Finite Group Databasebeta

← #77314

Subgroups — #77314

Order 384.

620 conjugacy classes of subgroups.

The subgroup lattice has 422 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#896C211yesmaximal, characteristic, complemented
2C2×#887C211yesmaximal, characteristic, complemented
3C2×#67⋊C4C211yesmaximal, characteristic
4C2×#878C211yesmaximal, characteristic, complemented
5C3×#396C211yesmaximal, characteristic, complemented
6#4994C211yesmaximal, characteristic, complemented
7#4994C211yesmaximal, characteristic, complemented
8#4976C211yesmaximal, characteristic, complemented
9#4975C211yesmaximal, characteristic, complemented
10#4957C211yesmaximal, characteristic
11#4957C211yesmaximal, characteristic
12#4948C211yesmaximal, characteristic, complemented
13#4942C211yesmaximal, characteristic, complemented
14#4667C211yesmaximal, characteristic, complemented
15#4647C211yesmaximal, characteristic, complemented
16#2538—31nomaximal, Sylow
17C22×#67C2211yescharacteristic
18C23×Dic3C2211yescharacteristic
19C22×C4×D3C2211yescharacteristic
20C22×Dic6C2211yescharacteristic
21C2×C3×C4⋊C4C2211yescharacteristic
22C2×C3×C4⋊C4C2211yescharacteristic
23C2×C3×C4⋊C4C2211yescharacteristic
24C2×C3×C22⋊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×C4⋊Dic3C2211yescharacteristic
35C2×C4⋊Dic3C2211yescharacteristic
36C2×Dic3⋊C4C2211yescharacteristic
37C2×Dic3⋊C4C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×Dic3⋊C4C2211yescharacteristic
40C2×Dic3⋊C4C2211yescharacteristic
41C2×Dic3⋊C4C2211yescharacteristic
42C2×Dic3⋊C4C2211yescharacteristic
43C2×Dic3⋊C4C2211yescharacteristic
44C2×Dic3⋊C4C2211yescharacteristic
45C2×C4×Dic3C2211yescharacteristic
46C2×C4×Dic3C2211yescharacteristic
47C2×C4×Dic3C2211yescharacteristic
48#896—24no—
49#887—24no—
50#67⋊C4—24no—
51#878—24no—
52C3×#95C2211yescharacteristic
53C3⋊#95C2211yescharacteristic
54C3⋊#95C2211yescharacteristic
55C3⋊#95C2211yescharacteristic
56C2×C22⋊Q8—31no—
57C2×C22⋊Q8—31no—
58C2×C4×Q8—31no—
59C2×C4×D4—31no—
60#396—31no—
61#396—31no—
62#396—31no—
63#396D311yescharacteristic, complemented
64#390—31no—
65#390—31no—
66#389—31no—
67#389—31no—
68#384—31no—
69#380—31no—
70#380—31no—
71C24×C3C2311yescharacteristic
72C23×D3C2311yescharacteristic
73C22×C3×C4C2311yescharacteristic
74C22×C3×C4C2311yescharacteristic
75C22×C3×C4C2311yescharacteristic
76C22×C3×C4C2311yescharacteristic
77C22×C3×C4C2311yescharacteristic
78C22×C3×C4C2311yescharacteristic
79C2×#67—22no—
80C2×#67—22no—
81C2×#67Q814yescomplemented
82C22×Dic3—22no—
83C22×Dic3—22no—
84C22×Dic3—22no—
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2311yescharacteristic
88C22×Dic3C2311yescharacteristic
89C22×Dic3C2311yescharacteristic
90C22×Dic3C2311yescharacteristic
91C22×Dic3C2311yescharacteristic
92C2×C4×D3—22no—
93C2×C4×D3—22no—
94C2×C4×D3—22no—
95C2×Dic6—22no—
96C2×Dic6—22no—
97C2×Dic6D414yescomplemented
98C3×C4⋊C4—22no—
99C3×C4⋊C4—22no—
100C3×C4⋊C4—41no—
101C3×C22⋊C4—22no—
102C3×C22⋊C4—22no—
103C3×C22⋊C4—41no—
104#217—22no—
105#217—41no—
106(C2×D3)⋊C4—22no—
107(C2×D3)⋊C4—22no—
108(C2×D3)⋊C4—22no—
109(C2×D3)⋊C4—41no—
110(C2×D3)⋊C4—41no—
111C4⋊Dic3—22no—
112C4⋊Dic3—41no—
113Dic3⋊C4—22no—
114Dic3⋊C4—22no—
115Dic3⋊C4—22no—
116Dic3⋊C4—22no—
117Dic3⋊C4—22no—
118Dic3⋊C4—22no—
119Dic3⋊C4—22no—
120Dic3⋊C4—22no—
121Dic3⋊C4—22no—
122C4×Dic3—22no—
123C4×Dic3—22no—
124C4×Dic3D414yes—
125C22×Q8—31no—
126C22×D4—31no—
127C23×C4—31no—
128C23×C4—31no—
129C22⋊Q8—64no—
130C22⋊Q8—64no—
131C4×Q8—64no—
132C4×D4—64no—
133C2×C4⋊C4—31no—
134C2×C4⋊C4—31no—
135C2×C4⋊C4—31no—
136C2×C4⋊C4—31no—
137C2×C4⋊C4—31no—
138C2×C4⋊C4—31no—
139C2×C4⋊C4—31no—
140C2×C4⋊C4—31no—
141C2×C4⋊C4—31no—
142C2×C4⋊C4—31no—
143C2×C4⋊C4—31no—
144C2×C4⋊C4C2×D311yescharacteristic
145C2×C4⋊C4C2×D311yescharacteristic
146C2×C4⋊C4C2×D311yescharacteristic
147C2×C22⋊C4—31no—
148C2×C22⋊C4—31no—
149C2×C22⋊C4—31no—
150C2×C22⋊C4—31no—
151C2×C22⋊C4—31no—
152C2×C22⋊C4—31no—
153C2×C22⋊C4—31no—
154C2×C22⋊C4C2×D311yescharacteristic
155C2×C22⋊C4C2×D311yescharacteristic
156C2×C22⋊C4C2×D311yescharacteristic
157C2×C42—31no—
158C2×C42—31no—
159C2×C42—31no—
160#95—31no—
161#95—31no—
162#95—31no—
163#95C2×D311yescharacteristic
164C23×C3—21no—
165C23×C3—21no—
166C23×C3—21no—
167C23×C3—21no—
168C23×C3—21no—
169C23×C3—21no—
170C23×C3C2×Q812yes—
171C23×C3C2411yescharacteristic
172C22×D3—21no—
173C22×D3—21no—
174C22×D3—21no—
175C22×D3—21no—
176C22×D3—21no—
177C22×D3—21no—
178C22×D3C2×Q812yes—
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—21no—
195C2×C3×C4C2×D412yes—
196C2×C3×C4C2×D412yes—
197#67—28no—
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×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—21no—
212C2×Dic3—44no—
213C2×Dic3C2×D412yes—
214C2×Dic3C2×D412yes—
215C2×Dic3C2×D412yes—
216C2×Dic3C2×D412yes—
217C2×Dic3C2×Q812yes—
218C2×Dic3#4112yes—
219C2×Dic3#4112yes—
220C4×D3—44no—
221Dic6—28no—
222C24—31no—
223C24C22×D311yescharacteristic
224C2×Q8—34no—
225C2×Q8—62no—
226C2×Q8—62no—
227C2×D4—34no—
228C2×D4—62no—
229C2×D4—62no—
230C22×C4—31no—
231C22×C4—31no—
232C22×C4—31no—
233C22×C4—31no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—31no—
237C22×C4—62no—
238C22×C4—62no—
239C22×C4—62no—
240C22×C4—62no—
241C22×C4—62no—
242C22×C4—62no—
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—41no—
252C4⋊C4—62no—
253C4⋊C4—62no—
254C4⋊C4—62no—
255C4⋊C4—62no—
256C4⋊C4—62no—
257C4⋊C4—62no—
258C4⋊C4—62no—
259C4⋊C4—62no—
260C4⋊C4—62no—
261C4⋊C4—62no—
262C4⋊C4—121no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—41no—
266C22⋊C4—62no—
267C22⋊C4—62no—
268C22⋊C4—62no—
269C22⋊C4—62no—
270C22⋊C4—121no—
271C22⋊C4—121no—
272C22⋊C4—121no—
273C42—34no—
274C42—62no—
275C42—62no—
276C22×C3—22no—
277C22×C3—22no—
278C22×C3—22no—
279C22×C3—41no—
280C22×C3—41no—
281C22×C3—41no—
282C22×C3—41no—
283C22×C3C22×D411yescharacteristic
284C22×C3C22×D411yescharacteristic
285C22×C3C22×Q811yescharacteristic
286C22×C3C2×#4111yescharacteristic
287C22×C3C2×#4111yescharacteristic
288C22×C3H(5,𝔽2)11yescharacteristic
289C22×C32−1+411yescharacteristic
290C2×D3—22no—
291C2×D3—22no—
292C2×D3—22no—
293C2×D3—41no—
294C2×D3—41no—
295C2×D3—41no—
296C2×D3—41no—
297C3×C4—22no—
298C3×C4—22no—
299C3×C4—41no—
300C3×C4—41no—
301C3×C4—41no—
302C3×C4—41no—
303Dic3—22no—
304Dic3—41no—
305Dic3—41no—
306Dic3—41no—
307Dic3—41no—
308Dic3#12114yes—
309Dic3C22⋊Q814yescomplemented
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—32no—
317C23—61no—
318C23—61no—
319C23—61no—
320C23—61no—
321C23—61no—
322C23—61no—
323C23D3×Q812yes—
324C23C23×D311yescharacteristic
325Q8—68no—
326D4—68no—
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—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—32no—
344C2×C4—32no—
345C2×C4—32no—
346C2×C4—32no—
347C2×C4—32no—
348C2×C4—32no—
349C2×C4—32no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—61no—
354C2×C4—61no—
355C2×C4—61no—
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—124no—
365C2×C4—124no—
366C2×C4D3×D412yes—
367C2×C4D3×D412yes—
368C2×C3—42no—
369C2×C3C2×#12111yescharacteristic
370C2×C3C2×C22⋊Q811yescharacteristic
371C2×C3#53711yescharacteristic
372C2×C3D4×Q811yescharacteristic
373C2×C3#55011yescharacteristic
374C2×C3#55411yescharacteristic
375C2×C3#55611yescharacteristic
376D3—42no—
377C22—22no—
378C22—22no—
379C22—22no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—41no—
384C22—62no—
385C22—62no—
386C22—62no—
387C22—121no—
388C22—121no—
389C22—121no—
390C22—121no—
391C22C2×D3×D411yescharacteristic
392C22C2×D3×D411yescharacteristic
393C22#100411yescharacteristic
394C22C2×D3×Q811yescharacteristic
395C22D3×#4111yescharacteristic
396C22D3×#4111yescharacteristic
397C22#101011yescharacteristic
398C4—22no—
399C4—22no—
400C4—34no—
401C4—34no—
402C4—41no—
403C4—41no—
404C4—41no—
405C4—41no—
406C4—62no—
407C4—121no—
408C4—121no—
409C4—121no—
410C4—121no—
411C3#253811yescharacteristic, complemented, Sylow
412C2—42no—
413C2—122no—
414C2#559711yescharacteristic
415C2#559911yescharacteristic
416C2D3×#12111yescharacteristic
417C2D3×C22⋊Q811yescharacteristic
418C2#563211yescharacteristic
419C2#563611yescharacteristic
420C2#570211yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC3×#396C2
prime radicalC22×#67C22
hypercenter#396D3
hypocenterC3#2538
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#896C2semidirect
2. C2×#887C2semidirect
4. C2×#878C2semidirect
5. C3×#396C2semidirect
6. #4994C2semidirect
7. #4994C2semidirect
8. #4976C2semidirect
9. #4975C2semidirect
12. #4948C2semidirect
13. #4942C2semidirect
14. #4667C2semidirect
15. #4647C2semidirect
63. #396D3semidirect
81. C2×#67Q8semidirect
97. C2×Dic6D4semidirect
309. Dic3C22⋊Q8semidirect
411. C3#2538semidirect

Maximal subgroups

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

subgroupindex
C2×#8962
C2×#8872
C2×#67⋊C42
C2×#8782
C3×#3962
#49942
#49762
#49752
#49572
#49482
#49422
#46672
#46472
#25383