Finite Group Databasebeta

← #11225568

Subgroups — #11225568

Order 768.

1570 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2399C211yesmaximal, characteristic
2#82483C212yesmaximal
3Q8×C4⋊Dic3C212yesmaximal
4#77917C214yesmaximal
5#77868C214yesmaximal
6#77192C214yesmaximal
7#77178C214yesmaximal
8#73717C212yesmaximal
9#73684C212yesmaximal
10#73682C212yesmaximal
11#73659C212yesmaximal
12#73390C211yesmaximal, characteristic
13#73390C211yesmaximal, characteristic
14#44982—31nomaximal, Sylow
15C2×C3×C4×Q8C2212yes—
16C2×Q8×Dic3C2214yes—
17C2×#890C2214yes—
18C2×#889C2214yes—
19C2×#870C2211yescharacteristic
20C2×#870C2211yescharacteristic
21C3×#398C2212yes—
22C3×#398C2212yes—
23C3×#395C2212yes—
24C3×#395C2212yes—
25C3×C4×C4⋊C4C2212yes—
26C3×C4×C4⋊C4C2211yescharacteristic
27C3×C4×C4⋊C4C2211yescharacteristic
28C3×#376C2211yescharacteristic
29#5230C2214yes—
30#4984C2214yes—
31#4984C2214yes—
32#4984C2214yes—
33#4984C2214yes—
34#4983C2214yes—
35#4982C2214yes—
36#4981C2214yes—
37#4980C2214yes—
38#4978C2214yes—
39#4976C2214yes—
40#4976C2214yes—
41#4976C2214yes—
42#4935C2211yescharacteristic
43#4935C2212yes—
44#4935C2212yes—
45#4935C2211yescharacteristic
46C4×C4⋊Dic3C2212yes—
47C4×C4⋊Dic3C2212yes—
48C4×C4⋊Dic3C2212yes—
49C4×C4⋊Dic3C2212yes—
50#4928C2212yes—
51#4928C2212yes—
52#4928C2212yes—
53#4928C2212yes—
54#4928C2212yes—
55#4928C2212yes—
56#4661C2214yes—
57#4661C2214yes—
58#4659C2214yes—
59#4657C2214yes—
60#4655C2214yes—
61#4655C2214yes—
62#4653C2214yes—
63#4652C2214yes—
64#4649C2214yes—
65#4647C2214yes—
66#4647C2214yes—
67#4647C2214yes—
68#2802—34no—
69#2726—34no—
70#2584—34no—
71#2584—32no—
72#2552—32no—
73#2507—32no—
74#2506—32no—
75#2462—34no—
76#2399D311yescharacteristic
77#2370—32no—
78Q8×C4⋊C4—32no—
79#2330—31no—
80#2330—31no—
81C22×C3×Q8C2312yes—
82C3×C4×Q8—28no—
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2312yes—
85C2×C3×C4⋊C4C2312yes—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C4⋊C4C2312yes—
88C2×C3×C4⋊C4C2312yes—
89C2×C3×C4⋊C4C2312yes—
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2312yes—
92C2×C3×C42C2312yes—
93C2×C3×C42C2311yescharacteristic
94C2×C3×C42C2311yescharacteristic
95Q8×Dic3—48no—
96C2×C4⋊Dic3C2314yes—
97C2×C4⋊Dic3C2312yes—
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2312yes—
100C2×C4⋊Dic3C2314yes—
101C2×C4⋊Dic3C2314yes—
102C2×C4⋊Dic3C2314yes—
103C2×C4⋊Dic3C2314yes—
104C2×C4⋊Dic3C2314yes—
105C2×C4⋊Dic3C2312yes—
106C2×Dic3⋊C4C2314yes—
107C2×Dic3⋊C4C2314yes—
108C2×Dic3⋊C4C2314yes—
109C2×Dic3⋊C4C2314yes—
110C2×Dic3⋊C4C2314yes—
111C2×Dic3⋊C4C2314yes—
112C2×Dic3⋊C4C2314yes—
113C2×Dic3⋊C4C2314yes—
114C2×C4×Dic3C2314yes—
115C2×C4×Dic3C2314yes—
116C2×C4×Dic3C2314yes—
117C2×C4×Dic3C2314yes—
118#890—48no—
119#889—48no—
120#870—24no—
121#870Q818yescomplemented
122C3×#95C2312yes—
123C3×#95C2311yescharacteristic
124C3×#95C2312yes—
125C3×#95C2312yes—
126C3×#95C2311yescharacteristic
127C3×#95C2312yes—
128C3×#95C2311yescharacteristic
129C3×#95C2311yescharacteristic
130C3⋊#95C2314yes—
131C3⋊#95C2314yes—
132C3⋊#95C2314yes—
133C3⋊#95C2314yes—
134C3⋊#95C2314yes—
135C3⋊#95C2314yes—
136C3⋊#95C2314yes—
137C3⋊#95C2314yes—
138C3⋊#95C2314yes—
139C3⋊#95C2314yes—
140C2×#125—34no—
141C2×#125—34no—
142C2×#125—31no—
143C2×#125—31no—
144C2×C4×Q8—34no—
145C2×C4×Q8C2×D312yes—
146#400—34no—
147#398—34no—
148#398—34no—
149#398—34no—
150#398—34no—
151#398—34no—
152#398—34no—
153#398C2×D312yes—
154#398C2×D312yes—
155#395—34no—
156#395C2×D312yes—
157#395C2×D312yes—
158#389—34no—
159#389—34no—
160#389—34no—
161#389—34no—
162#389—34no—
163#389—34no—
164#389—34no—
165#389—32no—
166#389—32no—
167#389—32no—
168#389—32no—
169#389—32no—
170#389—32no—
171#389—34no—
172#389—34no—
173#389—34no—
174#387—34no—
175#387—34no—
176#387—34no—
177#387—34no—
178#387—34no—
179#387—34no—
180#384—31no—
181#384—31no—
182#384—32no—
183#384—32no—
184C4×C4⋊C4—32no—
185C4×C4⋊C4—32no—
186C4×C4⋊C4—32no—
187C4×C4⋊C4—32no—
188C4×C4⋊C4—34no—
189C4×C4⋊C4C2×D312yes—
190C4×C4⋊C4C2×D311yescharacteristic
191C4×C4⋊C4C2×D311yescharacteristic
192#376C2×D311yescharacteristic
193C2×C3×Q8—24no—
194C2×C3×Q8—24no—
195C2×C3×Q8—24no—
196C22×C3×C4C2411yescharacteristic
197C22×C3×C4C2412yes—
198C22×C3×C4C2412yes—
199C22×C3×C4C2411yescharacteristic
200C22×C3×C4C2412yes—
201C22×C3×C4C2412yes—
202C22×C3×C4C2412yes—
203C22×C3×C4C2412yes—
204C22×C3×C4C2411yescharacteristic
205C22×Dic3C2414yes—
206C22×Dic3C2414yes—
207C22×Dic3C2414yes—
208C22×Dic3C2414yes—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—24no—
211C3×C4⋊C4—24no—
212C3×C4⋊C4—24no—
213C3×C4⋊C4—24no—
214C3×C4⋊C4—24no—
215C3×C4⋊C4—24no—
216C3×C42—24no—
217C3×C42—24no—
218C3×C42—22no—
219C3×C42C2×Q814yes—
220C3×C42#4114yes—
221C4⋊Dic3—24no—
222C4⋊Dic3—28no—
223C4⋊Dic3—24no—
224C4⋊Dic3—24no—
225C4⋊Dic3—44no—
226C4⋊Dic3—44no—
227C4⋊Dic3—44no—
228C4⋊Dic3—44no—
229C4⋊Dic3—44no—
230C4⋊Dic3C2×Q818yes—
231Dic3⋊C4—28no—
232Dic3⋊C4—28no—
233Dic3⋊C4—28no—
234Dic3⋊C4—44no—
235Dic3⋊C4—44no—
236Dic3⋊C4—44no—
237Dic3⋊C4—44no—
238Dic3⋊C4C2×Q8116yes—
239C4×Dic3—28no—
240C4×Dic3—44no—
241C4×Dic3—44no—
242C4×Dic3—44no—
243C22×Q8C22×D312yes—
244#125—38no—
245#125—64no—
246#125—128no—
247#125—128no—
248C4×Q8—28no—
249C4×Q8—128no—
250C2×C4⋊C4—32no—
251C2×C4⋊C4—34no—
252C2×C4⋊C4—34no—
253C2×C4⋊C4—34no—
254C2×C4⋊C4—32no—
255C2×C4⋊C4—34no—
256C2×C4⋊C4—32no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—34no—
259C2×C4⋊C4—34no—
260C2×C4⋊C4—34no—
261C2×C4⋊C4—34no—
262C2×C4⋊C4—34no—
263C2×C4⋊C4—34no—
264C2×C4⋊C4—34no—
265C2×C4⋊C4—34no—
266C2×C4⋊C4—34no—
267C2×C4⋊C4—34no—
268C2×C4⋊C4C22×D312yes—
269C2×C4⋊C4C22×D312yes—
270C2×C4⋊C4C22×D312yes—
271C2×C4⋊C4C22×D312yes—
272C2×C4⋊C4C22×D312yes—
273C2×C4⋊C4C22×D312yes—
274C2×C4⋊C4C22×D312yes—
275C2×C42—34no—
276C2×C42—34no—
277C2×C42—34no—
278C2×C42—34no—
279C2×C42C22×D312yes—
280C2×C42C22×D312yes—
281C2×C42C22×D311yescharacteristic
282C2×C42C22×D311yescharacteristic
283C2×C42C22×D311yescharacteristic
284#95—34no—
285#95—34no—
286#95—34no—
287#95—34no—
288#95—34no—
289#95—34no—
290#95—34no—
291#95—34no—
292#95—34no—
293#95—34no—
294#95C22×D312yes—
295#95C22×D312yes—
296#95C22×D312yes—
297#95C22×D311yescharacteristic
298#95C22×D312yes—
299#95C22×D311yescharacteristic
300#95C22×D311yescharacteristic
301#95C22×D311yescharacteristic
302C23×C3C2511yescharacteristic
303C3×Q8—216no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—22no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—22no—
310C2×C3×C4—22no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—22no—
314C2×C3×C4—22no—
315C2×C3×C4—22no—
316C2×C3×C4—22no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—22no—
320C2×C3×C4—22no—
321C2×C3×C4—22no—
322C2×C3×C4C22×Q814yes—
323C2×C3×C4C22×Q812yes—
324C2×C3×C4C2×#4114yes—
325C2×C3×C4C2×#4112yes—
326C2×C3×C4H(5,𝔽2)14yes—
327C2×C3×C42−1+412yes—
328C2×C3×C42−1+414yes—
329C2×Dic3—24no—
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—24no—
333C2×Dic3—24no—
334C2×Dic3—24no—
335C2×Dic3—24no—
336C2×Dic3—24no—
337C2×Dic3—24no—
338C2×Dic3—24no—
339C2×Dic3—24no—
340C2×Dic3C22×Q818yes—
341C2×Q8—24no—
342C2×Q8—24no—
343C2×Q8—24no—
344C22×C4—34no—
345C22×C4—34no—
346C22×C4—34no—
347C22×C4—34no—
348C22×C4C23×D311yescharacteristic
349C22×C4C23×D312yes—
350C22×C4C23×D312yes—
351C22×C4C23×D311yescharacteristic
352C22×C4C23×D312yes—
353C22×C4C23×D312yes—
354C22×C4C23×D312yes—
355C22×C4C23×D312yes—
356C22×C4C23×D311yescharacteristic
357C4⋊C4—24no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—24no—
362C4⋊C4—24no—
363C4⋊C4—24no—
364C4⋊C4—316no—
365C4⋊C4—38no—
366C4⋊C4—68no—
367C4⋊C4—64no—
368C4⋊C4—64no—
369C4⋊C4—64no—
370C4⋊C4—68no—
371C4⋊C4—68no—
372C4⋊C4—68no—
373C4⋊C4—124no—
374C4⋊C4—124no—
375C4⋊C4—124no—
376C4⋊C4—124no—
377C4⋊C4—124no—
378C4⋊C4—124no—
379C4⋊C4—124no—
380C4⋊C4—124no—
381C4⋊C4—124no—
382C42—24no—
383C42—24no—
384C42—22no—
385C42—68no—
386C42—124no—
387C42—124no—
388C42—124no—
389C42#23714yes—
390C42D3×Q814yes—
391C22×C3C23×Q811yescharacteristic
392C22×C3C22×#4111yescharacteristic
393C22×C3C2×H(5,𝔽2)12yes—
394C22×C3C2×2−1+411yescharacteristic
395C22×C3C2×2−1+412yes—
396C3×C4—24no—
397C3×C4—24no—
398C3×C4—24no—
399C3×C4—24no—
400C3×C4—42no—
401C3×C4—42no—
402C3×C4—41no—
403C3×C4—41no—
404C3×C4#55714yes—
405Dic3—28no—
406Dic3—44no—
407Dic3—44no—
408Dic3—44no—
409C23C24×D311yescharacteristic
410Q8—216no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—21no—
422C2×C4—21no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—22no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—38no—
430C2×C4—64no—
431C2×C4—64no—
432C2×C4—64no—
433C2×C4—64no—
434C2×C4—64no—
435C2×C4—64no—
436C2×C4—64no—
437C2×C4—64no—
438C2×C4—64no—
439C2×C4—64no—
440C2×C4—64no—
441C2×C4C2×#23714yes—
442C2×C4C2×#23712yes—
443C2×C4C2×D3×Q814yes—
444C2×C4C2×D3×Q812yes—
445C2×C4#100712yes—
446C2×C4#100914yes—
447C2×C4#101014yes—
448C2×C3C2×#55711yescharacteristic
449C2×C3#349212yes—
450C2×C3#350312yes—
451C2×C3#354811yescharacteristic
452C2×C3#355611yescharacteristic
453C22C22×#23711yescharacteristic
454C22C22×D3×Q811yescharacteristic
455C22C2×#100711yescharacteristic
456C22C2×#100912yes—
457C22C2×#101012yes—
458C4—24no—
459C4—24no—
460C4—24no—
461C4—24no—
462C4—42no—
463C4—42no—
464C4—41no—
465C4—41no—
466C4—68no—
467C4—124no—
468C4—124no—
469C4—124no—
470C4#572114yes—
471C3#4498211yescharacteristic, complemented, Sylow
472C2#8602412yes—
473C2C2×#572111yescharacteristic
474C2#8627011yescharacteristic
475C2#8628112yes—
476C2#8628811yescharacteristic
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, prime radical, cosocleC23×C3C25
FittingC3×#2399C2
hypercenter#2399D3
hypocenterC3#44982
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2399

Complements

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

NH ≅ G/Ntype
121. #870Q8semidirect
471. C3#44982semidirect

Maximal subgroups

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

subgroupindex
C3×#23992
#824832
Q8×C4⋊Dic32
#779172
#778682
#771922
#771782
#737172
#736842
#736822
#736592
#733902
#449823