Finite Group Databasebeta

← #82669

Subgroups — #82669

Order 384.

948 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#121C211yesmaximal, characteristic, complemented
2C2×#940C212yesmaximal, complemented
3C2×#940C211yesmaximal, characteristic, complemented
4C2×#938C211yesmaximal, characteristic, complemented
5C2×#930C211yesmaximal, characteristic, complemented
6C2×C4×#67C211yesmaximal, characteristic, complemented
7C2×#895C211yesmaximal, characteristic, complemented
8#5223C212yesmaximal, complemented
9Dic3×C4⋊C4C211yesmaximal, characteristic, complemented
10#4960C212yesmaximal, complemented
11#4957C212yesmaximal, complemented
12#2541—31nomaximal, Sylow
13C22×C3×D4C2211yescharacteristic
14C22×C3×D4C2212yes—
15C23×C3×C4C2211yescharacteristic
16C22×#67C2212yes—
17C22×#67C2212yes—
18C22×#67C2212yes—
19C22×#67C2211yescharacteristic
20C22×#67C2212yes—
21C22×D12C2211yescharacteristic
22C22×D12C2212yes—
23C22×C4×D3C2211yescharacteristic
24C3×#121—24no—
25C2×C3×C4⋊C4C2211yescharacteristic
26C2×C3×C22⋊C4C2212yes—
27C2×#217C2212yes—
28C2×#217C2211yescharacteristic
29#940—28no—
30#940—24no—
31#938—24no—
32#930—24no—
33C4×#67—24no—
34C2×(C2×D3)⋊C4C2212yes—
35C2×(C2×D3)⋊C4C2212yes—
36C2×(C2×D3)⋊C4C2211yescharacteristic
37C2×C4⋊Dic3C2211yescharacteristic
38C2×Dic3⋊C4C2211yescharacteristic
39C2×C4×Dic3C2212yes—
40C2×C4×Dic3C2211yescharacteristic
41#895—24no—
42C3⋊#95C2212yes—
43C2×#127—31no—
44C2×#127—32no—
45C2×#121—31no—
46C2×#121—31no—
47C2×#121—31no—
48C2×#121D311yescharacteristic, complemented
49C2×C4×D4—31no—
50#392—32no—
51#392—32no—
52#390—32no—
53C4×C4⋊C4—31no—
54C24×C3C2312yes—
55C24×C3C2311yescharacteristic
56C23×D3C2312yes—
57C23×D3C2311yescharacteristic
58C2×C3×D4—24no—
59C2×C3×D4—22no—
60C2×C3×D4—22no—
61C2×C3×D4—24no—
62C2×C3×D4—22no—
63C2×C3×D4—42no—
64C22×C3×C4—22no—
65C22×C3×C4—22no—
66C22×C3×C4—22no—
67C22×C3×C4C2311yescharacteristic
68C22×C3×C4C2312yes—
69C22×C3×C4C2311yescharacteristic
70C2×#67—24no—
71C2×#67—24no—
72C2×#67—22no—
73C2×#67—24no—
74C2×#67—24no—
75C2×#67—24no—
76C2×#67—24no—
77C2×#67—24no—
78C2×#67—22no—
79C2×#67—24no—
80C2×#67—42no—
81C2×#67—42no—
82C2×#67—42no—
83C2×#67—42no—
84C2×#67D414yescomplemented
85C22×Dic3C2311yescharacteristic
86C22×Dic3C2311yescharacteristic
87C22×Dic3C2312yes—
88C22×Dic3C2311yescharacteristic
89C2×D12—24no—
90C2×D12—24no—
91C2×D12—22no—
92C2×D12—22no—
93C2×D12—22no—
94C2×D12—42no—
95C2×C4×D3—22no—
96C2×C4×D3—22no—
97C2×C4×D3—22no—
98C3×C4⋊C4D414yescomplemented
99C3×C22⋊C4—24no—
100#217—22no—
101#217—24no—
102(C2×D3)⋊C4—24no—
103(C2×D3)⋊C4—22no—
104(C2×D3)⋊C4—24no—
105C4⋊Dic3D414yescomplemented
106Dic3⋊C4—22no—
107C4×Dic3—24no—
108C4×Dic3D414yescomplemented
109C22×D4—31no—
110C22×D4—32no—
111C22×D4—32no—
112C22×D4—32no—
113C22×D4—32no—
114C22×D4—31no—
115C22×D4—32no—
116C22×D4C2×D311yescharacteristic
117C22×D4C2×D312yes—
118C23×C4—31no—
119C23×C4C2×D311yescharacteristic
120#127—64no—
121#127—68no—
122#121—24no—
123#121—64no—
124#121—64no—
125#121—64no—
126C4×D4—64no—
127C2×C4⋊C4—31no—
128C2×C4⋊C4—31no—
129C2×C4⋊C4C2×D311yescharacteristic
130C2×C22⋊C4—31no—
131C2×C22⋊C4—32no—
132C2×C22⋊C4—31no—
133C2×C22⋊C4—32no—
134C2×C22⋊C4—32no—
135C2×C22⋊C4C2×D312yes—
136C2×C42—31no—
137C2×C42—32no—
138#95—32no—
139C23×C3—21no—
140C23×C3—21no—
141C23×C3—21no—
142C23×C3—21no—
143C23×C3—21no—
144C23×C3—21no—
145C23×C3—22no—
146C23×C3—22no—
147C23×C3—22no—
148C23×C3—22no—
149C23×C3—22no—
150C23×C3—22no—
151C23×C3—22no—
152C23×C3C2×D412yes—
153C23×C3C2411yescharacteristic
154C22×D3—21no—
155C22×D3—21no—
156C22×D3—21no—
157C22×D3—21no—
158C22×D3—21no—
159C22×D3—21no—
160C22×D3—22no—
161C22×D3—22no—
162C22×D3—22no—
163C22×D3—22no—
164C22×D3—22no—
165C22×D3—22no—
166C22×D3—22no—
167C22×D3C2×D412yes—
168C3×D4—48no—
169C3×D4—44no—
170C2×C3×C4—22no—
171C2×C3×C4—22no—
172C2×C3×C4—21no—
173C2×C3×C4—21no—
174C2×C3×C4—21no—
175C2×C3×C4—44no—
176C2×C3×C4C2×D412yes—
177C2×C3×C4C2×D412yes—
178C2×C3×C4C2×D414yes—
179C2×C3×C4C2×D412yes—
180#67—28no—
181#67—48no—
182#67—48no—
183#67—48no—
184#67—48no—
185C2×Dic3—22no—
186C2×Dic3—21no—
187C2×Dic3—21no—
188C2×Dic3—22no—
189C2×Dic3—21no—
190C2×Dic3—21no—
191C2×Dic3—21no—
192C2×Dic3C2×D412yes—
193C2×Dic3C2×D414yes—
194C2×Dic3C2×D412yes—
195C2×Dic3C2×D412yes—
196C2×Dic3#4112yes—
197D12—48no—
198D12—44no—
199C4×D3—44no—
200C24—31no—
201C24—32no—
202C24C22×D312yes—
203C24C22×D311yescharacteristic
204C2×D4—24no—
205C2×D4—22no—
206C2×D4—22no—
207C2×D4—24no—
208C2×D4—22no—
209C2×D4—34no—
210C2×D4—42no—
211C2×D4—64no—
212C2×D4—62no—
213C2×D4—62no—
214C2×D4—64no—
215C2×D4—64no—
216C2×D4—64no—
217C2×D4—64no—
218C2×D4—64no—
219C2×D4—64no—
220C2×D4—64no—
221C2×D4—64no—
222C2×D4—62no—
223C2×D4—62no—
224C2×D4—64no—
225C2×D4—62no—
226C2×D4—122no—
227C2×D4—122no—
228C2×D4—122no—
229C2×D4—122no—
230C2×D4—122no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—31no—
235C22×C4—31no—
236C22×C4—32no—
237C22×C4—31no—
238C22×C4—62no—
239C22×C4—62no—
240C22×C4—62no—
241C22×C4C22×D311yescharacteristic
242C22×C4C22×D312yes—
243C22×C4C22×D311yescharacteristic
244C4⋊C4—34no—
245C4⋊C4—62no—
246C4⋊C4#6714yescomplemented
247C22⋊C4—24no—
248C22⋊C4—62no—
249C22⋊C4—64no—
250C22⋊C4—62no—
251C22⋊C4—64no—
252C22⋊C4—64no—
253C42—34no—
254C42—64no—
255C22×C3—22no—
256C22×C3—22no—
257C22×C3—22no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—41no—
261C22×C3—41no—
262C22×C3—42no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3—42no—
266C22×C3—42no—
267C22×C3—42no—
268C22×C3—42no—
269C22×C3C22×D411yescharacteristic
270C22×C3C22×D411yescharacteristic
271C22×C3C22×D411yescharacteristic
272C22×C3C22×D411yescharacteristic
273C22×C3C2×#4111yescharacteristic
274C22×C3H(5,𝔽2)11yescharacteristic
275C22×C3H(5,𝔽2)11yescharacteristic
276C2×D3—22no—
277C2×D3—22no—
278C2×D3—22no—
279C2×D3—41no—
280C2×D3—41no—
281C2×D3—41no—
282C2×D3—41no—
283C2×D3—42no—
284C2×D3—42no—
285C2×D3—42no—
286C2×D3—42no—
287C2×D3—42no—
288C2×D3—42no—
289C2×D3—42no—
290C3×C4—24no—
291C3×C4—41no—
292C3×C4#12714yescomplemented
293Dic3—22no—
294Dic3—24no—
295Dic3—41no—
296Dic3#12114yescomplemented
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—22no—
304C23—22no—
305C23—22no—
306C23—22no—
307C23—22no—
308C23—22no—
309C23—22no—
310C23—32no—
311C23—61no—
312C23—61no—
313C23—61no—
314C23—61no—
315C23—61no—
316C23—61no—
317C23—62no—
318C23—62no—
319C23—62no—
320C23—62no—
321C23—62no—
322C23—62no—
323C23—62no—
324C23D3×D412yes—
325C23C23×D311yescharacteristic
326D4—48no—
327D4—44no—
328D4—68no—
329D4—128no—
330D4—128no—
331D4—128no—
332D4—128no—
333D4—128no—
334D4—124no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—32no—
341C2×C4—32no—
342C2×C4—32no—
343C2×C4—32no—
344C2×C4—34no—
345C2×C4—44no—
346C2×C4—62no—
347C2×C4—61no—
348C2×C4—61no—
349C2×C4—62no—
350C2×C4—61no—
351C2×C4—61no—
352C2×C4—61no—
353C2×C4—124no—
354C2×C4D3×D412yes—
355C2×C4D3×D412yes—
356C2×C4C2×#6714yes—
357C2×C4C2×#6712yes—
358C2×C3—42no—
359C2×C3—82no—
360C2×C3C2×#12111yescharacteristic
361C2×C3C2×#12711yescharacteristic
362C2×C3#53511yescharacteristic
363C2×C3D4211yescharacteristic
364C2×C3D4211yescharacteristic
365C2×C3#55011yescharacteristic
366C2×C3#55011yescharacteristic
367D3—42no—
368D3—82no—
369C22—22no—
370C22—22no—
371C22—22no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—42no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—62no—
384C22—62no—
385C22—62no—
386C22—121no—
387C22—121no—
388C22—121no—
389C22—121no—
390C22—122no—
391C22—122no—
392C22—122no—
393C22—122no—
394C22—122no—
395C22—122no—
396C22—122no—
397C22C2×D3×D411yescharacteristic
398C22C2×D3×D411yescharacteristic
399C22C2×D3×D411yescharacteristic
400C22#100411yescharacteristic
401C22D3×#4111yescharacteristic
402C22#100911yescharacteristic
403C22C22×#6711yescharacteristic
404C4—24no—
405C4—34no—
406C4—41no—
407C4—62no—
408C4—64no—
409C4—121no—
410C4#94014yescomplemented
411C3#254111yescharacteristic, complemented, Sylow
412C2—42no—
413C2—82no—
414C2—122no—
415C2—242no—
416C2#559911yescharacteristic
417C2D3×#12111yescharacteristic
418C2#560711yescharacteristic
419C2#560911yescharacteristic
420C2D4×#6711yescharacteristic
421C2C2×#94011yescharacteristic
422C2#583411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D3
commutator, socle, cosocleC23×C3C24
FittingC2×C3×#121C2
hypercenterC2×#121D3
hypocenterC3#2541
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×#121C2semidirect
2. C2×#940C2semidirect
3. C2×#940C2semidirect
4. C2×#938C2semidirect
5. C2×#930C2semidirect
6. C2×C4×#67C2semidirect
7. C2×#895C2semidirect
8. #5223C2semidirect
9. Dic3×C4⋊C4C2semidirect
10. #4960C2semidirect
11. #4957C2semidirect
48. C2×#121D3semidirect
84. C2×#67D4semidirect
98. C3×C4⋊C4D4semidirect
105. C4⋊Dic3D4semidirect
108. C4×Dic3D4semidirect
246. C4⋊C4#67semidirect
292. C3×C4#127semidirect
296. Dic3#121semidirect
410. C4#940semidirect
411. C3#2541semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#1212
C2×#9402
C2×#9382
C2×#9302
C2×C4×#672
C2×#8952
#52232
Dic3×C4⋊C42
#49602
#49572
#25413