Finite Group Databasebeta

← #10695639

Subgroups — #10695639

Order 768.

610 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 105 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×C3×#481C211yesmaximal, characteristic
2#82630C211yesmaximal, characteristic
3#82178C211yesmaximal, characteristic, complemented
4#71686C211yesmaximal, characteristic
5#71554C211yesmaximal, characteristic, complemented
6#69946C211yesmaximal, characteristic, complemented
7#67678C211yesmaximal, characteristic, complemented
8#9951—31nomaximal, Sylow
9C2×C3×#121C2211yescharacteristic
10C22×C3×C4⋊C4C2211yescharacteristic
11C2×#924C2211yescharacteristic
12C3×#481—22no—
13C3×#481—22no—
14C2×C3×#106—21no—
15C2×C3×#102—21no—
16C2×C3×C22⋊C8C2211yescharacteristic
17C2×#834—22no—
18#5218—21no—
19#5212C2211yescharacteristic
20#5210—22no—
21C4×#217C2211yescharacteristic
22C2×#830C2211yescharacteristic
23#5041—22no—
24#5040—21no—
25#5034—21no—
26#4983—22no—
27#4975—21no—
28#4973—21no—
29#4971—21no—
30#4529—22no—
31#4467—22no—
32#3187—31no—
33C2×#481D311yescharacteristic
34#2502—31no—
35#1964—31no—
36#1824—31no—
37#1520—31no—
38#1282—31no—
39C22×C3×D4—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C23×C3×C4C2311yescharacteristic
43C22×Dic6D411yescharacteristic
44C22×C3×C8—21no—
45C3×#121—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×#121—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4D412yes—
56C2×C3×C22⋊C4—21no—
57C2×#217—21no—
58C2×#217—21no—
59C2×#217D412yes—
60C2×#214—22no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3D411yescharacteristic
63#924—21no—
64#924—21no—
65#924—21no—
66#924—21no—
67C2×Dic3⋊C4—21no—
68C2×Dic3⋊C4—24no—
69C2×C4×Dic3—21no—
70C22×C3⋊C8D412yes—
71C3×#106—24no—
72C3×#102—24no—
73C3×C22⋊C8—21no—
74C3×C22⋊C8D412yes—
75#834—44no—
76#832—22no—
77C3⋊#95—24no—
78C3⋊#95—21no—
79C3⋊#95—21no—
80#830—21no—
81#830—21no—
82#810—22no—
83#808—22no—
84C2×C22⋊Q8—31no—
85C2×#121C2×D311yescharacteristic
86C22×C4⋊C4C2×D311yescharacteristic
87#483—61no—
88#481—22no—
89#481—22no—
90#481—61no—
91#465—62no—
92#450—61no—
93#448—61no—
94C2×#106—21no—
95C2×#102—21no—
96C2×#99—62no—
97C2×C22⋊C8—31no—
98C2×C22⋊C8C2×D311yescharacteristic
99#398—62no—
100#391—61no—
101#385—62no—
102#384—61no—
103#380—31no—
104C4×C22⋊C4—31no—
105#340—62no—
106#328—62no—
107C24×C3—21no—
108C24×C3C2×D411yescharacteristic
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—42no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—42no—
115C2×C3×D4—42no—
116C22×C3×C4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—42no—
119C22×C3×C4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—82no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C22×Dic3—22no—
129C22×Dic3—21no—
130C22×Dic3—21no—
131C22×Dic3—41no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×Dic6—42no—
135C2×C3×C8—22no—
136C2×C3×C8—42no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—41no—
142C3×C4⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C22⋊C4—41no—
145#217—22no—
146#217—41no—
147#217—41no—
148#217—42no—
149#217SD1614yescomplemented
150#214—82no—
151C4⋊Dic3—21no—
152C4⋊Dic3—21no—
153C4⋊Dic3—42no—
154Dic3⋊C4—41no—
155Dic3⋊C4—41no—
156Dic3⋊C4—84no—
157C4×Dic3—42no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—42no—
160C22×Q8—31no—
161C22×D4—21no—
162C22×D4#6711yescharacteristic
163C23×C4—21no—
164C23×C4C22×D311yescharacteristic
165C2×SD16—62no—
166C22×C8—21no—
167C22×C8—32no—
168C22⋊Q8—61no—
169C22⋊Q8—61no—
170C22⋊Q8—61no—
171C22⋊Q8—61no—
172#121—21no—
173#121—21no—
174#121—21no—
175#121—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—21no—
178C2×C4⋊C4—21no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4—31no—
182C2×C4⋊C4—61no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—64no—
185C2×C4⋊C4#6712yes—
186C2×C4⋊C4#6711yescharacteristic
187C2×C22⋊C4—21no—
188C2×C22⋊C4—32no—
189C2×C22⋊C4—61no—
190C2×C22⋊C4—61no—
191C2×C42—61no—
192#106—24no—
193#106—62no—
194#103—62no—
195#102—24no—
196#102—62no—
197#99—124no—
198C22⋊C8—21no—
199C22⋊C8—61no—
200C22⋊C8—61no—
201C22⋊C8#6712yes—
202#95—64no—
203#95—61no—
204#95—61no—
205C23×C3—41no—
206C23×C3—41no—
207C23×C3—41no—
208C23×C3—42no—
209C23×C3—81no—
210C23×C3—81no—
211C23×C3#12011yescharacteristic
212C23×C3#12111yescharacteristic
213C23×C3#12711yescharacteristic
214C23×C3C2×SD1612yes—
215C23×C3#13712yes—
216C3×D4—42no—
217C3×D4—42no—
218C3×D4—82no—
219C3×D4—81no—
220C3×D4—81no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—22no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—44no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—82no—
232C2×C3×C4—82no—
233C2×C3×C4#12011yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×Dic3—24no—
238C2×Dic3—42no—
239C2×Dic3—42no—
240C2×Dic3—42no—
241C2×Dic3—42no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—81no—
245C2×Dic3—81no—
246C2×Dic3—81no—
247Dic6—42no—
248Dic6—81no—
249C3×C8—42no—
250C3⋊C8—42no—
251C24—21no—
252C24D3×D411yescharacteristic
253C2×Q8—61no—
254C2×Q8—61no—
255C2×Q8—122no—
256C2×D4—21no—
257C2×D4—21no—
258C2×D4—42no—
259C2×D4—42no—
260C2×D4—42no—
261C2×D4—41no—
262C2×D4—41no—
263C22×C4—22no—
264C22×C4—21no—
265C22×C4—42no—
266C22×C4—41no—
267C22×C4—41no—
268C22×C4—62no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—82no—
272C22×C4—121no—
273C22×C4D3×D411yescharacteristic
274C22×C4D3×D411yescharacteristic
275C22×C4#23711yescharacteristic
276C22×C4C2×#6711yescharacteristic
277C22×C4C2×#6711yescharacteristic
278C22×C4C2×#6711yescharacteristic
279SD16—242no—
280C2×C8—22no—
281C2×C8—42no—
282C2×C8—62no—
283C2×C8—122no—
284C4⋊C4—21no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—21no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—122no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C4⋊C4—244no—
296C22⋊C4—34no—
297C22⋊C4—41no—
298C22⋊C4—41no—
299C22⋊C4—62no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C22⋊C4—122no—
303C42—122no—
304C22×C3—21no—
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—21no—
308C22×C3—21no—
309C22×C3—21no—
310C22×C3—42no—
311C22×C3—42no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3—162no—
316C22×C3#39211yescharacteristic
317C22×C3#44812yes—
318C22×C3#46112yes—
319C22×C3#49211yescharacteristic
320C22×C3#49711yescharacteristic
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—82no—
325C3×C4—81no—
326C3×C4#45812yes—
327Dic3—44no—
328Dic3—81no—
329Dic3—81no—
330Dic3—161no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—42no—
335C23—81no—
336C23—81no—
337C23D3×SD1612yes—
338C23#91512yes—
339C23#93711yescharacteristic
340C23#93911yescharacteristic
341C23#94011yescharacteristic
342Q8—122no—
343Q8—241no—
344D4—42no—
345D4—42no—
346D4—81no—
347D4—82no—
348D4—81no—
349C2×C4—22no—
350C2×C4—22no—
351C2×C4—22no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—44no—
356C2×C4—42no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—64no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—122no—
363C2×C4—122no—
364C2×C4—122no—
365C2×C4—122no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—241no—
369C2×C4—241no—
370C2×C4—241no—
371C2×C4#93711yescharacteristic
372C2×C4#93811yescharacteristic
373C2×C4#93911yescharacteristic
374C2×C4#95311yescharacteristic
375C8—42no—
376C8—122no—
377C2×C3—22no—
378C2×C3—21no—
379C2×C3—22no—
380C2×C3—21no—
381C2×C3—161no—
382C2×C3#200811yescharacteristic
383C2×C3#201311yescharacteristic
384C2×C3#202211yescharacteristic
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—42no—
392C22—42no—
393C22—81no—
394C22—81no—
395C22—81no—
396C22—162no—
397C22#516212yes—
398C22#516711yescharacteristic
399C22#517012yes—
400C22#517511yescharacteristic
401C22#522311yescharacteristic
402C4—21no—
403C4—21no—
404C4—21no—
405C4—81no—
406C4—82no—
407C4—124no—
408C4—241no—
409C4—241no—
410C4—481no—
411C4#505312yes—
412C3#995111yescharacteristic, complemented, Sylow
413C2—22no—
414C2—21no—
415C2—22no—
416C2—21no—
417C2—161no—
418C2#7170611yescharacteristic
419C2#7205711yescharacteristic
420C2#7209911yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#481C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenterC2×#481D3
hypocenterC3#9951
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#481

Complements

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

NH ≅ G/Ntype
3. #82178C2semidirect
5. #71554C2semidirect
6. #69946C2semidirect
7. #67678C2semidirect
149. #217SD16semidirect
412. C3#9951semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4812
#826302
#821782
#716862
#715542
#699462
#676782
#99513