Finite Group Databasebeta

← #10697635

Subgroups — #10697635

Order 768.

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

normal subgroups only

This group has 524 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
1C3×#3078C211yesmaximal, characteristic
2#83323C211yesmaximal, characteristic
3#82475C211yesmaximal, characteristic
4#71835C211yesmaximal, characteristic
5#71621C211yesmaximal, characteristic
6#70506C211yesmaximal, characteristic
7#67715C211yesmaximal, characteristic
8#10251—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C3×C4×Q8C2211yescharacteristic
11C2×#944—21no—
12C2×#869C2211yescharacteristic
13C3×#479—22no—
14C3×#477—21no—
15C3×#475—21no—
16C3×#428—21no—
17C3×#423C2211yescharacteristic
18C3×#419—21no—
19#5080—21no—
20#5079—21no—
21#5043—22no—
22#5020—22no—
23#4981—22no—
24#4973—21no—
25C4×C4⋊Dic3C2211yescharacteristic
26#4928—22no—
27#4925C2211yescharacteristic
28#4925C2211yescharacteristic
29#4538—22no—
30#4535—22no—
31#4482—22no—
32#3229—31no—
33#3078D311yescharacteristic
34#2546—31no—
35#1546—31no—
36#1503—31no—
37#1479—31no—
38#1303—31no—
39C22×C3×Q8D411yescharacteristic
40C22×Dic6—21no—
41C2×C3×M16—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×#117—21no—
54C2×C3×C4⋊C4—21no—
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C42—21no—
57C2×C3×C42C2311yescharacteristic
58#944—42no—
59#944—42no—
60C2×C4⋊Dic3—21no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3D412yes—
63C2×Dic3⋊C4—24no—
64C2×C4×Dic3—21no—
65#869—21no—
66#869—21no—
67#869—21no—
68#869—21no—
69C3×#107—22no—
70C3×#106—22no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8D412yes—
73C3×#103—22no—
74C3×#103—22no—
75#835—24no—
76C3⋊#95—24no—
77C3⋊#95D412yes—
78#830—21no—
79#830—21no—
80#808—24no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D412yes—
83C4×C3⋊C8—21no—
84C4×C3⋊C8—21no—
85#533C2×D311yescharacteristic
86C2×C4⋊Q8—31no—
87C2×C4⋊Q8—61no—
88C2×C4×Q8C2×D311yescharacteristic
89#498—61no—
90#487—61no—
91#483—62no—
92#479—22no—
93#477—21no—
94#475—21no—
95#475—62no—
96#431—31no—
97#431—31no—
98#428—21no—
99#423C2×D311yescharacteristic
100#419—21no—
101#398—62no—
102#391—61no—
103#389—62no—
104C4×C4⋊C4—31no—
105#332—62no—
106#328—62no—
107Q8⋊C8—62no—
108C2×C3×Q8—21no—
109C2×C3×Q8—21no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8SD1612yes—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×Dic3—22no—
122C22×Dic3—22no—
123C2×Dic6—41no—
124C2×Dic6—41no—
125C2×Dic6—41no—
126C2×Dic6—41no—
127C2×Dic6—41no—
128C2×Dic6—41no—
129C3×M16—41no—
130C3×M16—41no—
131C2×C3×C8—22no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—41no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42—41no—
144C3×C42—41no—
145C3×C42—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150C4⋊Dic3—24no—
151C4⋊Dic3—22no—
152C4⋊Dic3—22no—
153C4⋊Dic3—41no—
154C4⋊Dic3—41no—
155Dic3⋊C4—48no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—41no—
160C22×Q8—61no—
161C22×Q8#6711yescharacteristic
162C2×M16—21no—
163C4⋊Q8—61no—
164C4⋊Q8—61no—
165C4⋊Q8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—122no—
168C4⋊Q8—122no—
169C4⋊Q8#6711yescharacteristic
170C4⋊Q8#6711yescharacteristic
171#125—21no—
172C22⋊Q8—21no—
173C22⋊Q8—21no—
174C4×Q8—21no—
175C4×Q8—21no—
176C4×Q8—21no—
177C4×Q8—21no—
178C4×Q8—21no—
179C4×Q8—21no—
180#117—21no—
181C2×C4⋊C4—21no—
182C2×C4⋊C4—32no—
183C2×C4⋊C4—61no—
184C2×C4⋊C4—64no—
185C2×C4⋊C4—61no—
186C2×C4⋊C4#6711yescharacteristic
187C2×C42—21no—
188C2×C42—61no—
189C2×C42C22×D311yescharacteristic
190#107—22no—
191#106—22no—
192#106—64no—
193C4⋊C8—21no—
194C4⋊C8—32no—
195C4⋊C8—61no—
196C4⋊C8#6712yes—
197#103—22no—
198#103—22no—
199#103—64no—
200C22⋊C8—61no—
201C22⋊C8—61no—
202C4×C8—61no—
203C4×C8—61no—
204#95—32no—
205#95—64no—
206C23×C3#12011yescharacteristic
207C3×Q8—22no—
208C3×Q8—22no—
209C3×Q8—41no—
210C3×Q8—41no—
211C3×Q8—42no—
212C3×Q8—81no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4#12011yescharacteristic
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12111yescharacteristic
225C2×C3×C4#12111yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12711yescharacteristic
228C2×C3×C4C2×SD1611yescharacteristic
229C2×C3×C4#13511yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4#13711yescharacteristic
232C2×Dic3—24no—
233C2×Dic3—42no—
234C2×Dic3—42no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3—42no—
238Dic6—81no—
239Dic6—81no—
240Dic6—81no—
241Dic6—81no—
242C3×C8—42no—
243C3⋊C8—42no—
244C3⋊C8—81no—
245C2×Q8—21no—
246C2×Q8—21no—
247C2×Q8—21no—
248C2×Q8—21no—
249C2×Q8—21no—
250C2×Q8—21no—
251C2×Q8—21no—
252C2×Q8—121no—
253C2×Q8—121no—
254C2×Q8—121no—
255C2×Q8—121no—
256C2×Q8—121no—
257C2×Q8—121no—
258C2×Q8#21512yes—
259C22×C4—21no—
260C22×C4—21no—
261C22×C4—62no—
262C22×C4—62no—
263C22×C4D3×D411yescharacteristic
264C22×C4D3×D411yescharacteristic
265C22×C4C2×#6711yescharacteristic
266M16—41no—
267M16—41no—
268C2×C8—22no—
269C2×C8—62no—
270C2×C8—121no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—21no—
274C4⋊C4—21no—
275C4⋊C4—21no—
276C4⋊C4—21no—
277C4⋊C4—41no—
278C4⋊C4—41no—
279C4⋊C4—41no—
280C4⋊C4—41no—
281C4⋊C4—64no—
282C4⋊C4—62no—
283C4⋊C4—62no—
284C4⋊C4—128no—
285C4⋊C4—121no—
286C4⋊C4—121no—
287C22⋊C4—41no—
288C42—41no—
289C42—41no—
290C42—41no—
291C42—121no—
292C42—121no—
293C42D3×D411yescharacteristic
294C42#23711yescharacteristic
295C42C2×#6711yescharacteristic
296C42C2×#6711yescharacteristic
297C22×C3—21no—
298C22×C3—21no—
299C22×C3#39211yescharacteristic
300C22×C3#45011yescharacteristic
301C22×C3#45211yescharacteristic
302C3×C4—21no—
303C3×C4—21no—
304C3×C4—21no—
305C3×C4—42no—
306C3×C4—81no—
307C3×C4—81no—
308C3×C4—81no—
309C3×C4#44811yescharacteristic
310C3×C4#44911yescharacteristic
311C3×C4#45612yes—
312C3×C4#46511yescharacteristic
313C3×C4#46811yescharacteristic
314Dic3—44no—
315Dic3—82no—
316C23#93711yescharacteristic
317Q8—22no—
318Q8—22no—
319Q8—41no—
320Q8—41no—
321Q8—42no—
322Q8—81no—
323Q8—241no—
324Q8—241no—
325Q8—241no—
326Q8—241no—
327C2×C4—22no—
328C2×C4—22no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—42no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—64no—
337C2×C4—122no—
338C2×C4—122no—
339C2×C4—122no—
340C2×C4—122no—
341C2×C4—122no—
342C2×C4#93711yescharacteristic
343C2×C4#93811yescharacteristic
344C2×C4#93911yescharacteristic
345C2×C4#93911yescharacteristic
346C2×C4#94011yescharacteristic
347C2×C4C2×#21511yescharacteristic
348C2×C4#94211yescharacteristic
349C2×C4#94911yescharacteristic
350C2×C4#95011yescharacteristic
351C2×C4#95311yescharacteristic
352C8—42no—
353C8—122no—
354C8—241no—
355C2×C3—21no—
356C2×C3—21no—
357C2×C3#200011yescharacteristic
358C2×C3#200511yescharacteristic
359C2×C3#200911yescharacteristic
360C22—21no—
361C22—21no—
362C22#504611yescharacteristic
363C22#504711yescharacteristic
364C22#522311yescharacteristic
365C4—21no—
366C4—21no—
367C4—21no—
368C4—42no—
369C4—81no—
370C4—81no—
371C4—81no—
372C4—124no—
373C4—242no—
374C4#504811yescharacteristic
375C4#504911yescharacteristic
376C4#520512yes—
377C4#522711yescharacteristic
378C4#523811yescharacteristic
379C3#1025111yescharacteristic, complemented, Sylow
380C2—21no—
381C2—21no—
382C2#7169911yescharacteristic
383C2#7210811yescharacteristic
384C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3078C2
socleC22×C3#392
prime radicalC23×C3#120
hypercenter#3078D3
hypocenterC3#10251
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3078

Complements

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

NH ≅ G/Ntype
379. C3#10251semidirect

Maximal subgroups

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

subgroupindex
C3×#30782
#833232
#824752
#718352
#716212
#705062
#677152
#102513