Finite Group Databasebeta

← #10698042

Subgroups — #10698042

Order 768.

1066 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 1066 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3186C211yesmaximal, characteristic, complemented
2#82723C212yesmaximal, complemented
3#71675C212yesmaximal, complemented
4#69955C211yesmaximal, characteristic, complemented
5#67676C211yesmaximal, characteristic, complemented
6#10320—31nomaximal, Sylow
7C23×C4×D3C2211yescharacteristic
8C2×C3×#121C2212yes—
9C2×#938—22no—
10C2×#937—22no—
11C3×#480—22no—
12C3×#466—22no—
13C3×#447—22no—
14C2×C3×C22⋊C8C2211yescharacteristic
15#5220—22no—
16#5212C2212yes—
17C2×#830C2211yescharacteristic
18C2×(C2×D3)⋊C8—22no—
19#4987—22no—
20#4956—24no—
21#4537—24no—
22#3186D311yescharacteristic, complemented
23#2847—32no—
24#1779—31no—
25#1519—32no—
26#1289—31no—
27C24×D3D411yescharacteristic
28C22×C3×D4—22no—
29C22×C3×D4D412yescomplemented
30C23×C3×C4C2311yescharacteristic
31C22×#67—24no—
32C23×Dic3D411yescharacteristic
33C22×C4×D3—22no—
34C22×C4×D3—21no—
35C22×C4×D3—22no—
36C22×C4×D3—21no—
37C22×C4×D3—44no—
38C2×C3×D8—22no—
39C22×C3×C8D412yescomplemented
40C3×#121—22no—
41C3×#121—22no—
42C3×#121—22no—
43C3×#121—22no—
44C2×C3×C4⋊C4D412yescomplemented
45C2×C3×C22⋊C4—22no—
46C2×#217—24no—
47C2×#217—22no—
48#938—48no—
49#937—44no—
50#937—44no—
51C2×(C2×D3)⋊C4—24no—
52C2×C4⋊Dic3—22no—
53C22×C3⋊C8—21no—
54C3×#107—22no—
55C3×#102—24no—
56C3×C22⋊C8—22no—
57C3⋊#95—24no—
58C3⋊#95D414yescomplemented
59#830—22no—
60(C2×D3)⋊C8—44no—
61C24×C4—31no—
62C2×#121—62no—
63C2×#121C2×D312yes—
64C2×#120—62no—
65#480—22no—
66#466—22no—
67#447—22no—
68C2×C22⋊C8—31no—
69C2×C22⋊C8—62no—
70C2×C22⋊C8C2×D311yescharacteristic
71#386—64no—
72#386—62no—
73#380—32no—
74#380—62no—
75#327—64no—
76C24×C3—22no—
77C24×C3C2×D411yescharacteristic
78C23×D3—22no—
79C23×D3—21no—
80C23×D3—21no—
81C23×D3—22no—
82C23×D3—42no—
83C23×D3—42no—
84C23×D3D812yescomplemented
85C2×C3×D4—22no—
86C2×C3×D4—22no—
87C2×C3×D4—44no—
88C2×C3×D4—42no—
89C2×C3×D4—42no—
90C2×C3×D4—42no—
91C2×C3×D4—42no—
92C2×C3×D4—42no—
93C2×C3×D4—42no—
94C22×C3×C4—22no—
95C22×C3×C4—42no—
96C22×C3×C4C2×D412yes—
97C22×C3×C4C2×D412yes—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4#4111yescharacteristic
100C2×#67—48no—
101C2×#67—48no—
102C2×#67—48no—
103C22×Dic3—22no—
104C22×Dic3—21no—
105C22×Dic3—24no—
106C22×Dic3—42no—
107C2×C4×D3—24no—
108C2×C4×D3—28no—
109C2×C4×D3—44no—
110C2×C4×D3—42no—
111C2×C4×D3—42no—
112C2×C4×D3—44no—
113C2×C4×D3—44no—
114C2×C4×D3—42no—
115C2×C4×D3—41no—
116C2×C4×D3—42no—
117C2×C4×D3—41no—
118C3×D8—82no—
119C2×C3×C8—22no—
120C2×C3×C8—42no—
121C3×C4⋊C4—22no—
122C3×C4⋊C4—22no—
123C3×C22⋊C4—42no—
124C3×C22⋊C4—42no—
125#217—48no—
126#217—42no—
127#217—42no—
128(C2×D3)⋊C4—48no—
129C4⋊Dic3—44no—
130C2×C3⋊C8—42no—
131C2×C3⋊C8—41no—
132C25—31no—
133C22×D4—22no—
134C22×D4—64no—
135C22×D4#6712yescomplemented
136C23×C4—31no—
137C23×C4—62no—
138C23×C4—61no—
139C23×C4—62no—
140C23×C4—61no—
141C23×C4—124no—
142C23×C4C22×D311yescharacteristic
143C2×D8—22no—
144C22×C8—61no—
145C22×C8#6712yescomplemented
146#121—22no—
147#121—22no—
148#121—22no—
149#121—22no—
150#121—128no—
151#120—124no—
152#120—124no—
153C2×C4⋊C4—62no—
154C2×C4⋊C4#6712yescomplemented
155C2×C22⋊C4—22no—
156C2×C22⋊C4—64no—
157C2×C22⋊C4—64no—
158C2×C22⋊C4—62no—
159#107—22no—
160#102—24no—
161C22⋊C8—22no—
162C22⋊C8—62no—
163C22⋊C8—124no—
164#95—34no—
165#95—64no—
166C23×C3—41no—
167C23×C3—41no—
168C23×C3—44no—
169C23×C3—44no—
170C23×C3—42no—
171C23×C3—42no—
172C23×C3—42no—
173C23×C3#12012yes—
174C23×C3#12111yescharacteristic
175C23×C3C2×D811yescharacteristic
176C23×C3#13511yescharacteristic
177C23×C3AGL(1,ℤ/8)12yes—
178C22×D3—24no—
179C22×D3—22no—
180C22×D3—24no—
181C22×D3—22no—
182C22×D3—24no—
183C22×D3—22no—
184C22×D3—44no—
185C22×D3—48no—
186C22×D3—44no—
187C22×D3—42no—
188C22×D3—41no—
189C22×D3—42no—
190C22×D3—41no—
191C22×D3—41no—
192C22×D3—42no—
193C22×D3—41no—
194C3×D4—44no—
195C3×D4—82no—
196C3×D4—82no—
197C3×D4—82no—
198C3×D4—82no—
199C3×D4—82no—
200C2×C3×C4—22no—
201C2×C3×C4—24no—
202C2×C3×C4—42no—
203C2×C3×C4—41no—
204C2×C3×C4—42no—
205C2×C3×C4—42no—
206C2×C3×C4—42no—
207C2×C3×C4#12011yescharacteristic
208C2×C3×C4#12112yes—
209C2×C3×C4#12711yescharacteristic
210#67—816no—
211C2×Dic3—24no—
212C2×Dic3—22no—
213C2×Dic3—42no—
214C2×Dic3—41no—
215C2×Dic3—41no—
216C2×Dic3—44no—
217C2×Dic3—44no—
218C2×Dic3—44no—
219C4×D3—44no—
220C4×D3—48no—
221C4×D3—44no—
222C3×C8—42no—
223C3⋊C8—81no—
224C24—22no—
225C24—32no—
226C24—62no—
227C24—61no—
228C24—62no—
229C24—61no—
230C24—122no—
231C24—122no—
232C24D3×D411yescharacteristic
233C2×D4—22no—
234C2×D4—22no—
235C2×D4—44no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—42no—
239C2×D4—42no—
240C2×D4—42no—
241C2×D4—42no—
242C2×D4—128no—
243C2×D4—128no—
244C2×D4—128no—
245C22×C4—22no—
246C22×C4—42no—
247C22×C4—64no—
248C22×C4—68no—
249C22×C4—62no—
250C22×C4—61no—
251C22×C4—64no—
252C22×C4—124no—
253C22×C4—122no—
254C22×C4—124no—
255C22×C4—122no—
256C22×C4—124no—
257C22×C4—122no—
258C22×C4—122no—
259C22×C4—122no—
260C22×C4—121no—
261C22×C4—121no—
262C22×C4D3×D412yes—
263C22×C4#23711yescharacteristic
264C22×C4C2×#6712yes—
265C22×C4C2×#6711yescharacteristic
266D8—82no—
267C2×C8—22no—
268C2×C8—42no—
269C2×C8—122no—
270C2×C8—121no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—124no—
274C22⋊C4—42no—
275C22⋊C4—42no—
276C22⋊C4—128no—
277C22⋊C4—128no—
278C22⋊C4—122no—
279C22⋊C4—122no—
280C22×C3—22no—
281C22×C3—21no—
282C22×C3—22no—
283C22×C3—21no—
284C22×C3—41no—
285C22×C3—41no—
286C22×C3—42no—
287C22×C3—84no—
288C22×C3—84no—
289C22×C3—82no—
290C22×C3—82no—
291C22×C3—82no—
292C22×C3#39211yescharacteristic
293C22×C3#44712yes—
294C22×C3#44912yes—
295C22×C3#46611yescharacteristic
296C22×C3#46911yescharacteristic
297C2×D3—28no—
298C2×D3—24no—
299C2×D3—44no—
300C2×D3—42no—
301C2×D3—44no—
302C2×D3—44no—
303C2×D3—42no—
304C2×D3—42no—
305C2×D3—42no—
306C2×D3—42no—
307C2×D3—42no—
308C3×C4—21no—
309C3×C4—22no—
310C3×C4—82no—
311C3×C4B(4,𝔽2)12yes—
312Dic3—42no—
313Dic3—84no—
314C23—41no—
315C23—41no—
316C23—44no—
317C23—44no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—64no—
322C23—62no—
323C23—64no—
324C23—62no—
325C23—64no—
326C23—62no—
327C23—122no—
328C23—121no—
329C23—122no—
330C23—121no—
331C23—124no—
332C23—128no—
333C23—124no—
334C23—122no—
335C23—121no—
336C23—121no—
337C23D3×D811yescharacteristic
338C23#91112yes—
339C23#91211yescharacteristic
340C23#93712yes—
341C23#93811yescharacteristic
342D4—44no—
343D4—82no—
344D4—82no—
345D4—82no—
346D4—82no—
347D4—82no—
348D4—2416no—
349C2×C4—24no—
350C2×C4—22no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—42no—
354C2×C4—42no—
355C2×C4—42no—
356C2×C4—64no—
357C2×C4—62no—
358C2×C4—124no—
359C2×C4—128no—
360C2×C4—122no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—124no—
364C2×C4—124no—
365C2×C4—124no—
366C2×C4—124no—
367C2×C4#93912yes—
368C2×C4#94011yescharacteristic
369C2×C4#95311yescharacteristic
370C8—42no—
371C8—241no—
372C2×C3—22no—
373C2×C3—21no—
374C2×C3—22no—
375C2×C3—21no—
376C2×C3—162no—
377C2×C3#199911yescharacteristic
378C2×C3#201011yescharacteristic
379C2×C3#202111yescharacteristic
380D3—44no—
381C22—22no—
382C22—21no—
383C22—22no—
384C22—21no—
385C22—41no—
386C22—42no—
387C22—41no—
388C22—68no—
389C22—64no—
390C22—84no—
391C22—84no—
392C22—82no—
393C22—82no—
394C22—82no—
395C22—124no—
396C22—122no—
397C22—124no—
398C22—124no—
399C22—122no—
400C22—122no—
401C22—122no—
402C22—122no—
403C22—122no—
404C22#515612yes—
405C22#515711yescharacteristic
406C22#515812yes—
407C22#515911yescharacteristic
408C22#522311yescharacteristic
409C4—21no—
410C4—22no—
411C4—82no—
412C4—122no—
413C4—244no—
414C4#524412yes—
415C3#1032011yescharacteristic, complemented, Sylow
416C2—22no—
417C2—21no—
418C2—22no—
419C2—21no—
420C2—124no—
421C2—162no—
422C2#7185111yescharacteristic
423C2#7204311yescharacteristic
424C2#7211811yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#3186C2
socleC22×C3#392
hypercenter#3186D3
hypocenterC3#10320
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3186

Complements

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

NH ≅ G/Ntype
1. C3×#3186C2semidirect
2. #82723C2semidirect
3. #71675C2semidirect
4. #69955C2semidirect
5. #67676C2semidirect
22. #3186D3semidirect
29. C22×C3×D4D4semidirect
39. C22×C3×C8D4semidirect
44. C2×C3×C4⋊C4D4semidirect
58. C3⋊#95D4semidirect
84. C23×D3D8semidirect
135. C22×D4#67semidirect
145. C22×C8#67semidirect
154. C2×C4⋊C4#67semidirect
415. C3#10320semidirect

Maximal subgroups

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

subgroupindex
C3×#31862
#827232
#716752
#699552
#676762
#103203