Finite Group Databasebeta

← #10769313

Subgroups — #10769313

Order 768.

1058 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1058 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×#5483C211yesmaximal, characteristic
2C3×#2561C211yesmaximal, characteristic
3#82742C212yesmaximal, complemented
4#71539C211yesmaximal, characteristic
5#69710C214yesmaximal, complemented
6#69701C214yesmaximal, complemented
7#69445C212yesmaximal, complemented
8#18319—31nomaximal, Sylow
9C2×C3×C22⋊Q8C2211yescharacteristic
10C22×C3×C22⋊C4C2211yescharacteristic
11C22×#217—21no—
12C2×#924—21no—
13C2×#924C2212yes—
14#5483—22no—
15#5483—24no—
16#5483C2214yes—
17C2×#878—22no—
18C2×#878—22no—
19C3×#397C2212yes—
20C3×#393C2212yes—
21C3×#388—22no—
22C3×#381—21no—
23C3×#379—21no—
24C3×C4×C22⋊C4C2211yescharacteristic
25#5228C2214yescomplemented
26#5207C2212yes—
27#5052—24no—
28#5050—22no—
29#5043—22no—
30#4971C2214yescomplemented
31#4525C2214yes—
32#4452C2218yescomplemented
33C2×#543—31no—
34#3195—32no—
35#2561D311yescharacteristic
36#1608—34no—
37#1602—34no—
38#1508—32no—
39#1462—31no—
40C25×C3D411yescharacteristic
41C22×C3×Q8C2311yescharacteristic
42C23×C3×C4D411yescharacteristic
43C23×C3×C4C2311yescharacteristic
44C23×Dic3—21no—
45C23×Dic3—21no—
46C22×Dic6—21no—
47C22×Dic6D412yes—
48C3×C22⋊Q8—22no—
49C3×C22⋊Q8—22no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4C2311yescharacteristic
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4—22no—
54C2×C3×C22⋊C4—21no—
55C2×C3×C22⋊C4—21no—
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4D412yes—
59C2×C3×C22⋊C4C2312yes—
60C2×C3×C42—21no—
61C2×#217—22no—
62C2×#217—24no—
63C2×#217—24no—
64C2×#217—22no—
65C2×#216—24no—
66C2×C4⋊Dic3—21no—
67C2×C4⋊Dic3D412yes—
68#924—24no—
69#924—22no—
70#924—22no—
71#924—24no—
72#924D414yes—
73#924C2314yes—
74C2×Dic3⋊C4—22no—
75C2×Dic3⋊C4—22no—
76C2×#208—22no—
77#878—28no—
78#878—28no—
79#878—28no—
80#878—28no—
81C3×#95—22no—
82C3×#95—22no—
83C3×#95C2312yes—
84#835—24no—
85#830—22no—
86#830C2314yes—
87#810—24no—
88#808—24no—
89#543—34no—
90#543—64no—
91#543—62no—
92C2×C22⋊Q8—32no—
93C2×C22⋊Q8—62no—
94C2×C22⋊Q8—62no—
95C2×C22⋊Q8—61no—
96C2×C22⋊Q8C2×D311yescharacteristic
97C22×C22⋊C4—61no—
98C22×C22⋊C4C2×D311yescharacteristic
99#483—34no—
100#483—62no—
101#470—64no—
102#451—34no—
103#451—62no—
104#407—32no—
105#397C2×D312yes—
106#393C2×D312yes—
107#388—22no—
108#381—21no—
109#379—21no—
110C4×C22⋊C4C2×D311yescharacteristic
111#328—38no—
112#323—34no—
113C24×C3—21no—
114C24×C3—21no—
115C24×C3—24no—
116C24×C3—42no—
117C24×C3—81no—
118C24×C3C2×D411yescharacteristic
119C24×C3C2×D412yes—
120C2×C3×Q8—22no—
121C2×C3×Q8—81no—
122C22×C3×C4—21no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—42no—
128C22×C3×C4—81no—
129C22×C3×C4C2×D412yes—
130C22×C3×C4C2×D412yes—
131C22×C3×C4C2×D412yes—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2411yescharacteristic
134C22×Dic3—22no—
135C22×Dic3—22no—
136C22×Dic3—24no—
137C22×Dic3—24no—
138C22×Dic3—82no—
139C22×Dic3—82no—
140C2×Dic6—24no—
141C2×Dic6—22no—
142C2×Dic6—82no—
143C2×Dic6—82no—
144C2×Dic6C2×D414yes—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—42no—
147C3×C22⋊C4—24no—
148C3×C22⋊C4—24no—
149C3×C22⋊C4—42no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—44no—
153C3×C22⋊C4—82no—
154C3×C42—81no—
155#217—28no—
156#217—24no—
157#217—24no—
158#216—84no—
159C4⋊Dic3—24no—
160C4⋊Dic3—22no—
161C4⋊Dic3C2×D414yes—
162Dic3⋊C4—28no—
163Dic3⋊C4—24no—
164Dic3⋊C4—24no—
165#208—82no—
166C2×C3⋊C8—24no—
167C25#6711yescharacteristic
168C22×Q8—32no—
169C22×Q8—61no—
170C22×Q8C22×D311yescharacteristic
171C23×C4—61no—
172C23×C4—61no—
173C23×C4#6711yescharacteristic
174C23×C4C22×D311yescharacteristic
175C2×Q16—64no—
176C2×M16—62no—
177C22⋊Q8—22no—
178C22⋊Q8—22no—
179C22⋊Q8—34no—
180C22⋊Q8—34no—
181C22⋊Q8—68no—
182C22⋊Q8—68no—
183C22⋊Q8—68no—
184C22⋊Q8—68no—
185C22⋊Q8—64no—
186C22⋊Q8—62no—
187C22⋊Q8—62no—
188C22⋊Q8—64no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—32no—
191C2×C4⋊C4—62no—
192C2×C4⋊C4—61no—
193C2×C4⋊C4—62no—
194C2×C4⋊C4C22×D311yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—62no—
202C2×C22⋊C4—64no—
203C2×C22⋊C4—62no—
204C2×C22⋊C4—64no—
205C2×C22⋊C4#6712yes—
206C2×C22⋊C4C22×D312yes—
207C2×C42—21no—
208#106—64no—
209#103—64no—
210#103—64no—
211C22⋊C8—34no—
212C22⋊C8—62no—
213#95—22no—
214#95—22no—
215#95C22×D312yes—
216C23×C3—24no—
217C23×C3—24no—
218C23×C3—22no—
219C23×C3—22no—
220C23×C3—22no—
221C23×C3—22no—
222C23×C3—41no—
223C23×C3—42no—
224C23×C3—42no—
225C23×C3—42no—
226C23×C3—42no—
227C23×C3—81no—
228C23×C3—81no—
229C23×C3—88no—
230C23×C3#12011yescharacteristic
231C23×C3#12011yescharacteristic
232C23×C3#13714yes—
233C23×C3C22×D411yescharacteristic
234C3×Q8—44no—
235C2×C3×C4—22no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
238C2×C3×C4—42no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—42no—
244C2×C3×C4—42no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—42no—
249C2×C3×C4—81no—
250C2×C3×C4#12012yes—
251C2×C3×C4C22×D412yes—
252C2×Dic3—24no—
253C2×Dic3—24no—
254C2×Dic3—44no—
255C2×Dic3—44no—
256C2×Dic3—84no—
257C2×Dic3—84no—
258Dic6—44no—
259Dic6—82no—
260Dic6—84no—
261C3⋊C8—44no—
262C24—21no—
263C24—21no—
264C24—24no—
265C24—42no—
266C24—81no—
267C24C2×#6711yescharacteristic
268C24C2×#6712yes—
269C2×Q8—22no—
270C2×Q8—34no—
271C2×Q8—64no—
272C2×Q8—62no—
273C2×Q8—81no—
274C2×Q8—242no—
275C2×Q8—242no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—22no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—42no—
282C22×C4—62no—
283C22×C4—62no—
284C22×C4—64no—
285C22×C4—64no—
286C22×C4—81no—
287C22×C4—242no—
288C22×C4—242no—
289C22×C4D3×D412yes—
290C22×C4D3×D412yes—
291C22×C4C2×#6712yes—
292C22×C4C2×#6711yescharacteristic
293C22×C4C23×D311yescharacteristic
294Q16—244no—
295M16—242no—
296C2×C8—64no—
297C4⋊C4—22no—
298C4⋊C4—34no—
299C4⋊C4—42no—
300C4⋊C4—68no—
301C4⋊C4—64no—
302C4⋊C4—62no—
303C4⋊C4—64no—
304C4⋊C4—64no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—42no—
308C22⋊C4—41no—
309C22⋊C4—41no—
310C22⋊C4—44no—
311C22⋊C4—68no—
312C22⋊C4—64no—
313C22⋊C4—64no—
314C22⋊C4—82no—
315C42—81no—
316C22×C3—21no—
317C22×C3—21no—
318C22×C3—21no—
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—24no—
322C22×C3—21no—
323C22×C3—21no—
324C22×C3—44no—
325C22×C3—44no—
326C22×C3—48no—
327C22×C3—42no—
328C22×C3—42no—
329C22×C3—82no—
330C22×C3—84no—
331C22×C3#45512yes—
332C22×C3C2×#12011yescharacteristic
333C22×C3C2×#13712yes—
334C3×C4—22no—
335C3×C4—22no—
336C3×C4—42no—
337C3×C4—82no—
338C3×C4—81no—
339C3×C4—81no—
340Dic3—44no—
341Dic3—44no—
342C23—24no—
343C23—24no—
344C23—22no—
345C23—22no—
346C23—22no—
347C23—22no—
348C23—41no—
349C23—42no—
350C23—42no—
351C23—42no—
352C23—42no—
353C23—81no—
354C23—81no—
355C23—88no—
356C23#93711yescharacteristic
357C23#93711yescharacteristic
358C23#95114yes—
359C23C22×#6711yescharacteristic
360Q8—44no—
361Q8—124no—
362Q8—242no—
363Q8—244no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—42no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—42no—
377C2×C4—42no—
378C2×C4—64no—
379C2×C4—64no—
380C2×C4—81no—
381C2×C4—124no—
382C2×C4—124no—
383C2×C4—244no—
384C2×C4—244no—
385C2×C4#93712yes—
386C2×C4C2×D3×D412yes—
387C8—124no—
388C2×C3—22no—
389C2×C3—22no—
390C2×C3—41no—
391C2×C3—44no—
392C2×C3#300411yescharacteristic
393C2×C3C2×#45511yescharacteristic
394C2×C3#302611yescharacteristic
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—21no—
400C22—21no—
401C22—21no—
402C22—24no—
403C22—44no—
404C22—44no—
405C22—48no—
406C22—42no—
407C22—42no—
408C22—82no—
409C22—84no—
410C22#506112yes—
411C22C2×#93711yescharacteristic
412C22C2×#95112yes—
413C4—22no—
414C4—22no—
415C4—42no—
416C4—82no—
417C4—81no—
418C4—81no—
419C4—124no—
420C4—124no—
421C3#1831911yescharacteristic, complemented, Sylow
422C2—22no—
423C2—22no—
424C2—41no—
425C2—44no—
426C2#8293311yescharacteristic
427C2C2×#506111yescharacteristic
428C2#8337511yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2561C2
socleC22×C3C2×#120
prime radicalC25×C3D4
hypercenter#2561D3
hypocenterC3#18319
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2561

Complements

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

NH ≅ G/Ntype
3. #82742C2semidirect
5. #69710C2semidirect
6. #69701C2semidirect
7. #69445C2semidirect
25. #5228C22semidirect
30. #4971C22semidirect
32. #4452C22semidirect
421. C3#18319semidirect

Maximal subgroups

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

subgroupindex
C2×#54832
C3×#25612
#827422
#715392
#697102
#697012
#694452
#183193