Finite Group Databasebeta

← #10767641

Subgroups — #10767641

Order 768.

1170 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1170 conjugacy classes of subgroups — too many to draw. Only its 167 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5500C211yesmaximal, characteristic, complemented
2C3×#2406C211yesmaximal, characteristic
3#84084C212yesmaximal
4#71552C211yesmaximal, characteristic, complemented
5#69707C214yesmaximal, complemented
6#69704C214yesmaximal, complemented
7#69533C212yesmaximal, complemented
8#18143—31nomaximal, Sylow
9C22×C3×#41C2211yescharacteristic
10C2×C3×#123—21no—
11C22×C3×C4⋊C4C2211yescharacteristic
12C2×#951—22no—
13C2×#926—21no—
14C2×#924—21no—
15C2×#924C2212yes—
16#5500—22no—
17#5500—24no—
18#5500C2214yes—
19C2×#890—22no—
20C2×#888—22no—
21C3×#395C2212yes—
22C3×#394C2212yes—
23C3×#386—21no—
24C3×#385—22no—
25C3×#381C2211yescharacteristic
26#5239—24no—
27#5232—22no—
28#5228C2214yes—
29#5218C2214yes—
30#5207C2212yes—
31#4528C2214yes—
32#4452C2218yes—
33C2×#544—31no—
34#3007—32no—
35#2406D311yescharacteristic
36#1605—34no—
37#1599—34no—
38#1511—32no—
39#1463—31no—
40C2×C3×#41—22no—
41C2×C3×#41—22no—
42C2×C3×#41—44no—
43C22×C3×Q8C2311yescharacteristic
44C22×C3×D4—21no—
45C22×C3×D4C2311yescharacteristic
46C23×C3×C4—21no—
47C23×C3×C4D411yescharacteristic
48C23×C3×C4D411yescharacteristic
49C23×C3×C4C2311yescharacteristic
50C22×Dic6—21no—
51C22×Dic6D412yes—
52C3×#123—44no—
53C2×C3×C4⋊C4—22no—
54C2×C3×C4⋊C4—22no—
55C2×C3×C4⋊C4—22no—
56C2×C3×C4⋊C4D412yes—
57C2×C3×C4⋊C4C2312yes—
58C2×C3×C22⋊C4—21no—
59C2×C3×C22⋊C4—22no—
60C2×C3×C22⋊C4—22no—
61C2×#217—22no—
62#951—48no—
63C2×#216—24no—
64C2×#214—24no—
65#926—24no—
66#926—24no—
67C2×C4⋊Dic3—21no—
68C2×C4⋊Dic3—21no—
69C2×C4⋊Dic3—21no—
70C2×C4⋊Dic3D412yes—
71#924—24no—
72#924—22no—
73#924—22no—
74#924—24no—
75#924D414yes—
76#924C2314yes—
77C2×Dic3⋊C4—22no—
78C2×Dic3⋊C4—22no—
79C2×C4×Dic3—22no—
80C2×#208—22no—
81#890—28no—
82#890—28no—
83#888—28no—
84#888—28no—
85C3×#95—21no—
86C3×#95—22no—
87C3×#95C2312yes—
88#835—24no—
89#832—24no—
90#830—22no—
91#830C2314yes—
92C22×#41C2×D311yescharacteristic
93C2×#137—62no—
94#544—34no—
95#544—64no—
96#544—62no—
97C2×C4⋊Q8—62no—
98C2×#125—62no—
99C2×#123—21no—
100C2×C22⋊Q8—32no—
101C2×C22⋊Q8—61no—
102C2×#117—61no—
103C22×C4⋊C4C2×D311yescharacteristic
104#452—64no—
105#451—34no—
106#450—34no—
107#417—62no—
108#407—32no—
109#395C2×D312yes—
110#394C2×D312yes—
111#386—21no—
112#385—22no—
113#381C2×D311yescharacteristic
114#328—38no—
115#324—34no—
116C24×C3—21no—
117C24×C3C2×D411yescharacteristic
118C3×#41—28no—
119C3×#41—44no—
120C3×#41—44no—
121C3×#41—44no—
122C2×C3×Q8—22no—
123C2×C3×Q8—42no—
124C2×C3×D4—24no—
125C2×C3×D4—22no—
126C2×C3×D4—42no—
127C2×C3×D4—42no—
128C2×C3×D4—41no—
129C2×C3×D4—41no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4—24no—
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—22no—
137C22×C3×C4—42no—
138C22×C3×C4—42no—
139C22×C3×C4—42no—
140C22×C3×C4—42no—
141C22×C3×C4—42no—
142C22×C3×C4C2×D412yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4C2×D412yes—
145C22×C3×C4C2×D412yes—
146C22×C3×C4C2×D411yescharacteristic
147C22×C3×C4C2411yescharacteristic
148C22×Dic3—22no—
149C22×Dic3—22no—
150C2×Dic6—24no—
151C2×Dic6—22no—
152C2×Dic6—44no—
153C2×Dic6—82no—
154C2×Dic6C2×D414yes—
155C3×C4⋊C4—24no—
156C3×C4⋊C4—24no—
157C3×C4⋊C4—44no—
158C3×C22⋊C4—44no—
159C3×C22⋊C4—42no—
160C3×C22⋊C4—44no—
161#217—24no—
162#217—24no—
163#216—48no—
164#214—48no—
165C4⋊Dic3—24no—
166C4⋊Dic3—24no—
167C4⋊Dic3—24no—
168C4⋊Dic3—22no—
169C4⋊Dic3C2×D414yes—
170Dic3⋊C4—28no—
171Dic3⋊C4—24no—
172Dic3⋊C4—24no—
173C4×Dic3—28no—
174#208—44no—
175C2×C3⋊C8—24no—
176C2×#41—22no—
177C2×#41—22no—
178C2×#41—44no—
179C22×Q8—32no—
180C22×Q8—61no—
181C22×Q8C22×D311yescharacteristic
182C22×D4—21no—
183C22×D4C22×D311yescharacteristic
184C23×C4—21no—
185C23×C4#6711yescharacteristic
186C23×C4#6711yescharacteristic
187C23×C4C22×D311yescharacteristic
188#137—128no—
189C2×Q16—64no—
190C2×SD16—64no—
191C2×M16—62no—
192C4⋊Q8—68no—
193C4⋊Q8—68no—
194#125—68no—
195#125—68no—
196#123—44no—
197C22⋊Q8—34no—
198C22⋊Q8—34no—
199C22⋊Q8—64no—
200C22⋊Q8—62no—
201C22⋊Q8—62no—
202C22⋊Q8—64no—
203#117—64no—
204#117—64no—
205C2×C4⋊C4—22no—
206C2×C4⋊C4—22no—
207C2×C4⋊C4—22no—
208C2×C4⋊C4—32no—
209C2×C4⋊C4—61no—
210C2×C4⋊C4—61no—
211C2×C4⋊C4—61no—
212C2×C4⋊C4—62no—
213C2×C4⋊C4—62no—
214C2×C4⋊C4#6712yes—
215C2×C4⋊C4C22×D312yes—
216C2×C22⋊C4—21no—
217C2×C22⋊C4—22no—
218C2×C22⋊C4—22no—
219C2×C22⋊C4—62no—
220C2×C42—62no—
221#103—64no—
222#102—64no—
223C22⋊C8—34no—
224C22⋊C8—62no—
225#95—21no—
226#95—22no—
227#95C22×D312yes—
228C23×C3—22no—
229C23×C3—22no—
230C23×C3—22no—
231C23×C3—41no—
232C23×C3—41no—
233C23×C3—42no—
234C23×C3—41no—
235C23×C3—42no—
236C23×C3#12011yescharacteristic
237C23×C3#12011yescharacteristic
238C23×C3C22×D411yescharacteristic
239C3×Q8—24no—
240C3×Q8—42no—
241C3×D4—24no—
242C3×D4—42no—
243C3×D4—44no—
244C3×D4—44no—
245C2×C3×C4—24no—
246C2×C3×C4—24no—
247C2×C3×C4—22no—
248C2×C3×C4—22no—
249C2×C3×C4—24no—
250C2×C3×C4—22no—
251C2×C3×C4—22no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×C3×C4—41no—
255C2×C3×C4—48no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—42no—
259C2×C3×C4—42no—
260C2×C3×C4—42no—
261C2×C3×C4—42no—
262C2×C3×C4—42no—
263C2×C3×C4—81no—
264C2×C3×C4#12012yes—
265C2×C3×C4#13714yescomplemented
266C2×C3×C4C22×D412yes—
267C2×Dic3—24no—
268C2×Dic3—24no—
269C2×Dic3—44no—
270C2×Dic3—44no—
271Dic6—44no—
272Dic6—82no—
273Dic6—84no—
274C3⋊C8—44no—
275C24—21no—
276C24C2×#6711yescharacteristic
277#41—28no—
278#41—44no—
279#41—44no—
280#41—44no—
281C2×Q8—22no—
282C2×Q8—34no—
283C2×Q8—42no—
284C2×Q8—64no—
285C2×Q8—62no—
286C2×Q8—124no—
287C2×Q8—242no—
288C2×D4—24no—
289C2×D4—22no—
290C2×D4—42no—
291C2×D4—42no—
292C2×D4—41no—
293C2×D4—41no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—24no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4—42no—
304C22×C4—42no—
305C22×C4—42no—
306C22×C4—62no—
307C22×C4—62no—
308C22×C4D3×D412yes—
309C22×C4D3×D412yes—
310C22×C4C2×#6712yes—
311C22×C4C2×#6712yes—
312C22×C4C2×#6711yescharacteristic
313C22×C4C23×D311yescharacteristic
314Q16—128no—
315SD16—128no—
316M16—124no—
317C2×C8—64no—
318C4⋊C4—24no—
319C4⋊C4—24no—
320C4⋊C4—34no—
321C4⋊C4—44no—
322C4⋊C4—64no—
323C4⋊C4—64no—
324C4⋊C4—64no—
325C4⋊C4—68no—
326C4⋊C4—62no—
327C4⋊C4—64no—
328C4⋊C4—64no—
329C22⋊C4—44no—
330C22⋊C4—42no—
331C22⋊C4—44no—
332C22⋊C4—64no—
333C22⋊C4—64no—
334C42—68no—
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—21no—
338C22×C3—21no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—44no—
343C22×C3—44no—
344C22×C3—42no—
345C22×C3—82no—
346C22×C3—81no—
347C22×C3—81no—
348C22×C3B(4,𝔽2)12yes—
349C22×C3C2×#12011yescharacteristic
350C22×C3C2×#13712yes—
351C3×C4—24no—
352C3×C4—22no—
353C3×C4—22no—
354C3×C4—42no—
355C3×C4—42no—
356C3×C4—82no—
357Dic3—44no—
358Dic3—44no—
359C23—22no—
360C23—22no—
361C23—22no—
362C23—41no—
363C23—41no—
364C23—42no—
365C23—41no—
366C23—42no—
367C23#93711yescharacteristic
368C23#93711yescharacteristic
369C23C22×#6711yescharacteristic
370Q8—24no—
371Q8—42no—
372Q8—124no—
373Q8—242no—
374Q8—244no—
375D4—24no—
376D4—42no—
377D4—44no—
378D4—44no—
379C2×C4—24no—
380C2×C4—24no—
381C2×C4—22no—
382C2×C4—24no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—48no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—42no—
394C2×C4—42no—
395C2×C4—42no—
396C2×C4—42no—
397C2×C4—64no—
398C2×C4—64no—
399C2×C4—81no—
400C2×C4—124no—
401C2×C4—124no—
402C2×C4#93712yes—
403C2×C4#95114yescomplemented
404C2×C4C2×D3×D412yes—
405C8—124no—
406C2×C3—22no—
407C2×C3—22no—
408C2×C3—41no—
409C2×C3—82no—
410C2×C3#300411yescharacteristic
411C2×C3#302011yescharacteristic
412C2×C3C2×B(4,𝔽2)11yescharacteristic
413C22—21no—
414C22—21no—
415C22—21no—
416C22—21no—
417C22—21no—
418C22—21no—
419C22—21no—
420C22—44no—
421C22—44no—
422C22—42no—
423C22—82no—
424C22—81no—
425C22—81no—
426C22#503212yes—
427C22C2×#93711yescharacteristic
428C22C2×#95112yes—
429C4—22no—
430C4—22no—
431C4—24no—
432C4—42no—
433C4—42no—
434C4—82no—
435C4—124no—
436C4—124no—
437C3#1814311yescharacteristic, complemented, Sylow
438C2—22no—
439C2—22no—
440C2—41no—
441C2—82no—
442C2C2×#503211yescharacteristic
443C2#8293311yescharacteristic
444C2#8341511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2406C2
socleC22×C3C2×#120
prime radicalC22×C3×#41C22
hypercenter#2406D3
hypocenterC3#18143
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2406

Complements

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

NH ≅ G/Ntype
1. C2×#5500C2semidirect
4. #71552C2semidirect
5. #69707C2semidirect
6. #69704C2semidirect
7. #69533C2semidirect
265. C2×C3×C4#137semidirect
403. C2×C4#951semidirect
437. C3#18143semidirect

Maximal subgroups

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

subgroupindex
C2×#55002
C3×#24062
#840842
#715522
#697072
#697042
#695332
#181433