#380—31no—
68C2×#95—31no—
69C24×C3C2311yescharacteristic
70C23×D3C2311yescharacteristic
71C22×C3×C4—22no—
72C22×C3×C4—22no—
73C22×C3×C4—41no—
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C22×C3×C4C2311yescharacteristic
80C2×#67—22no—
81C2×#67—22no—
82C2×#67D414yes—
83C22×Dic3—22no—
84C22×Dic3—22no—
85C22×Dic3—22no—
86C22×Dic3—22no—
87C22×Dic3—22no—
88C22×Dic3—41no—
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C22×Dic3C2311yescharacteristic
93C22×Dic3C2311yescharacteristic
94C22×Dic3C2311yescharacteristic
95C22×Dic3C2311yescharacteristic
96C2×C4×D3—22no—
97C2×C4×D3—22no—
98C2×C4×D3—41no—
99C3×C4⋊C4—22no—
100C3×C4⋊C4—41no—
101C3×C22⋊C4—22no—
102C3×C22⋊C4—22no—
103#217—22no—
104(C2×D3)⋊C4—22no—
105(C2×D3)⋊C4—22no—
106(C2×D3)⋊C4—22no—
107(C2×D3)⋊C4—22no—
108(C2×D3)⋊C4—41no—
109C4⋊Dic3—22no—
110C4⋊Dic3—22no—
111Dic3⋊C4—22no—
112Dic3⋊C4—22no—
113Dic3⋊C4—41no—
114C4×Dic3—22no—
115C22×D4—31no—
116C23×C4—31no—
117C23×C4—31no—
118C23×C4—31no—
119C23×C4C2×D311yescharacteristic
120#123—64no—
121C4×D4—64no—
122C2×C4⋊C4—31no—
123C2×C4⋊C4—31no—
124C2×C4⋊C4—31no—
125C2×C4⋊C4—31no—
126C2×C4⋊C4—31no—
127C2×C4⋊C4C2×D311yescharacteristic
128C2×C4⋊C4C2×D311yescharacteristic
129C2×C22⋊C4—31no—
130C2×C22⋊C4—31no—
131C2×C22⋊C4—31no—
132C2×C22⋊C4—31no—
133C2×C22⋊C4—31no—
134C2×C22⋊C4—31no—
135C2×C22⋊C4C2×D311yescharacteristic
136C2×C22⋊C4C2×D311yescharacteristic
137C2×C42—31no—
138#95—31no—
139#95—31no—
140#95—31no—
141#95—31no—
142#95—31no—
143#95—31no—
144#95—31no—
145#95—31no—
146#95—31no—
147#95—31no—
148#95—31no—
149#95—31no—
150#95C2×D311yescharacteristic
151#95C2×D311yescharacteristic
152C23×C3—21no—
153C23×C3—21no—
154C23×C3—21no—
155C23×C3—21no—
156C23×C3C2×D412yes—
157C23×C3#4112yes—
158C23×C3#4112yes—
159C23×C3C2411yescharacteristic
160C22×D3—21no—
161C22×D3—21no—
162C22×D3—21no—
163C22×D3—21no—
164C22×D3—21no—
165C22×D3—21no—
166C22×D3C2×D412yes—
167C2×C3×C4—21no—
168C2×C3×C4—21no—
169C2×C3×C4—21no—
170C2×C3×C4—21no—
171C2×C3×C4—21no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—21no—
177C2×C3×C4—21no—
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—44no—
185C2×C3×C4#4112yes—
186#67—28no—
187C2×Dic3—21no—
188C2×Dic3—21no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3—21no—
193C2×Dic3—21no—
194C2×Dic3—21no—
195C2×Dic3—21no—
196C2×Dic3—21no—
197C2×Dic3—21no—
198C2×Dic3—21no—
199C2×Dic3—21no—
200C2×Dic3—21no—
201C2×Dic3—21no—
202C2×Dic3—21no—
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—44no—
207C2×Dic3—44no—
208C2×Dic3C2×D412yes—
209C2×Dic3#4112yes—
210C4×D3—44no—
211C24—31no—
212C24C22×D311yescharacteristic
213C2×D4—34no—
214C2×D4—62no—
215C2×D4—62no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—31no—
219C22×C4—31no—
220C22×C4—31no—
221C22×C4—31no—
222C22×C4—31no—
223C22×C4—31no—
224C22×C4—31no—
225C22×C4—41no—
226C22×C4—62no—
227C22×C4—62no—
228C22×C4—62no—
229C22×C4—62no—
230C22×C4—62no—
231C22×C4—62no—
232C22×C4—62no—
233C22×C4—121no—
234C22×C4—121no—
235C22×C4C22×D311yescharacteristic
236C22×C4C22×D311yescharacteristic
237C22×C4C22×D311yescharacteristic
238C22×C4C22×D311yescharacteristic
239C22×C4C22×D311yescharacteristic
240C22×C4C22×D311yescharacteristic
241C4⋊C4—22no—
242C4⋊C4—41no—
243C4⋊C4—62no—
244C4⋊C4—62no—
245C4⋊C4—62no—
246C4⋊C4—62no—
247C4⋊C4—121no—
248C22⋊C4—22no—
249C22⋊C4—22no—
250C22⋊C4—62no—
251C22⋊C4—62no—
252C22⋊C4—62no—
253C22⋊C4—62no—
254C22⋊C4—62no—
255C22⋊C4—121no—
256C42—62no—
257C22×C3—22no—
258C22×C3—22no—
259C22×C3—22no—
260C22×C3—22no—
261C22×C3—22no—
262C22×C3—22no—
263C22×C3#12314yes—
264C22×C3C22×D411yescharacteristic
265C22×C3C2×#4111yescharacteristic
266C22×C3C2×#4111yescharacteristic
267C22×C3C2×#4111yescharacteristic
268C22×C3C2×#4111yescharacteristic
269C22×C3H(5,𝔽2)11yescharacteristic
270C22×C32−1+411yescharacteristic
271C2×D3—22no—
272C2×D3—22no—
273C2×D3—22no—
274C2×D3—41no—
275C2×D3—41no—
276C2×D3—41no—
277C2×D3—41no—
278C3×C4—22no—
279C3×C4—41no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4—41no—
283C3×C4—41no—
284Dic3—22no—
285Dic3—22no—
286Dic3—41no—
287Dic3—41no—
288Dic3—41no—
289Dic3—41no—
290Dic3—41no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—32no—
296C23—61no—
297C23—61no—
298C23—61no—
299C23—61no—
300C23—61no—
301C23—61no—
302C23#23512yes—
303C23D3×D412yes—
304C23#23912yes—
305C23C23×D311yescharacteristic
306D4—68no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—32no—
325C2×C4—32no—
326C2×C4—44no—
327C2×C4—61no—
328C2×C4—61no—
329C2×C4—61no—
330C2×C4—61no—
331C2×C4—61no—
332C2×C4—61no—
333C2×C4—61no—
334C2×C4—61no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—124no—
347C2×C4—124no—
348C2×C4—124no—
349C2×C4#23712yes—
350C2×C3—24no—
351C2×C3C2×#12311yescharacteristic
352C2×C3#52911yescharacteristic
353C2×C3#53911yescharacteristic
354C2×C3#54611yescharacteristic
355C2×C3#54711yescharacteristic
356C2×C3#55111yescharacteristic
357C2×C3#55211yescharacteristic
358D3—42no—
359C22—22no—
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—62no—
366C22—62no—
367C22—62no—
368C22—121no—
369C22—121no—
370C22—121no—
371C22—121no—
372C22#89414yes—
373C22C2×#23511yescharacteristic
374C22C2×D3×D411yescharacteristic
375C22C2×#23711yescharacteristic
376C22#100411yescharacteristic
377C22C2×#23911yescharacteristic
378C22D3×#4111yescharacteristic
379C22#101011yescharacteristic
380C4—22no—
381C4—41no—
382C4—41no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—62no—
387C4—62no—
388C4—121no—
389C4—121no—
390C4—121no—
391C4—121no—
392C4—121no—
393C3#241911yescharacteristic, complemented, Sylow
394C2—24no—
395C2—122no—
396C2C2×#89411yescharacteristic
397C2#553811yescharacteristic
398C2#556411yescharacteristic
399C2#559111yescharacteristic
400C2#563711yescharacteristic
401C2#564211yescharacteristic
402C2#565511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#385C2
prime radicalC22×#67C22
hypercenter#385D3
hypocenterC3#2419
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
3. C3×#385C2semidirect
4. C2×C3⋊#95C2semidirect
5. #4992C2semidirect
6. #4989C2semidirect
7. #4982C2semidirect
8. #4980C2semidirect
10. #4943C2semidirect
11. #4942C2semidirect
12. #4675C2semidirect
13. #4664C2semidirect
14. #4660C2semidirect
15. #4655C2semidirect
64. #385D3semidirect
393. C3#2419semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#672
C2×#8862
C3×#3852
C2×C3⋊#952
#49922
#49892
#49822
#49802
#49582
#49432
#49422
#46752
#46642
#46602
#46552
#24193