Finite Group Databasebeta

← #78344

Subgroups — #78344

Order 384.

584 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#896C211yesmaximal, characteristic, complemented
2C2×#893C211yesmaximal, characteristic
3C3×#400C211yesmaximal, characteristic, complemented
4#4987C211yesmaximal, characteristic, complemented
5#4982C211yesmaximal, characteristic, complemented
6#4676C211yesmaximal, characteristic, complemented
7#4673C211yesmaximal, characteristic, complemented
8#4668C211yesmaximal, characteristic, complemented
9#4668C211yesmaximal, characteristic, complemented
10#4668C211yesmaximal, characteristic, complemented
11#4665C211yesmaximal, characteristic
12#4665C211yesmaximal, characteristic
13D3×#95C211yesmaximal, characteristic, complemented
14#4654C211yesmaximal, characteristic, complemented
15#4646C211yesmaximal, characteristic, complemented
16#2450—31nomaximal, Sylow
17C22×D12C2211yescharacteristic
18C22×C4×D3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C22×C4×D3C2211yescharacteristic
21C22×C4×D3C2211yescharacteristic
22C22×Dic6C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C4⋊C4C2211yescharacteristic
25C2×(C2×D3)⋊C4C2211yescharacteristic
26C2×(C2×D3)⋊C4C2211yescharacteristic
27C2×(C2×D3)⋊C4C2211yescharacteristic
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×Dic3⋊C4C2211yescharacteristic
39C2×C4×Dic3C2211yescharacteristic
40#896—24no—
41#893—24no—
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×C22⋊Q8—31no—
55C2×C4×D4—31no—
56#400D311yescharacteristic, complemented
57#398—31no—
58#396—31no—
59#393—31no—
60#391—31no—
61#387—31no—
62#386—31no—
63#386—31no—
64#385—31no—
65#385—31no—
66#385—31no—
67#380—31no—
68C2×#95—31no—
69C23×D3C2311yescharacteristic
70C23×D3C2311yescharacteristic
71C22×C3×C4C2311yescharacteristic
72C22×C3×C4C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×Dic3C2311yescharacteristic
79C22×Dic3C2311yescharacteristic
80C22×Dic3C2311yescharacteristic
81C22×Dic3C2311yescharacteristic
82C22×Dic3C2311yescharacteristic
83C22×Dic3C2311yescharacteristic
84C2×D12—22no—
85C2×D12—22no—
86C2×D12Q814yescomplemented
87C2×C4×D3—22no—
88C2×C4×D3—22no—
89C2×C4×D3—22no—
90C2×C4×D3—22no—
91C2×C4×D3—22no—
92C2×C4×D3—22no—
93C2×C4×D3—22no—
94C2×C4×D3—41no—
95C2×C4×D3—41no—
96C2×C4×D3—41no—
97C2×C4×D3—41no—
98C2×C4×D3—41no—
99C2×Dic6—22no—
100C2×Dic6—22no—
101C2×Dic6—41no—
102C3×C4⋊C4—22no—
103C3×C4⋊C4—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—
109(C2×D3)⋊C4—41no—
110(C2×D3)⋊C4—41no—
111(C2×D3)⋊C4D414yes—
112C4⋊Dic3—22no—
113C4⋊Dic3—41no—
114Dic3⋊C4—22no—
115Dic3⋊C4—22no—
116Dic3⋊C4—41no—
117Dic3⋊C4—41no—
118C4×Dic3—22no—
119C22×Q8—31no—
120C22×D4—31no—
121C23×C4—31no—
122C23×C4—31no—
123C23×C4—31no—
124C23×C4—31no—
125C22⋊Q8—64no—
126C4×D4—64no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4—31no—
130C2×C4⋊C4—31no—
131C2×C4⋊C4—31no—
132C2×C4⋊C4—31no—
133C2×C4⋊C4C2×D311yescharacteristic
134C2×C4⋊C4C2×D311yescharacteristic
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⋊C4—31no—
143C2×C42—31no—
144#95—31no—
145#95—31no—
146#95—31no—
147#95—31no—
148#95—31no—
149#95—31no—
150#95—31no—
151#95C2×D311yescharacteristic
152#95C2×D311yescharacteristic
153#95C2×D311yescharacteristic
154#95C2×D311yescharacteristic
155#95C2×D311yescharacteristic
156C23×C3C2411yescharacteristic
157C22×D3—21no—
158C22×D3—21no—
159C22×D3—21no—
160C22×D3—21no—
161C22×D3—21no—
162C22×D3—21no—
163C22×D3—21no—
164C22×D3—21no—
165C22×D3—21no—
166C22×D3—21no—
167C22×D3C2×D412yes—
168C22×D3C2×Q812yes—
169C22×D3C2×Q812yes—
170C22×D3#4112yes—
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—21no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4C2×D412yes—
191C2×C3×C4C2×Q812yes—
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—21no—
207C2×Dic3C2×D412yes—
208C2×Dic3#4112yes—
209C2×Dic3#4112yes—
210D12—28no—
211C4×D3—44no—
212C4×D3—44no—
213C4×D3—44no—
214C4×D3—44no—
215Dic6—44no—
216C24—31no—
217C24—31no—
218C2×Q8—62no—
219C2×Q8—62no—
220C2×Q8—121no—
221C2×D4—34no—
222C2×D4—62no—
223C2×D4—62no—
224C22×C4—31no—
225C22×C4—31no—
226C22×C4—31no—
227C22×C4—31no—
228C22×C4—31no—
229C22×C4—31no—
230C22×C4—62no—
231C22×C4—62no—
232C22×C4—62no—
233C22×C4—62no—
234C22×C4—62no—
235C22×C4—62no—
236C22×C4—62no—
237C22×C4—121no—
238C22×C4—121no—
239C22×C4—121no—
240C22×C4—121no—
241C22×C4—121no—
242C22×C4C22×D311yescharacteristic
243C22×C4C22×D311yescharacteristic
244C22×C4C22×D311yescharacteristic
245C22×C4C22×D311yescharacteristic
246C22×C4C22×D311yescharacteristic
247C22×C4C22×D311yescharacteristic
248C22×C4C22×D311yescharacteristic
249C4⋊C4—22no—
250C4⋊C4—22no—
251C4⋊C4—62no—
252C4⋊C4—62no—
253C4⋊C4—62no—
254C4⋊C4—121no—
255C4⋊C4—121no—
256C4⋊C4—121no—
257C22⋊C4—34no—
258C22⋊C4—62no—
259C22⋊C4—62no—
260C22⋊C4—62no—
261C22⋊C4—62no—
262C22⋊C4—121no—
263C22⋊C4—121no—
264C22⋊C4—121no—
265C42—62no—
266C22×C3C22×D411yescharacteristic
267C22×C3C22×Q811yescharacteristic
268C22×C3C2×#4111yescharacteristic
269C22×C3C2×#4111yescharacteristic
270C22×C3C2×#4111yescharacteristic
271C22×C3H(5,𝔽2)11yescharacteristic
272C22×C3H(5,𝔽2)11yescharacteristic
273C2×D3—22no—
274C2×D3—22no—
275C2×D3—22no—
276C2×D3—22no—
277C2×D3—22no—
278C2×D3—22no—
279C2×D3—22no—
280C2×D3—22no—
281C2×D3—22no—
282C2×D3—41no—
283C2×D3—41no—
284C2×D3—41no—
285C2×D3—41no—
286C2×D3C22⋊Q814yes—
287C3×C4—22no—
288C3×C4—22no—
289C3×C4—41no—
290C3×C4—41no—
291C3×C4—41no—
292C3×C4—41no—
293C3×C4—41no—
294Dic3—22no—
295Dic3—22no—
296Dic3—22no—
297Dic3—41no—
298Dic3—41no—
299Dic3—41no—
300C23—32no—
301C23—32no—
302C23—32no—
303C23—32no—
304C23—61no—
305C23—61no—
306C23—61no—
307C23—61no—
308C23—61no—
309C23—61no—
310C23—61no—
311C23—61no—
312C23—61no—
313C23—61no—
314C23C23×D311yescharacteristic
315Q8—124no—
316D4—68no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—32no—
337C2×C4—32no—
338C2×C4—32no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—61no—
344C2×C4—61no—
345C2×C4—61no—
346C2×C4—61no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—61no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—124no—
355C2×C4—124no—
356C2×C4—124no—
357C2×C4—124no—
358C2×C4D3×D412yes—
359C2×C4D3×Q812yes—
360C2×C3C2×C22⋊Q811yescharacteristic
361C2×C3#52911yescharacteristic
362C2×C3#53811yescharacteristic
363C2×C3#54611yescharacteristic
364C2×C3#54611yescharacteristic
365C2×C3#55411yescharacteristic
366C2×C3#55411yescharacteristic
367D3—24no—
368D3—42no—
369C22—34no—
370C22—62no—
371C22—62no—
372C22—62no—
373C22—62no—
374C22—62no—
375C22—62no—
376C22—62no—
377C22—62no—
378C22—62no—
379C22—121no—
380C22—121no—
381C22—121no—
382C22—121no—
383C22C2×D3×D411yescharacteristic
384C22C2×D3×Q811yescharacteristic
385C22D3×#4111yescharacteristic
386C22D3×#4111yescharacteristic
387C22D3×#4111yescharacteristic
388C22#100911yescharacteristic
389C22#100911yescharacteristic
390C4—22no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—62no—
398C4—62no—
399C4—62no—
400C4—121no—
401C4—121no—
402C4—121no—
403C3#245011yescharacteristic, complemented, Sylow
404C2—64no—
405C2—122no—
406C2D3×C22⋊Q811yescharacteristic
407C2#565411yescharacteristic
408C2#567611yescharacteristic
409C2#569011yescharacteristic
410C2#569011yescharacteristic
411C2#570411yescharacteristic
412C2#570611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#400C2
prime radicalC22×D12C22
hypercenter#400D3
hypocenterC3#2450
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
1. C2×#896C2semidirect
3. C3×#400C2semidirect
4. #4987C2semidirect
5. #4982C2semidirect
6. #4676C2semidirect
7. #4673C2semidirect
8. #4668C2semidirect
9. #4668C2semidirect
10. #4668C2semidirect
13. D3×#95C2semidirect
14. #4654C2semidirect
15. #4646C2semidirect
56. #400D3semidirect
86. C2×D12Q8semidirect
403. C3#2450semidirect

Maximal subgroups

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

subgroupindex
C2×#8962
C2×#8932
C3×#4002
#49872
#49822
#46762
#46732
#46682
#46652
D3×#952
#46542
#46462
#24503