Finite Group Databasebeta

← #10695629

Subgroups — #10695629

Order 768.

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

normal subgroups only

This group has 786 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
1#84065C211yesmaximal, characteristic, complemented
2C2×#5227C211yesmaximal, characteristic
3#78612C211yesmaximal, characteristic
4C3×#1787C211yesmaximal, characteristic, complemented
5#69528C211yesmaximal, characteristic
6#69510C211yesmaximal, characteristic, complemented
7#67677C211yesmaximal, characteristic, complemented
8#9949—31nomaximal, Sylow
9C23×C3×Q8C2211yescharacteristic
10C2×#944—21no—
11C22×#215—21no—
12C2×#930C2211yescharacteristic
13C2×#924C2211yescharacteristic
14C2×C3×#103—22no—
15C2×C3×C22⋊C8C2211yescharacteristic
16C3×#380C2211yescharacteristic
17#5230—22no—
18#5229—21no—
19#5227—22no—
20#5227—22no—
21C2×#835—21no—
22#5210—22no—
23C4×#217C2211yescharacteristic
24C2×#830C2211yescharacteristic
25#5114—22no—
26C2×#806—22no—
27#4738—21no—
28#4735—21no—
29#4731—21no—
30#4721—21no—
31#4452—22no—
32#3187—31no—
33C2×#448—31no—
34#2434—31no—
35#1964—31no—
36#1787D311yescharacteristic, complemented
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—
46C23×C3×C4—21no—
47C23×C3×C4C2311yescharacteristic
48C22×#67—21no—
49C22×D12D411yescharacteristic
50C22×Dic6D411yescharacteristic
51C22×C3×C8D412yes—
52C2×C3×C4⋊C4—21no—
53C2×C3×C22⋊C4—21no—
54C2×#217—21no—
55C2×#217D412yes—
56#944—44no—
57C2×#215—24no—
58C2×#215—44no—
59#930—21no—
60#930—21no—
61#930—21no—
62#930—21no—
63C2×(C2×D3)⋊C4—21no—
64C2×C4⋊Dic3D411yescharacteristic
65C2×C4⋊Dic3D411yescharacteristic
66#924—21no—
67#924—21no—
68#924—21no—
69#924—21no—
70C2×Dic3⋊C4—24no—
71C2×Dic3⋊C4—21no—
72C2×C4×Dic3—21no—
73C22×C3⋊C8—21no—
74C2×#204—22no—
75C3×#103—44no—
76C3×C22⋊C8—21no—
77C3×C22⋊C8—21no—
78C3×#95—21no—
79#835—24no—
80C3⋊#95—24no—
81C3⋊#95—21no—
82#830—21no—
83#830D412yes—
84#821—22no—
85#817—22no—
86#816—22no—
87#806—44no—
88C23×Q8C2×D311yescharacteristic
89C22×SD16—61no—
90C2×C4⋊Q8—61no—
91C2×C22⋊Q8—31no—
92C2×#121—31no—
93#483—61no—
94#481—61no—
95#465—62no—
96#450—61no—
97#448—61no—
98#448—62no—
99#448—62no—
100C2×#103—22no—
101C2×#103—61no—
102C2×#99—62no—
103C2×C22⋊C8—31no—
104C2×C22⋊C8C2×D311yescharacteristic
105#395—62no—
106#391—61no—
107#385—62no—
108#380C2×D311yescharacteristic
109C4×C22⋊C4—31no—
110#328—62no—
111C24×C3C2×D411yescharacteristic
112C23×D3—21no—
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—
126C22×C3×C4—22no—
127C22×C3×C4—42no—
128C22×C3×C4—44no—
129C22×C3×C4—41no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C2×#67—41no—
139C2×#67—41no—
140C2×#67—42no—
141C2×#67—42no—
142C22×Dic3—22no—
143C22×Dic3—21no—
144C22×Dic3—21no—
145C22×Dic3—21no—
146C2×D12—21no—
147C2×D12—21no—
148C2×D12—42no—
149C2×Dic6—21no—
150C2×Dic6—21no—
151C2×Dic6—42no—
152C2×C3×C8—22no—
153C2×C3×C8—42no—
154C3×C4⋊C4—42no—
155C3×C22⋊C4—42no—
156#217—22no—
157#217—41no—
158#217—41no—
159#217SD1614yescomplemented
160#215—48no—
161(C2×D3)⋊C4—41no—
162(C2×D3)⋊C4—41no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167Dic3⋊C4—48no—
168Dic3⋊C4—41no—
169Dic3⋊C4—41no—
170C4×Dic3—42no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—42no—
173#204—82no—
174C22×Q8—21no—
175C22×Q8—21no—
176C22×Q8—21no—
177C22×Q8—21no—
178C22×Q8—21no—
179C22×Q8—31no—
180C22×Q8—44no—
181C22×Q8D1212yes—
182C22×D4—31no—
183C22×D4—61no—
184C23×C4—21no—
185C23×C4C22×D311yescharacteristic
186C2×SD16—62no—
187C2×SD16—64no—
188C2×SD16—124no—
189C22×C8—61no—
190C22×C8#6712yes—
191C4⋊Q8—124no—
192C22⋊Q8—61no—
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195C22⋊Q8—61no—
196#121—61no—
197#121—61no—
198#121—61no—
199#121—61no—
200C2×C4⋊C4—21no—
201C2×C4⋊C4—31no—
202C2×C4⋊C4—31no—
203C2×C4⋊C4—64no—
204C2×C4⋊C4—61no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—32no—
207C2×C22⋊C4—61no—
208C2×C22⋊C4—61no—
209C2×C42—61no—
210#106—62no—
211#103—44no—
212#103—62no—
213#103—64no—
214#102—62no—
215#99—124no—
216C22⋊C8—21no—
217C22⋊C8—21no—
218C22⋊C8—32no—
219C22⋊C8—61no—
220#95—21no—
221#95—64no—
222#95—61no—
223C23×C3—41no—
224C23×C3—41no—
225C23×C3#12011yescharacteristic
226C23×C3#12111yescharacteristic
227C23×C3#12711yescharacteristic
228C23×C3C2×SD1612yes—
229C23×C3#13712yes—
230C22×D3—41no—
231C22×D3—42no—
232C22×D3—81no—
233C22×D3—81no—
234C3×Q8—24no—
235C3×Q8—24no—
236C3×Q8—42no—
237C3×Q8—44no—
238C3×Q8—42no—
239C3×Q8—44no—
240C2×C3×C4—22no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—24no—
244C2×C3×C4—41no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—44no—
249C2×C3×C4—44no—
250C2×C3×C4—42no—
251C2×C3×C4—81no—
252C2×C3×C4—81no—
253C2×C3×C4—81no—
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12111yescharacteristic
258#67—82no—
259#67—81no—
260#67—81no—
261C2×Dic3—24no—
262C2×Dic3—42no—
263C2×Dic3—42no—
264C2×Dic3—42no—
265C2×Dic3—42no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269C2×Dic3—42no—
270D12—42no—
271D12—42no—
272Dic6—42no—
273Dic6—81no—
274C3×C8—42no—
275C3⋊C8—42no—
276C24—61no—
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—61no—
295C2×D4—61no—
296C2×D4—122no—
297C2×D4—121no—
298C2×D4—122no—
299C2×D4—122no—
300C2×D4—121no—
301C22×C4—22no—
302C22×C4—42no—
303C22×C4—44no—
304C22×C4—41no—
305C22×C4—41no—
306C22×C4—41no—
307C22×C4—62no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4C2×D1211yescharacteristic
312C22×C4#23511yescharacteristic
313C22×C4D3×D411yescharacteristic
314C22×C4D3×D411yescharacteristic
315C22×C4D3×D411yescharacteristic
316C22×C4C2×#6711yescharacteristic
317SD16—128no—
318SD16—242no—
319C2×C8—22no—
320C2×C8—42no—
321C2×C8—62no—
322C2×C8—122no—
323C4⋊C4—42no—
324C4⋊C4—61no—
325C4⋊C4—61no—
326C4⋊C4—61no—
327C4⋊C4—61no—
328C4⋊C4—128no—
329C4⋊C4—121no—
330C4⋊C4—121no—
331C22⋊C4—34no—
332C22⋊C4—42no—
333C22⋊C4—62no—
334C22⋊C4—121no—
335C22⋊C4—121no—
336C22⋊C4—121no—
337C22⋊C4—121no—
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#39211yescharacteristic
348C22×C3#44812yes—
349C22×C3#46112yes—
350C22×C3#49211yescharacteristic
351C22×C3#49711yescharacteristic
352C2×D3—81no—
353C2×D3—81no—
354C2×D3—81no—
355C2×D3—162no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—21no—
359C3×C4—44no—
360C3×C4—161no—
361C3×C4B(4,𝔽2)12yes—
362Dic3—44no—
363Dic3—81no—
364Dic3—81no—
365Dic3—81no—
366C23—41no—
367C23—41no—
368C23—121no—
369C23—122no—
370C23—241no—
371C23—241no—
372C23#88211yescharacteristic
373C23#88411yescharacteristic
374C23D3×SD1612yes—
375C23#91812yes—
376C23#94011yescharacteristic
377Q8—24no—
378Q8—24no—
379Q8—42no—
380Q8—44no—
381Q8—42no—
382Q8—44no—
383Q8—122no—
384Q8—241no—
385D4—122no—
386D4—122no—
387D4—241no—
388D4—241no—
389D4—242no—
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—64no—
402C2×C4—81no—
403C2×C4—81no—
404C2×C4—81no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—122no—
408C2×C4—122no—
409C2×C4—121no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—122no—
413C2×C4#88211yescharacteristic
414C2×C4#88411yescharacteristic
415C2×C4#93011yescharacteristic
416C2×C4#93711yescharacteristic
417C8—42no—
418C8—122no—
419C2×C3—22no—
420C2×C3—21no—
421C2×C3—22no—
422C2×C3—21no—
423C2×C3#200911yescharacteristic
424C2×C3#201311yescharacteristic
425C2×C3#202111yescharacteristic
426D3—161no—
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—42no—
434C22—42no—
435C22—241no—
436C22—241no—
437C22—241no—
438C22—482no—
439C22#480612yes—
440C22#481812yes—
441C22#496011yescharacteristic
442C22#517511yescharacteristic
443C22#519011yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—44no—
448C4—124no—
449C4—161no—
450C4—241no—
451C4—241no—
452C4—241no—
453C4#503212yes—
454C3#994911yescharacteristic, complemented, Sylow
455C2—22no—
456C2—21no—
457C2—22no—
458C2—21no—
459C2—481no—
460C2#6950011yescharacteristic
461C2#7028211yescharacteristic
462C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC23×C3×C4C23
FrattiniC23×C4C22×D3
FittingC3×#1787C2
socleC22×C3#392
prime radicalC2×#930C22
hypercenter#1787D3
hypocenterC3#9949
LCSG ▷ C23×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
1. #84065C2semidirect
4. C3×#1787C2semidirect
6. #69510C2semidirect
7. #67677C2semidirect
36. #1787D3semidirect
159. #217SD16semidirect
454. C3#9949semidirect

Maximal subgroups

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

subgroupindex
#840652
C2×#52272
#786122
C3×#17872
#695282
#695102
#676772
#99493