Finite Group Databasebeta

← #10695460

Subgroups — #10695460

Order 768.

810 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#395C211yesmaximal, characteristic
2C2×#5228C211yesmaximal, characteristic
3#82183C211yesmaximal, characteristic
4#72132C211yesmaximal, characteristic
5C2×#4528C211yesmaximal, characteristic
6C2×#4452C211yesmaximal, characteristic
7#67612C211yesmaximal, characteristic
8#9925—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×C3×C4⋊C4—21no—
11C22×C3×C4⋊C4C2211yescharacteristic
12C22×#216—21no—
13C2×#924C2211yescharacteristic
14C3×#395—22no—
15C3×#395—22no—
16C2×C3×#95—21no—
17C2×C3×#95C2211yescharacteristic
18#5228—22no—
19#5228—22no—
20C2×#835—21no—
21C2×#835—22no—
22#5212C2211yescharacteristic
23#5211—21no—
24#5210—21no—
25C2×#830C2211yescharacteristic
26C2×#830C2211yescharacteristic
27#4986—21no—
28#4982—22no—
29#4979—21no—
30#4528—22no—
31#4452—22no—
32C2×#451—31no—
33#2627—31no—
34C2×#395D311yescharacteristic
35#1787—31no—
36C2×#328—31no—
37C2×#324—31no—
38#1294—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—44no—
45C22×C3×Q8D412yes—
46C23×C3×C4—21no—
47C23×C3×C4—21no—
48C23×C3×C4D411yescharacteristic
49C23×C3×C4D411yescharacteristic
50C23×C3×C4C2311yescharacteristic
51C22×Dic6D411yescharacteristic
52C2×C3×C4⋊C4—24no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C4⋊C4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—44no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#216—24no—
63C2×#216—44no—
64C2×C4⋊Dic3D411yescharacteristic
65C2×C4⋊Dic3D412yes—
66#924—21no—
67#924—21no—
68#924—21no—
69#924—21no—
70C2×Dic3⋊C4—21no—
71C2×Dic3⋊C4—22no—
72C22×C3⋊C8—21no—
73C22×C3⋊C8—21no—
74C3×#95—24no—
75C3×#95—21no—
76C3×#95D412yes—
77#835—24no—
78#835—44no—
79C3⋊#95—22no—
80C3⋊#95—22no—
81C3⋊#95—22no—
82C3⋊#95—21no—
83#830—21no—
84#830—21no—
85#830—21no—
86#830D412yes—
87C23×Q8C2×D311yescharacteristic
88C22×Q16—61no—
89C2×C22⋊Q8—31no—
90C22×C4⋊C4—21no—
91C22×C4⋊C4C2×D311yescharacteristic
92#451—62no—
93#451—62no—
94C2×#103—61no—
95C2×#103—62no—
96C2×C22⋊C8—31no—
97C2×C22⋊C8—31no—
98#400—61no—
99#398—62no—
100#395—22no—
101#395—22no—
102#395—61no—
103#385—61no—
104#381—61no—
105#380—31no—
106C2×#95—21no—
107C2×#95C2×D311yescharacteristic
108#328—62no—
109#324—62no—
110C24×C3C2×D411yescharacteristic
111C2×C3×Q8—24no—
112C2×C3×Q8—24no—
113C2×C3×Q8—22no—
114C2×C3×Q8—22no—
115C2×C3×Q8—44no—
116C2×C3×Q8—42no—
117C2×C3×Q8—42no—
118C2×C3×Q8—44no—
119C2×C3×Q8—44no—
120C2×C3×Q8—44no—
121C2×C3×Q8—41no—
122C2×C3×Q8—41no—
123C22×C3×C4—24no—
124C22×C3×C4—24no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—22no—
128C22×C3×C4—22no—
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—42no—
137C22×C3×C4—42no—
138C22×C3×C4—42no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4—41no—
143C22×C3×C4—42no—
144C22×C3×C4Q1612yescomplemented
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2×D411yescharacteristic
148C22×C3×C4C2×D411yescharacteristic
149C22×C3×C4C2×D411yescharacteristic
150C22×C3×C4#4111yescharacteristic
151C22×Dic3—21no—
152C22×Dic3—22no—
153C22×Dic3—21no—
154C2×Dic6—21no—
155C2×Dic6—21no—
156C2×Dic6—42no—
157C3×C4⋊C4—48no—
158C3×C4⋊C4—41no—
159C3×C4⋊C4—41no—
160C3×C4⋊C4—41no—
161C3×C4⋊C4—41no—
162#217—41no—
163#217—41no—
164#217—41no—
165#217—41no—
166#216—48no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—42no—
170Dic3⋊C4—41no—
171Dic3⋊C4—41no—
172Dic3⋊C4—82no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—42no—
175C2×C3⋊C8—42no—
176C2×C3⋊C8—41no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—21no—
180C22×Q8—21no—
181C22×Q8—21no—
182C22×Q8—31no—
183C22×Q8—44no—
184C22×Q8#6712yes—
185C23×C4—21no—
186C23×C4—21no—
187C23×C4#6711yescharacteristic
188C23×C4#6711yescharacteristic
189C23×C4C22×D311yescharacteristic
190C2×Q16—64no—
191C2×Q16—124no—
192C22×C8—61no—
193C22×C8—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196C22⋊Q8—61no—
197C22⋊Q8—61no—
198C2×C4⋊C4—24no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—21no—
204C2×C4⋊C4—21no—
205C2×C4⋊C4—31no—
206C2×C4⋊C4—32no—
207C2×C4⋊C4—44no—
208C2×C4⋊C4—61no—
209C2×C4⋊C4—62no—
210C2×C22⋊C4—61no—
211C2×C22⋊C4—61no—
212#103—64no—
213#103—124no—
214C22⋊C8—32no—
215C22⋊C8—61no—
216C22⋊C8—61no—
217C22⋊C8—61no—
218#95—24no—
219#95—21no—
220#95—62no—
221#95—62no—
222#95—62no—
223#95—61no—
224#95#6712yes—
225C23×C3—22no—
226C23×C3—22no—
227C23×C3#12011yescharacteristic
228C23×C3#12011yescharacteristic
229C23×C3#12111yescharacteristic
230C23×C3C2×Q1611yescharacteristic
231C23×C3#13511yescharacteristic
232C23×C3#13711yescharacteristic
233C23×C3#13711yescharacteristic
234C3×Q8—24no—
235C3×Q8—24no—
236C3×Q8—42no—
237C3×Q8—44no—
238C3×Q8—44no—
239C3×Q8—42no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—24no—
243C2×C3×C4—24no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—41no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—42no—
252C2×C3×C4—44no—
253C2×C3×C4—44no—
254C2×C3×C4—42no—
255C2×C3×C4—44no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—42no—
261C2×C3×C4—44no—
262C2×C3×C4—41no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×C3×C4—81no—
267C2×C3×C4#12011yescharacteristic
268C2×C3×C4#12111yescharacteristic
269C2×C3×C4#12111yescharacteristic
270C2×C3×C4#12711yescharacteristic
271C2×Dic3—41no—
272C2×Dic3—41no—
273C2×Dic3—42no—
274C2×Dic3—42no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277C2×Dic3—42no—
278Dic6—42no—
279Dic6—81no—
280C3⋊C8—42no—
281C3⋊C8—81no—
282C24C2×#6711yescharacteristic
283C2×Q8—24no—
284C2×Q8—24no—
285C2×Q8—22no—
286C2×Q8—22no—
287C2×Q8—44no—
288C2×Q8—42no—
289C2×Q8—42no—
290C2×Q8—44no—
291C2×Q8—44no—
292C2×Q8—44no—
293C2×Q8—41no—
294C2×Q8—41no—
295C2×Q8—61no—
296C2×Q8—61no—
297C2×Q8—122no—
298C22×C4—24no—
299C22×C4—24no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—22no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—42no—
312C22×C4—42no—
313C22×C4—42no—
314C22×C4—41no—
315C22×C4—41no—
316C22×C4—41no—
317C22×C4—41no—
318C22×C4—42no—
319C22×C4—61no—
320C22×C4—62no—
321C22×C4—61no—
322C22×C4#21612yescomplemented
323C22×C4D3×D411yescharacteristic
324C22×C4D3×D411yescharacteristic
325C22×C4D3×D411yescharacteristic
326C22×C4#23711yescharacteristic
327C22×C4C2×#6711yescharacteristic
328C22×C4C2×#6711yescharacteristic
329Q16—128no—
330C2×C8—62no—
331C2×C8—122no—
332C2×C8—122no—
333C2×C8—121no—
334C4⋊C4—48no—
335C4⋊C4—41no—
336C4⋊C4—41no—
337C4⋊C4—41no—
338C4⋊C4—41no—
339C4⋊C4—61no—
340C4⋊C4—61no—
341C4⋊C4—122no—
342C4⋊C4—121no—
343C4⋊C4—121no—
344C4⋊C4—242no—
345C22⋊C4—121no—
346C22⋊C4—121no—
347C22⋊C4—121no—
348C22⋊C4—121no—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—21no—
354C22×C3—21no—
355C22×C3—21no—
356C22×C3—21no—
357C22×C3#39211yescharacteristic
358C22×C3#45111yescharacteristic
359C22×C3#45111yescharacteristic
360C22×C3#45211yescharacteristic
361C22×C3#45211yescharacteristic
362C22×C3#45612yes—
363C22×C3B(4,𝔽2)12yes—
364C22×C3#46711yescharacteristic
365C22×C3#47011yescharacteristic
366C3×C4—21no—
367C3×C4—21no—
368C3×C4—21no—
369C3×C4—21no—
370C3×C4—44no—
371C3×C4—42no—
372C3×C4—44no—
373C3×C4—81no—
374Dic3—82no—
375Dic3—81no—
376Dic3—81no—
377C23—22no—
378C23—22no—
379C23#93711yescharacteristic
380C23#93911yescharacteristic
381C23#94211yescharacteristic
382C23C2×#21611yescharacteristic
383C23#95011yescharacteristic
384C23#95111yescharacteristic
385C23#95311yescharacteristic
386Q8—24no—
387Q8—24no—
388Q8—42no—
389Q8—44no—
390Q8—44no—
391Q8—42no—
392Q8—122no—
393Q8—241no—
394C2×C4—22no—
395C2×C4—22no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—22no—
399C2×C4—22no—
400C2×C4—22no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—44no—
408C2×C4—44no—
409C2×C4—44no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—41no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—42no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—81no—
421C2×C4—121no—
422C2×C4—121no—
423C2×C4—122no—
424C2×C4—122no—
425C2×C4—122no—
426C2×C4—122no—
427C2×C4—122no—
428C2×C4#93711yescharacteristic
429C2×C4#93811yescharacteristic
430C2×C4#93911yescharacteristic
431C2×C4#94011yescharacteristic
432C8—122no—
433C8—241no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3#160312yes—
439C2×C3#160512yes—
440C2×C3#200111yescharacteristic
441C2×C3#200511yescharacteristic
442C2×C3#202111yescharacteristic
443C22—22no—
444C22—22no—
445C22—22no—
446C22—22no—
447C22—21no—
448C22—21no—
449C22—21no—
450C22—21no—
451C22#503212yes—
452C22#504711yescharacteristic
453C22#505011yescharacteristic
454C22#505111yescharacteristic
455C22#505211yescharacteristic
456C22#509512yes—
457C22#522311yescharacteristic
458C22#522811yescharacteristic
459C22#523911yescharacteristic
460C4—21no—
461C4—21no—
462C4—21no—
463C4—21no—
464C4—44no—
465C4—42no—
466C4—44no—
467C4—81no—
468C4—242no—
469C4—241no—
470C4—241no—
471C3#992511yescharacteristic, complemented, Sylow
472C2—21no—
473C2—21no—
474C2—21no—
475C2—21no—
476C2#6970512yes—
477C2#6970712yes—
478C2#7167111yescharacteristic
479C2#7170011yescharacteristic
480C2#7179511yescharacteristic
subgroupG/H
centerC23#937
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#395C2
socleC23×C3#120
prime radicalC24×C3C2×D4
hypercenterC2×#395D3
hypocenterC3#9925
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#395

Complements

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

NH ≅ G/Ntype
144. C22×C3×C4Q16semidirect
322. C22×C4#216semidirect
471. C3#9925semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#3952
C2×#52282
#821832
#721322
C2×#45282
C2×#44522
#676122
#99253