Finite Group Databasebeta

← #11225562

Subgroups — #11225562

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×#2330C211yesmaximal, characteristic
2#83025C212yesmaximal
3#76128C214yesmaximal
4#76079C214yesmaximal
5#76035C212yesmaximal
6#76031C212yesmaximal
7#75274C214yesmaximal
8#74998C214yesmaximal
9#74336C212yesmaximal
10#73727C212yesmaximal
11C4⋊C4×Dic6C212yesmaximal
12#73397C211yesmaximal, characteristic
13#73390C211yesmaximal, characteristic
14#44982—31nomaximal, Sylow
15C2×C3×#125C2211yescharacteristic
16C2×#890C2214yes—
17C2×#890C2214yes—
18C2×#870C2211yescharacteristic
19C2×C4×Dic6C2212yes—
20C2×C4×Dic6C2214yes—
21C3×#389C2212yes—
22C3×#389C2214yes—
23C3×#387C2214yes—
24C3×#384C2211yescharacteristic
25C3×C4×C4⋊C4C2212yes—
26C3×#376C2211yescharacteristic
27#4984C2212yes—
28#4984C2212yes—
29#4984C2214yes—
30#4984C2212yes—
31#4984C2212yes—
32#4982C2214yes—
33#4982C2214yes—
34#4982C2214yes—
35#4982C2214yes—
36#4980C2214yes—
37#4980C2214yes—
38#4975C2212yes—
39#4975C2212yes—
40Dic3×C4⋊C4C2212yes—
41Dic3×C4⋊C4C2214yes—
42Dic3×C4⋊C4C2212yes—
43#4972C2214yes—
44#4972C2214yes—
45#4935C2211yescharacteristic
46C4×C4⋊Dic3C2211yescharacteristic
47C4×C4⋊Dic3C2212yes—
48C4×C4⋊Dic3C2212yes—
49C4×C4⋊Dic3C2211yescharacteristic
50#4928C2212yes—
51#4928C2214yes—
52#4928C2214yes—
53#4928C2214yes—
54#4661C2214yes—
55#4661C2212yes—
56#4661C2214yes—
57#4661C2212yes—
58#4660C2214yes—
59#4656C2212yes—
60#4656C2214yes—
61#4656C2212yes—
62#4655C2214yes—
63#4655C2214yes—
64#4654C2214yes—
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#2399—31no—
77#2370—32no—
78Q8×C4⋊C4—32no—
79#2330—31no—
80#2330D311yescharacteristic
81C22×Dic6C2312yes—
82C3×#125Q818yescomplemented
83C2×C3×C4⋊C4C2312yes—
84C2×C3×C4⋊C4C2314yes—
85C2×C3×C4⋊C4C2314yes—
86C2×C3×C4⋊C4C2312yes—
87C2×C3×C4⋊C4C2314yes—
88C2×C3×C42C2311yescharacteristic
89C2×C3×C42C2314yes—
90C2×C3×C42C2311yescharacteristic
91C2×C3×C42C2311yescharacteristic
92C2×C4⋊Dic3C2312yes—
93C2×C4⋊Dic3C2314yes—
94C2×C4⋊Dic3C2314yes—
95C2×C4⋊Dic3C2312yes—
96C2×C4⋊Dic3C2312yes—
97C2×C4⋊Dic3C2312yes—
98C2×C4⋊Dic3C2312yes—
99C2×C4⋊Dic3C2312yes—
100C2×C4⋊Dic3C2312yes—
101C2×C4⋊Dic3C2314yes—
102C2×C4⋊Dic3C2314yes—
103C2×C4⋊Dic3C2314yes—
104C2×Dic3⋊C4C2312yes—
105C2×Dic3⋊C4C2312yes—
106C2×Dic3⋊C4C2314yes—
107C2×Dic3⋊C4C2314yes—
108C2×Dic3⋊C4C2314yes—
109C2×Dic3⋊C4C2314yes—
110C2×Dic3⋊C4C2314yes—
111C2×Dic3⋊C4C2314yes—
112C2×C4×Dic3C2314yes—
113C2×C4×Dic3C2312yes—
114C2×C4×Dic3C2312yes—
115C2×C4×Dic3C2314yes—
116C2×C4×Dic3C2314yes—
117#890—48no—
118#890—48no—
119#870—24no—
120C4×Dic6—28no—
121C4×Dic6—48no—
122C3×#95C2312yes—
123C3×#95C2314yes—
124C3×#95C2314yes—
125C3×#95C2312yes—
126C3⋊#95C2314yes—
127C3⋊#95C2314yes—
128C3⋊#95C2314yes—
129C3⋊#95C2314yes—
130C3⋊#95C2314yes—
131C3⋊#95C2314yes—
132C3⋊#95C2311yescharacteristic
133C3⋊#95C2314yes—
134C3⋊#95C2311yescharacteristic
135C3⋊#95C2312yes—
136C3⋊#95C2311yescharacteristic
137C3⋊#95C2314yes—
138C3⋊#95C2312yes—
139C3⋊#95C2311yescharacteristic
140C2×#125—34no—
141C2×#125—34no—
142C2×#125—31no—
143C2×#125C2×D311yescharacteristic
144C2×C4×Q8—32no—
145C2×C4×Q8—34no—
146#400—34no—
147#398—34no—
148#398—34no—
149#398—34no—
150#398—34no—
151#398—34no—
152#398—32no—
153#398—32no—
154#398—34no—
155#395—32no—
156#395—34no—
157#395—32no—
158#389—34no—
159#389—32no—
160#389—32no—
161#389—34no—
162#389—34no—
163#389—32no—
164#389—32no—
165#389—34no—
166#389—34no—
167#389—32no—
168#389—34no—
169#389—34no—
170#389—34no—
171#389—34no—
172#389C2×D314yes—
173#389C2×D312yes—
174#387—34no—
175#387—34no—
176#387—34no—
177#387—34no—
178#387—34no—
179#387C2×D314yes—
180#384—32no—
181#384—32no—
182#384—31no—
183#384C2×D311yescharacteristic
184C4×C4⋊C4—34no—
185C4×C4⋊C4—32no—
186C4×C4⋊C4—32no—
187C4×C4⋊C4—32no—
188C4×C4⋊C4—32no—
189C4×C4⋊C4—31no—
190C4×C4⋊C4—31no—
191C4×C4⋊C4C2×D312yes—
192#376C2×D311yescharacteristic
193C22×C3×C4C2411yescharacteristic
194C22×C3×C4C2414yes—
195C22×C3×C4C2411yescharacteristic
196C22×C3×C4C2414yes—
197C22×C3×C4C2412yes—
198C22×C3×C4C2412yes—
199C22×C3×C4C2411yescharacteristic
200C22×Dic3C2412yes—
201C22×Dic3C2412yes—
202C22×Dic3C2414yes—
203C22×Dic3C2414yes—
204C22×Dic3C2412yes—
205C22×Dic3C2412yes—
206C2×Dic6—24no—
207C2×Dic6—24no—
208C2×Dic6—24no—
209C3×C4⋊C4—24no—
210C3×C4⋊C4—44no—
211C3×C4⋊C4—44no—
212C3×C4⋊C4C2×Q8116yes—
213C3×C4⋊C4C2×Q818yes—
214C3×C42—22no—
215C3×C42—44no—
216C3×C42C2×Q814yes—
217C3×C42#4114yes—
218C4⋊Dic3—24no—
219C4⋊Dic3—24no—
220C4⋊Dic3—24no—
221C4⋊Dic3—28no—
222C4⋊Dic3—24no—
223C4⋊Dic3—24no—
224C4⋊Dic3—28no—
225C4⋊Dic3—24no—
226C4⋊Dic3—24no—
227C4⋊Dic3—44no—
228C4⋊Dic3—44no—
229C4⋊Dic3—44no—
230Dic3⋊C4—24no—
231Dic3⋊C4—24no—
232Dic3⋊C4—28no—
233Dic3⋊C4—28no—
234Dic3⋊C4—44no—
235Dic3⋊C4—44no—
236Dic3⋊C4—44no—
237Dic3⋊C4—44no—
238C4×Dic3—24no—
239C4×Dic3—24no—
240C4×Dic3—28no—
241C4×Dic3—44no—
242C4×Dic3—44no—
243C22×Q8—32no—
244#125—64no—
245#125—128no—
246#125—128no—
247#125Dic618yescomplemented
248C4×Q8—68no—
249C4×Q8—128no—
250C2×C4⋊C4—34no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—32no—
253C2×C4⋊C4—32no—
254C2×C4⋊C4—32no—
255C2×C4⋊C4—32no—
256C2×C4⋊C4—34no—
257C2×C4⋊C4—32no—
258C2×C4⋊C4—34no—
259C2×C4⋊C4—32no—
260C2×C4⋊C4—32no—
261C2×C4⋊C4—34no—
262C2×C4⋊C4—32no—
263C2×C4⋊C4—34no—
264C2×C4⋊C4—34no—
265C2×C4⋊C4—34no—
266C2×C4⋊C4—34no—
267C2×C4⋊C4—34no—
268C2×C4⋊C4—34no—
269C2×C4⋊C4—34no—
270C2×C4⋊C4C22×D312yes—
271C2×C4⋊C4C22×D314yes—
272C2×C4⋊C4C22×D314yes—
273C2×C4⋊C4C22×D312yes—
274C2×C4⋊C4C22×D314yes—
275C2×C42—34no—
276C2×C42—32no—
277C2×C42—32no—
278C2×C42—34no—
279C2×C42—34no—
280C2×C42C22×D311yescharacteristic
281C2×C42C22×D314yes—
282C2×C42C22×D311yescharacteristic
283C2×C42C22×D311yescharacteristic
284#95—32no—
285#95—31no—
286#95—34no—
287#95—34no—
288#95—34no—
289#95—32no—
290#95—34no—
291#95—31no—
292#95—31no—
293#95—31no—
294#95—34no—
295#95—34no—
296#95—34no—
297#95—34no—
298#95C22×D312yes—
299#95C22×D312yes—
300#95C22×D314yes—
301#95C22×D314yes—
302C23×C3C2511yescharacteristic
303C2×C3×C4—24no—
304C2×C3×C4—22no—
305C2×C3×C4—22no—
306C2×C3×C4—24no—
307C2×C3×C4—22no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—24no—
311C2×C3×C4—22no—
312C2×C3×C4—22no—
313C2×C3×C4—21no—
314C2×C3×C4—21no—
315C2×C3×C4—24no—
316C2×C3×C4C22×Q814yes—
317C2×C3×C4C22×Q818yes—
318C2×C3×C4C22×Q812yes—
319C2×C3×C4C2×#4114yes—
320C2×C3×C4C2×#4112yes—
321C2×C3×C42−1+412yes—
322C2×Dic3—22no—
323C2×Dic3—22no—
324C2×Dic3—22no—
325C2×Dic3—22no—
326C2×Dic3—22no—
327C2×Dic3—22no—
328C2×Dic3—24no—
329C2×Dic3—24no—
330C2×Dic3—24no—
331C2×Dic3—24no—
332C2×Dic3—22no—
333C2×Dic3—22no—
334C2×Dic3—24no—
335C2×Dic3—22no—
336C2×Dic3—22no—
337C2×Dic3—24no—
338C2×Dic3H(5,𝔽2)14yes—
339C2×Dic32−1+414yes—
340Dic6—216no—
341C2×Q8—64no—
342C2×Q8—64no—
343C2×Q8—64no—
344C22×C4—32no—
345C22×C4—32no—
346C22×C4—32no—
347C22×C4—32no—
348C22×C4—34no—
349C22×C4—34no—
350C22×C4C23×D311yescharacteristic
351C22×C4C23×D311yescharacteristic
352C22×C4C23×D314yes—
353C22×C4C23×D314yes—
354C22×C4C23×D312yes—
355C22×C4C23×D312yes—
356C22×C4C23×D311yescharacteristic
357C4⋊C4—24no—
358C4⋊C4—44no—
359C4⋊C4—44no—
360C4⋊C4—68no—
361C4⋊C4—64no—
362C4⋊C4—68no—
363C4⋊C4—64no—
364C4⋊C4—68no—
365C4⋊C4—64no—
366C4⋊C4—64no—
367C4⋊C4—64no—
368C4⋊C4—64no—
369C4⋊C4—64no—
370C4⋊C4—68no—
371C4⋊C4—64no—
372C4⋊C4—64no—
373C4⋊C4—124no—
374C4⋊C4—124no—
375C4⋊C4—124no—
376C4⋊C4—124no—
377C4⋊C4—124no—
378C4⋊C4—124no—
379C4⋊C4—124no—
380C4⋊C4C2×Dic6116yes—
381C4⋊C4C2×Dic618yes—
382C42—22no—
383C42—44no—
384C42—68no—
385C42—64no—
386C42—64no—
387C42—124no—
388C42—124no—
389C42C2×Dic614yes—
390C42#23914yes—
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—28no—
398C3×C4—24no—
399C3×C4—44no—
400C3×C4—41no—
401C3×C4—41no—
402C3×C4#55714yes—
403Dic3—24no—
404Dic3—24no—
405Dic3—42no—
406Dic3—42no—
407Dic3—44no—
408Dic3—44no—
409C23C24×D311yescharacteristic
410Q8—616no—
411C2×C4—22no—
412C2×C4—24no—
413C2×C4—22no—
414C2×C4—24no—
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—24no—
420C2×C4—24no—
421C2×C4—22no—
422C2×C4—24no—
423C2×C4—22no—
424C2×C4—34no—
425C2×C4—34no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—64no—
433C2×C4—64no—
434C2×C4—64no—
435C2×C4—64no—
436C2×C4—64no—
437C2×C4—62no—
438C2×C4—62no—
439C2×C4—62no—
440C2×C4—62no—
441C2×C4—64no—
442C2×C4C22×Dic614yes—
443C2×C4C22×Dic618yes—
444C2×C4C22×Dic612yes—
445C2×C4C2×#23914yes—
446C2×C4C2×#23912yes—
447C2×C4#101012yes—
448C2×C3C2×#55711yescharacteristic
449C2×C3#349212yes—
450C2×C3#350312yes—
451C2×C3#354811yescharacteristic
452C2×C3#355611yescharacteristic
453C22C23×Dic611yescharacteristic
454C22C22×#23911yescharacteristic
455C22C2×#101011yescharacteristic
456C22D3×H(5,𝔽2)12yes—
457C22D3×2−1+412yes—
458C4—24no—
459C4—28no—
460C4—24no—
461C4—41no—
462C4—44no—
463C4—41no—
464C4—64no—
465C4—64no—
466C4—122no—
467C4—122no—
468C4—124no—
469C4—124no—
470C4#557014yes—
471C3#4498211yescharacteristic, complemented, Sylow
472C2C2×#557011yescharacteristic
473C2#8588912yes—
474C2#8647611yescharacteristic
475C2#8650211yescharacteristic
476C2#8659612yes—
subgroupG/H
center, FrattiniC23C24×D3
commutator, socle, prime radical, cosocleC23×C3C25
FittingC3×#2330C2
hypercenter#2330D3
hypocenterC3#44982
LCSG ▷ C23×C3 ▷ C3
UCSC1 ◁ C23 ◁ #2330

Complements

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

NH ≅ G/Ntype
82. C3×#125Q8semidirect
247. #125Dic6semidirect
471. C3#44982semidirect

Maximal subgroups

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

subgroupindex
C3×#23302
#830252
#761282
#760792
#760352
#760312
#752742
#749982
#743362
#737272
C4⋊C4×Dic62
#733972
#733902
#449823