Finite Group Databasebeta

← #10695625

Subgroups — #10695625

Order 768.

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

normal subgroups only

This group has 786 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×#448C211yesmaximal, characteristic
2#84065C211yesmaximal, characteristic, complemented
3#82630C211yesmaximal, characteristic
4#72132C211yesmaximal, characteristic, complemented
5#71686C211yesmaximal, characteristic
6#69946C211yesmaximal, characteristic, complemented
7#67678C211yesmaximal, characteristic, complemented
8#9949—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C22×C3×SD16—21no—
11C2×C3×#121C2211yescharacteristic
12C2×#944—21no—
13C2×#924C2211yescharacteristic
14C3×#448—22no—
15C3×#448—22no—
16C2×C3×#103—21no—
17C2×C3×C22⋊C8C2211yescharacteristic
18#5230—22no—
19#5229—21no—
20C2×#835—22no—
21C2×#834—22no—
22#5218—21no—
23#5212C2211yescharacteristic
24#5210—22no—
25C4×#217C2211yescharacteristic
26C2×#830C2211yescharacteristic
27#5041—22no—
28#5040—21no—
29#5034—21no—
30#4971—21no—
31#4467—22no—
32#3187—31no—
33C2×#448D311yescharacteristic
34#2434—31no—
35#1964—31no—
36#1787—31no—
37#1520—31no—
38#1282—31no—
39C22×C3×Q8—21no—
40C22×C3×Q8—21no—
41C22×C3×Q8—21no—
42C22×C3×Q8—21no—
43C22×C3×Q8—21no—
44C22×C3×Q8—44no—
45C22×C3×Q8D412yes—
46C22×C3×D4—21no—
47C22×C3×D4D411yescharacteristic
48C23×C3×C4—21no—
49C23×C3×C4C2311yescharacteristic
50C22×Dic6D411yescharacteristic
51C2×C3×SD16—24no—
52C2×C3×SD16—44no—
53C22×C3×C8—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×#121—21no—
57C3×#121—21no—
58C2×C3×C4⋊C4D411yescharacteristic
59C2×C3×C22⋊C4—21no—
60C2×#217—21no—
61C2×#217—21no—
62C2×#217D412yes—
63#944—44no—
64C2×#214—22no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3D411yescharacteristic
67#924—21no—
68#924—21no—
69#924—21no—
70#924—21no—
71C2×Dic3⋊C4—24no—
72C2×Dic3⋊C4—21no—
73C2×C4×Dic3—21no—
74C22×C3⋊C8D412yes—
75C3×#103—24no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8D412yes—
78#835—44no—
79#834—44no—
80#832—22no—
81C3⋊#95—24no—
82C3⋊#95—21no—
83C3⋊#95—21no—
84#830—21no—
85#830—21no—
86#810—22no—
87#808—22no—
88C23×Q8C2×D311yescharacteristic
89C22×SD16—21no—
90C2×C4⋊Q8—61no—
91C2×C22⋊Q8—31no—
92C2×#121C2×D311yescharacteristic
93#483—61no—
94#481—61no—
95#465—62no—
96#450—61no—
97#448—22no—
98#448—22no—
99#448—61no—
100C2×#103—21no—
101C2×#103—62no—
102C2×#99—62no—
103C2×C22⋊C8—31no—
104C2×C22⋊C8C2×D311yescharacteristic
105#395—62no—
106#391—61no—
107#385—62no—
108#380—31no—
109C4×C22⋊C4—31no—
110#328—62no—
111C24×C3—21no—
112C24×C3C2×D411yescharacteristic
113C2×C3×Q8—24no—
114C2×C3×Q8—24no—
115C2×C3×Q8—24no—
116C2×C3×Q8—22no—
117C2×C3×Q8—22no—
118C2×C3×Q8—44no—
119C2×C3×Q8—42no—
120C2×C3×Q8—42no—
121C2×C3×Q8—42no—
122C2×C3×Q8—44no—
123C2×C3×Q8—44no—
124C2×C3×Q8—41no—
125C2×C3×Q8—41no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—42no—
129C2×C3×D4—41no—
130C2×C3×D4—41no—
131C2×C3×D4—42no—
132C2×C3×D4—42no—
133C22×C3×C4—22no—
134C22×C3×C4—21no—
135C22×C3×C4—42no—
136C22×C3×C4—44no—
137C22×C3×C4—41no—
138C22×C3×C4—41no—
139C22×C3×C4C2×D411yescharacteristic
140C22×C3×C4C2×D411yescharacteristic
141C22×C3×C4C2×D411yescharacteristic
142C22×C3×C4C2×D411yescharacteristic
143C22×C3×C4C2×D411yescharacteristic
144C22×C3×C4#4111yescharacteristic
145C22×Dic3—22no—
146C22×Dic3—21no—
147C22×Dic3—21no—
148C22×Dic3—41no—
149C2×Dic6—21no—
150C2×Dic6—21no—
151C2×Dic6—42no—
152C3×SD16—48no—
153C2×C3×C8—22no—
154C2×C3×C8—42no—
155C3×C4⋊C4—21no—
156C3×C4⋊C4—21no—
157C3×C22⋊C4—41no—
158C3×C22⋊C4—41no—
159#217—22no—
160#217—41no—
161#217—41no—
162#217—42no—
163#217SD1614yescomplemented
164#214—82no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167C4⋊Dic3—42no—
168Dic3⋊C4—48no—
169Dic3⋊C4—41no—
170Dic3⋊C4—41no—
171C4×Dic3—42no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—42no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—31no—
180C22×Q8—44no—
181C22×Q8#6712yes—
182C22×D4—21no—
183C22×D4#6711yescharacteristic
184C23×C4—21no—
185C23×C4C22×D311yescharacteristic
186C2×SD16—24no—
187C2×SD16—44no—
188C2×SD16—62no—
189C22×C8—21no—
190C22×C8—32no—
191C4⋊Q8—124no—
192C22⋊Q8—61no—
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196#121—21no—
197#121—21no—
198#121—21no—
199#121—21no—
200C2×C4⋊C4—31no—
201C2×C4⋊C4—64no—
202C2×C4⋊C4—61no—
203C2×C4⋊C4—61no—
204C2×C4⋊C4#6711yescharacteristic
205C2×C22⋊C4—21no—
206C2×C22⋊C4—32no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C42—61no—
210#106—62no—
211#103—24no—
212#103—62no—
213#103—124no—
214#102—62no—
215#99—124no—
216C22⋊C8—21no—
217C22⋊C8—61no—
218C22⋊C8—61no—
219C22⋊C8#6712yes—
220#95—64no—
221#95—61no—
222#95—61no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3—41no—
226C23×C3—42no—
227C23×C3—81no—
228C23×C3—81no—
229C23×C3#12011yescharacteristic
230C23×C3#12111yescharacteristic
231C23×C3#12711yescharacteristic
232C23×C3C2×SD1612yes—
233C23×C3#13712yes—
234C3×Q8—24no—
235C3×Q8—24no—
236C3×Q8—42no—
237C3×Q8—44no—
238C3×Q8—42no—
239C3×Q8—44no—
240C3×D4—42no—
241C3×D4—42no—
242C3×D4—82no—
243C3×D4—81no—
244C3×D4—81no—
245C2×C3×C4—22no—
246C2×C3×C4—22no—
247C2×C3×C4—22no—
248C2×C3×C4—24no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—41no—
252C2×C3×C4—42no—
253C2×C3×C4—44no—
254C2×C3×C4—44no—
255C2×C3×C4—42no—
256C2×C3×C4—41no—
257C2×C3×C4—42no—
258C2×C3×C4#12011yescharacteristic
259C2×C3×C4#12011yescharacteristic
260C2×C3×C4#12111yescharacteristic
261C2×C3×C4#12111yescharacteristic
262C2×Dic3—24no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—42no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—81no—
270C2×Dic3—81no—
271C2×Dic3—81no—
272Dic6—42no—
273Dic6—81no—
274C3×C8—42no—
275C3⋊C8—42no—
276C24—21no—
277C24D3×D411yescharacteristic
278C2×Q8—24no—
279C2×Q8—24no—
280C2×Q8—24no—
281C2×Q8—22no—
282C2×Q8—22no—
283C2×Q8—44no—
284C2×Q8—42no—
285C2×Q8—42no—
286C2×Q8—42no—
287C2×Q8—44no—
288C2×Q8—44no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—61no—
292C2×Q8—61no—
293C2×Q8—122no—
294C2×D4—21no—
295C2×D4—21no—
296C2×D4—42no—
297C2×D4—42no—
298C2×D4—42no—
299C2×D4—41no—
300C2×D4—41no—
301C22×C4—22no—
302C22×C4—21no—
303C22×C4—42no—
304C22×C4—44no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—62no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—121no—
311C22×C4D3×D411yescharacteristic
312C22×C4D3×D411yescharacteristic
313C22×C4#23711yescharacteristic
314C22×C4C2×#6711yescharacteristic
315C22×C4C2×#6711yescharacteristic
316C22×C4C2×#6711yescharacteristic
317SD16—48no—
318SD16—242no—
319C2×C8—22no—
320C2×C8—42no—
321C2×C8—62no—
322C2×C8—122no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—128no—
328C4⋊C4—122no—
329C4⋊C4—121no—
330C4⋊C4—121no—
331C22⋊C4—34no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—62no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C22⋊C4—122no—
338C42—122no—
339C22×C3—21no—
340C22×C3—21no—
341C22×C3—21no—
342C22×C3—21no—
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—42no—
346C22×C3—42no—
347C22×C3—81no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3—162no—
351C22×C3#39211yescharacteristic
352C22×C3#44812yes—
353C22×C3#46112yes—
354C22×C3#49211yescharacteristic
355C22×C3#49711yescharacteristic
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—44no—
360C3×C4—81no—
361C3×C4B(4,𝔽2)12yes—
362Dic3—44no—
363Dic3—81no—
364Dic3—81no—
365Dic3—161no—
366C23—41no—
367C23—41no—
368C23—41no—
369C23—42no—
370C23—81no—
371C23—81no—
372C23D3×SD1612yes—
373C23#91512yes—
374C23#93711yescharacteristic
375C23#93911yescharacteristic
376C23#94011yescharacteristic
377Q8—24no—
378Q8—24no—
379Q8—42no—
380Q8—44no—
381Q8—42no—
382Q8—44no—
383Q8—122no—
384Q8—241no—
385D4—42no—
386D4—42no—
387D4—81no—
388D4—82no—
389D4—81no—
390C2×C4—22no—
391C2×C4—22no—
392C2×C4—22no—
393C2×C4—24no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—44no—
399C2×C4—44no—
400C2×C4—42no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—64no—
404C2×C4—122no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—122no—
408C2×C4—121no—
409C2×C4—121no—
410C2×C4—241no—
411C2×C4—241no—
412C2×C4—241no—
413C2×C4#93711yescharacteristic
414C2×C4#93811yescharacteristic
415C2×C4#93911yescharacteristic
416C2×C4#95311yescharacteristic
417C8—42no—
418C8—122no—
419C2×C3—22no—
420C2×C3—21no—
421C2×C3—22no—
422C2×C3—21no—
423C2×C3—161no—
424C2×C3#200911yescharacteristic
425C2×C3#201311yescharacteristic
426C2×C3#202111yescharacteristic
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—42no—
434C22—42no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—162no—
439C22#516212yes—
440C22#516711yescharacteristic
441C22#517012yes—
442C22#517511yescharacteristic
443C22#522311yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—44no—
448C4—81no—
449C4—124no—
450C4—241no—
451C4—241no—
452C4—481no—
453C4#503212yes—
454C3#994911yescharacteristic, complemented, Sylow
455C2—22no—
456C2—21no—
457C2—22no—
458C2—21no—
459C2—161no—
460C2#7167111yescharacteristic
461C2#7205711yescharacteristic
462C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC2×C3×#448C2
socleC22×C3#392
prime radicalC2×C3×#121C22
hypercenterC2×#448D3
hypocenterC3#9949
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#448

Complements

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

NH ≅ G/Ntype
2. #84065C2semidirect
4. #72132C2semidirect
6. #69946C2semidirect
7. #67678C2semidirect
163. #217SD16semidirect
454. C3#9949semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4482
#840652
#826302
#721322
#716862
#699462
#676782
#99493