Finite Group Databasebeta

← #10767973

Subgroups — #10767973

Order 768.

1106 conjugacy classes of subgroups; only the 183 normal subgroups are shown.

normal subgroups only

This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 183 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
1C3×#3501C211yesmaximal, characteristic, complemented
2#83353C212yesmaximal, complemented
3#82347C211yesmaximal, characteristic
4#71892C214yesmaximal, complemented
5#71891C214yesmaximal, complemented
6#71836C212yesmaximal, complemented
7#71603C211yesmaximal, characteristic, complemented
8#18180—31nomaximal, Sylow
9C3×#554—22no—
10C3×#554—24no—
11C3×D4×Q8—22no—
12C3×#544—21no—
13C3×#543C2214yes—
14C3×#533C2212yes—
15C2×C3×C22⋊Q8—22no—
16C3×#518—21no—
17C2×C3×C4×D4C2211yescharacteristic
18C2×C4×#67—22no—
19C2×C4×D12C2211yescharacteristic
20#5223—21no—
21#5087—22no—
22#5083—22no—
23#5046C2214yes—
24#5044C2214yes—
25#5026—24no—
26#4993—21no—
27#4960C2212yes—
28#4949C2212yes—
29#4932C2211yescharacteristic
30#4926C2212yes—
31#4538C2218yes—
32#4525C2214yes—
33#3501D311yescharacteristic, complemented
34#3110—32no—
35#2397—31no—
36#1614—34no—
37#1596—34no—
38#1506—32no—
39#1466—31no—
40C22×C3×Q8—22no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4D412yes—
43C22×#67—22no—
44C22×D12C2311yescharacteristic
45C22×C4×D3—21no—
46C3×C4⋊Q8—24no—
47C3×C4⋊Q8D412yescomplemented
48C3×C4⋊Q8D412yescomplemented
49C3×#125—24no—
50C3×#125—22no—
51C3×C22⋊Q8—24no—
52C3×C22⋊Q8—24no—
53C3×C22⋊Q8—28no—
54C3×C22⋊Q8—22no—
55C3×C22⋊Q8—24no—
56C3×C22⋊Q8—28no—
57C3×C22⋊Q8C2314yes—
58C3×C4×Q8—22no—
59C3×C4×Q8D414yescomplemented
60C3×C4×D4—24no—
61C3×C4×D4—24no—
62C3×C4×D4—22no—
63C3×C4×D4—22no—
64C3×#117—22no—
65C2×C3×C4⋊C4—22no—
66C2×C3×C4⋊C4—24no—
67C2×C3×C4⋊C4D411yescharacteristic
68C2×C3×C22⋊C4—24no—
69C2×C3×C22⋊C4C2312yes—
70C2×C3×C42C2311yescharacteristic
71C2×#217—22no—
72C2×#215—24no—
73C4×#67—48no—
74C2×(C2×D3)⋊C4—22no—
75C2×(C2×D3)⋊C4—21no—
76C2×C4⋊Dic3C2311yescharacteristic
77C2×Dic3⋊C4—22no—
78C2×C4×Dic3—21no—
79C4×D12—22no—
80C4×D12—22no—
81#835—24no—
82C3⋊#95—21no—
83C3⋊#95C2312yes—
84#830C2314yes—
85#809—24no—
86#807—24no—
87C4⋊(C3⋊C8)—22no—
88#803—22no—
89#554—22no—
90#554—24no—
91D4×Q8—22no—
92#544—21no—
93#543C2×D314yes—
94#533C2×D312yes—
95C2×C22⋊Q8—22no—
96#518—21no—
97C2×C4×D4—31no—
98C2×C4×D4—62no—
99C2×C4×D4C2×D311yescharacteristic
100#484—34no—
101#474—62no—
102#461—62no—
103#450—34no—
104#440—64no—
105#432—32no—
106#399—32no—
107#399—61no—
108#392—32no—
109#392—61no—
110#376—31no—
111#328—38no—
112#323—34no—
113C24×C3C2×D412yes—
114C23×D3—21no—
115C2×C3×Q8—24no—
116C2×C3×Q8—22no—
117C2×C3×Q8—84no—
118C2×C3×Q8C2×D414yes—
119C2×C3×D4—24no—
120C2×C3×D4—22no—
121C22×C3×C4—24no—
122C22×C3×C4—24no—
123C22×C3×C4—22no—
124C22×C3×C4—22no—
125C22×C3×C4—44no—
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D412yes—
129C22×C3×C4C2411yescharacteristic
130C2×#67—44no—
131C2×#67—44no—
132C2×#67—44no—
133C22×Dic3—22no—
134C22×Dic3—21no—
135C2×D12—22no—
136C2×D12—42no—
137C2×C4×D3—44no—
138C2×C4×D3—41no—
139C2×C4×D3—41no—
140C3×C4⋊C4—24no—
141C3×C4⋊C4—28no—
142C3×C4⋊C4—22no—
143C3×C4⋊C4—24no—
144C3×C4⋊C4—22no—
145C3×C4⋊C4—22no—
146C3×C4⋊C4—22no—
147C3×C4⋊C4—24no—
148C3×C4⋊C4C2×D414yes—
149C3×C22⋊C4—28no—
150C3×C22⋊C4—24no—
151C3×C22⋊C4—24no—
152C3×C42—24no—
153C3×C42C2×D412yes—
154C3×C42C2×D412yes—
155#217—44no—
156#215—48no—
157(C2×D3)⋊C4—44no—
158(C2×D3)⋊C4—42no—
159C4⋊Dic3—22no—
160Dic3⋊C4—44no—
161C4×Dic3—24no—
162C2×C3⋊C8—24no—
163C22×Q8—22no—
164C22×D4—31no—
165C22×D4—62no—
166C22×D4#6711yescharacteristic
167C23×C4—61no—
168C23×C4#6712yes—
169C2×SD16—64no—
170C4⋊Q8—24no—
171C4⋊Q8#6712yescomplemented
172C4⋊Q8#6712yescomplemented
173#125—24no—
174#125—22no—
175C22⋊Q8—24no—
176C22⋊Q8—24no—
177C22⋊Q8—28no—
178C22⋊Q8—22no—
179C22⋊Q8—24no—
180C22⋊Q8—28no—
181C22⋊Q8C22×D314yes—
182C4×Q8—22no—
183C4×Q8#6714yescomplemented
184C4×D4—24no—
185C4×D4—24no—
186C4×D4—22no—
187C4×D4—22no—
188C4×D4—62no—
189C4×D4—62no—
190C4×D4—128no—
191#117—22no—
192C2×C4⋊C4—22no—
193C2×C4⋊C4—24no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—62no—
196C2×C4⋊C4#6711yescharacteristic
197C2×C22⋊C4—24no—
198C2×C22⋊C4—62no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4—62no—
201C2×C22⋊C4C22×D312yes—
202C2×C42—61no—
203C2×C42C22×D311yescharacteristic
204#107—64no—
205C4⋊C8—62no—
206#103—64no—
207#102—64no—
208C22⋊C8—34no—
209#97—62no—
210#95—32no—
211#95—61no—
212C23×C3—22no—
213C23×C3—22no—
214C23×C3—42no—
215C23×C3—42no—
216C23×C3#13714yes—
217C23×C3C22×D411yescharacteristic
218C22×D3—22no—
219C22×D3—42no—
220C22×D3—41no—
221C22×D3—41no—
222C22×D3—42no—
223C3×Q8—44no—
224C3×Q8—44no—
225C3×Q8—84no—
226C3×D4—28no—
227C2×C3×C4—22no—
228C2×C3×C4—22no—
229C2×C3×C4—24no—
230C2×C3×C4—24no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—44no—
234C2×C3×C4—42no—
235C2×C3×C4—82no—
236C2×C3×C4—84no—
237C2×C3×C4—84no—
238C2×C3×C4#12012yes—
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12011yescharacteristic
241C2×C3×C4C22×D412yes—
242#67—416no—
243C2×Dic3—24no—
244C2×Dic3—42no—
245C2×Dic3—42no—
246C2×Dic3—42no—
247C2×Dic3—41no—
248D12—24no—
249D12—42no—
250C4×D3—82no—
251C4×D3—82no—
252C3⋊C8—44no—
253C24—61no—
254C24C2×#6712yes—
255C2×Q8—24no—
256C2×Q8—22no—
257C2×Q8—84no—
258C2×Q8C2×#6714yes—
259C2×D4—24no—
260C2×D4—22no—
261C2×D4—62no—
262C2×D4—122no—
263C2×D4—124no—
264C2×D4—124no—
265C2×D4—124no—
266C22×C4—24no—
267C22×C4—24no—
268C22×C4—22no—
269C22×C4—22no—
270C22×C4—44no—
271C22×C4—62no—
272C22×C4—61no—
273C22×C4—124no—
274C22×C4—121no—
275C22×C4—121no—
276C22×C4C2×#6711yescharacteristic
277C22×C4C2×#6711yescharacteristic
278C22×C4C2×#6712yes—
279C22×C4C23×D311yescharacteristic
280SD16—128no—
281C2×C8—64no—
282C4⋊C4—24no—
283C4⋊C4—28no—
284C4⋊C4—22no—
285C4⋊C4—24no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—24no—
290C4⋊C4—62no—
291C4⋊C4—124no—
292C4⋊C4C2×#6714yes—
293C22⋊C4—28no—
294C22⋊C4—24no—
295C22⋊C4—24no—
296C22⋊C4—124no—
297C22⋊C4—124no—
298C22⋊C4—122no—
299C42—24no—
300C42—64no—
301C42C2×#6712yes—
302C42C2×#6712yes—
303C22×C3—21no—
304C22×C3—21no—
305C22×C3—24no—
306C22×C3—44no—
307C22×C3—42no—
308C22×C3—42no—
309C22×C3—82no—
310C22×C3C2×#12011yescharacteristic
311C22×C3C2×#13712yes—
312C2×D3—44no—
313C2×D3—42no—
314C2×D3—82no—
315C2×D3—81no—
316C2×D3—81no—
317C3×C4—22no—
318C3×C4—21no—
319C3×C4—22no—
320C3×C4—42no—
321C3×C4—44no—
322C3×C4—44no—
323C3×C4#45312yes—
324Dic3—44no—
325Dic3—81no—
326C23—22no—
327C23—22no—
328C23—42no—
329C23—42no—
330C23—62no—
331C23—122no—
332C23—121no—
333C23—122no—
334C23—121no—
335C23#94214yes—
336C23C22×#6711yescharacteristic
337Q8—44no—
338Q8—44no—
339Q8—84no—
340D4—28no—
341D4—64no—
342D4—122no—
343D4—1216no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—24no—
347C2×C4—24no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—44no—
351C2×C4—42no—
352C2×C4—64no—
353C2×C4—82no—
354C2×C4—84no—
355C2×C4—84no—
356C2×C4—122no—
357C2×C4—122no—
358C2×C4—122no—
359C2×C4—121no—
360C2×C4—242no—
361C2×C4—242no—
362C2×C4#95312yes—
363C2×C4#95311yescharacteristic
364C2×C4#95311yescharacteristic
365C2×C4C22×#6712yes—
366C8—124no—
367C2×C3—22no—
368C2×C3—44no—
369C2×C3#300411yescharacteristic
370C2×C3C2×#45311yescharacteristic
371C2×C3#302511yescharacteristic
372D3—82no—
373C22—21no—
374C22—21no—
375C22—24no—
376C22—44no—
377C22—42no—
378C22—42no—
379C22—82no—
380C22—124no—
381C22—122no—
382C22—242no—
383C22—241no—
384C22—241no—
385C22C2×#94212yes—
386C22C2×#95311yescharacteristic
387C4—22no—
388C4—21no—
389C4—22no—
390C4—42no—
391C4—44no—
392C4—44no—
393C4—124no—
394C4—241no—
395C4#524112yes—
396C3#1818011yescharacteristic, complemented, Sylow
397C2—22no—
398C2—44no—
399C2—242no—
400C2#8405911yescharacteristic
401C2C2×#524111yescharacteristic
402C2#8412511yescharacteristic
subgroupG/H
centerC22C2×#953
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#3501C2
socleC22×C3C2×#120
prime radical#82347C2
hypercenter#3501D3
hypocenterC3#18180
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3501

Complements

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

NH ≅ G/Ntype
1. C3×#3501C2semidirect
2. #83353C2semidirect
4. #71892C2semidirect
5. #71891C2semidirect
6. #71836C2semidirect
7. #71603C2semidirect
33. #3501D3semidirect
47. C3×C4⋊Q8D4semidirect
48. C3×C4⋊Q8D4semidirect
59. C3×C4×Q8D4semidirect
171. C4⋊Q8#67semidirect
172. C4⋊Q8#67semidirect
183. C4×Q8#67semidirect
396. C3#18180semidirect

Maximal subgroups

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

subgroupindex
C3×#35012
#833532
#823472
#718922
#718912
#718362
#716032
#181803