Finite Group Databasebeta

← #10767983

Subgroups — #10767983

Order 768.

970 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 970 conjugacy classes of subgroups — too many to draw. Only its 167 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#85766C211yesmaximal, characteristic
2#82348C211yesmaximal, characteristic
3#79356C212yesmaximal, complemented
4C3×#1466C211yesmaximal, characteristic
5#70836C212yesmaximal, complemented
6#68116C214yesmaximal, complemented
7#68115C214yesmaximal, complemented
8#18181—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5798—21no—
11C2×#924—22no—
12#5543—22no—
13#5539—24no—
14D4×Dic6—22no—
15#5500—21no—
16#5483C2214yes—
17#5470C2212yes—
18C2×C4×Dic6C2211yescharacteristic
19C3×#432C2212yes—
20#5222—21no—
21C4×#217—22no—
22#4985—21no—
23#4953C2212yes—
24#4947C2212yes—
25#4932C2211yescharacteristic
26#4851—22no—
27#4849—22no—
28#4739C2214yes—
29#4721C2214yes—
30#4716—24no—
31C3×#323C2214yes—
32#4467C2218yes—
33#3501—31no—
34#3113—32no—
35#2398—31no—
36#1608—34no—
37#1602—34no—
38#1506—32no—
39#1466D311yescharacteristic
40C22×C3×D4D411yescharacteristic
41C23×C3×C4D412yes—
42C22×Dic6—22no—
43C22×Dic6C2311yescharacteristic
44C3×C4×D4—22no—
45C3×C4×D4—22no—
46C2×C3×C4⋊C4D411yescharacteristic
47C2×C3×C22⋊C4C2312yes—
48C2×C3×C42C2311yescharacteristic
49C2×#217—24no—
50C2×#217—22no—
51C2×#217—22no—
52D4×Dic3—24no—
53D4×Dic3—24no—
54#926—22no—
55C2×C4⋊Dic3—22no—
56C2×C4⋊Dic3C2311yescharacteristic
57#924—24no—
58#924—24no—
59#924—22no—
60#924—24no—
61#924C2314yes—
62C2×Dic3⋊C4—24no—
63C2×Dic3⋊C4—21no—
64C2×C4×Dic3—21no—
65C2×C4×Dic3—21no—
66C2×Dic12—24no—
67#890—24no—
68#888—24no—
69#878—28no—
70#878—28no—
71#870—22no—
72#869D412yes—
73#869D412yes—
74C4×Dic6—22no—
75C4×Dic6—22no—
76C4×Dic6—22no—
77C4×Dic6D414yes—
78C3×C4⋊C8—22no—
79C3×C22⋊C8C2314yes—
80C3×#97—22no—
81C3⋊#95—21no—
82C3⋊#95—22no—
83C3⋊#95—22no—
84C3⋊#95C2312yes—
85#817—24no—
86#816—24no—
87#816—24no—
88#554—64no—
89#554—62no—
90D4×Q8—62no—
91#544—61no—
92#543—34no—
93#533—32no—
94C2×C22⋊Q8—62no—
95#518—61no—
96C2×C4×Q8—31no—
97C2×C4×D4C2×D311yescharacteristic
98#483—34no—
99#475—62no—
100#462—62no—
101#451—34no—
102#441—64no—
103#432C2×D312yes—
104#400—61no—
105#397—32no—
106#394—61no—
107#393—32no—
108C4×C22⋊C4—62no—
109#376—31no—
110#328—38no—
111#323C2×D314yes—
112C24×C3C2×D412yes—
113C2×C3×D4—24no—
114C2×C3×D4—22no—
115C22×C3×C4—22no—
116C22×C3×C4—22no—
117C22×C3×C4—44no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D412yes—
121C22×C3×C4C2411yescharacteristic
122C22×Dic3—21no—
123C22×Dic3—24no—
124C22×Dic3—24no—
125C22×Dic3—22no—
126C22×Dic3—21no—
127C2×Dic6—22no—
128C2×Dic6—22no—
129C2×Dic6—24no—
130C2×Dic6—42no—
131C2×Dic6—84no—
132C2×Dic6C2×D414yes—
133C2×C3×C8—24no—
134C3×C4⋊C4—22no—
135C3×C22⋊C4—24no—
136C3×C42C2×D412yes—
137C3×C42C2×D412yes—
138#217—28no—
139#217—24no—
140#217—44no—
141#217—44no—
142C4⋊Dic3—24no—
143C4⋊Dic3—22no—
144C4⋊Dic3—22no—
145C4⋊Dic3—22no—
146C4⋊Dic3—22no—
147C4⋊Dic3C2×D414yes—
148Dic3⋊C4—28no—
149Dic3⋊C4—24no—
150Dic3⋊C4—24no—
151Dic3⋊C4—42no—
152C4×Dic3—24no—
153C4×Dic3—42no—
154C4×Dic3—41no—
155C4×Dic3—41no—
156Dic12—48no—
157C22×Q8—31no—
158C22×Q8—62no—
159C22×D4D1211yescharacteristic
160C23×C4D1212yes—
161C2×Q16—64no—
162C4⋊Q8—32no—
163C4⋊Q8—32no—
164C4⋊Q8—64no—
165#125—64no—
166#125—62no—
167C22⋊Q8—34no—
168C22⋊Q8—68no—
169C22⋊Q8—68no—
170C22⋊Q8—64no—
171C22⋊Q8—64no—
172C22⋊Q8—62no—
173C22⋊Q8—64no—
174C4×Q8—34no—
175C4×Q8—62no—
176C4×Q8—62no—
177C4×Q8—62no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—64no—
181C4×D4—64no—
182#117—62no—
183C2×C4⋊C4—31no—
184C2×C4⋊C4—64no—
185C2×C4⋊C4—62no—
186C2×C4⋊C4—61no—
187C2×C4⋊C4D1211yescharacteristic
188C2×C22⋊C4—64no—
189C2×C22⋊C4—62no—
190C2×C22⋊C4—62no—
191C2×C22⋊C4C22×D312yes—
192C2×C42—61no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195#106—64no—
196C4⋊C8—22no—
197#103—64no—
198#103—64no—
199C22⋊C8C22×D314yes—
200#97—22no—
201#95—32no—
202#95—61no—
203#95—62no—
204#95—62no—
205C23×C3—22no—
206C23×C3—22no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3#13714yes—
210C23×C3C22×D411yescharacteristic
211C3×D4—28no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—22no—
215C2×C3×C4—41no—
216C2×C3×C4—44no—
217C2×C3×C4—42no—
218C2×C3×C4—82no—
219C2×C3×C4#12012yes—
220C2×C3×C4#12011yescharacteristic
221C2×C3×C4#12011yescharacteristic
222C2×C3×C4C22×D412yes—
223C2×Dic3—22no—
224C2×Dic3—24no—
225C2×Dic3—24no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—42no—
229C2×Dic3—42no—
230C2×Dic3—41no—
231C2×Dic3—42no—
232C2×Dic3—42no—
233C2×Dic3—84no—
234C2×Dic3—84no—
235Dic6—24no—
236Dic6—42no—
237Dic6—44no—
238Dic6—44no—
239Dic6—84no—
240C3×C8—44no—
241C24C2×D1212yes—
242C2×Q8—34no—
243C2×Q8—64no—
244C2×Q8—62no—
245C2×Q8—62no—
246C2×Q8—122no—
247C2×Q8—244no—
248C2×D4—24no—
249C2×D4—22no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—44no—
253C22×C4—61no—
254C22×C4—64no—
255C22×C4—64no—
256C22×C4—62no—
257C22×C4—61no—
258C22×C4C2×D1211yescharacteristic
259C22×C4C2×D1211yescharacteristic
260C22×C4C2×D1212yes—
261C22×C4C23×D311yescharacteristic
262Q16—128no—
263C2×C8—24no—
264C4⋊C4—22no—
265C4⋊C4—34no—
266C4⋊C4—64no—
267C4⋊C4—68no—
268C4⋊C4—64no—
269C4⋊C4—62no—
270C4⋊C4—62no—
271C4⋊C4—62no—
272C4⋊C4—64no—
273C4⋊C4—62no—
274C4⋊C4—122no—
275C22⋊C4—24no—
276C22⋊C4—68no—
277C22⋊C4—64no—
278C22⋊C4—124no—
279C22⋊C4—124no—
280C42—64no—
281C42—122no—
282C42—121no—
283C42—121no—
284C42D3×D412yes—
285C42D3×D412yes—
286C22×C3—21no—
287C22×C3—21no—
288C22×C3—24no—
289C22×C3—44no—
290C22×C3—42no—
291C22×C3—42no—
292C22×C3—82no—
293C22×C3C2×#12011yescharacteristic
294C22×C3C2×#13712yes—
295C3×C4—22no—
296C3×C4—21no—
297C3×C4—22no—
298C3×C4—42no—
299C3×C4#45512yes—
300Dic3—44no—
301Dic3—42no—
302Dic3—44no—
303Dic3—82no—
304Dic3—81no—
305C23—22no—
306C23—22no—
307C23—42no—
308C23—42no—
309C23#90914yes—
310C23C22×D1211yescharacteristic
311Q8—64no—
312Q8—124no—
313Q8—122no—
314Q8—124no—
315Q8—244no—
316D4—28no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—44no—
322C2×C4—42no—
323C2×C4—62no—
324C2×C4—64no—
325C2×C4—64no—
326C2×C4—82no—
327C2×C4—121no—
328C2×C4—121no—
329C2×C4—122no—
330C2×C4—122no—
331C2×C4—121no—
332C2×C4—122no—
333C2×C4—122no—
334C2×C4—244no—
335C2×C4—244no—
336C2×C4#88212yes—
337C2×C4#88211yescharacteristic
338C2×C4#88211yescharacteristic
339C2×C4C2×D3×D412yes—
340C8—44no—
341C2×C3—22no—
342C2×C3—44no—
343C2×C3#300411yescharacteristic
344C2×C3C2×#45511yescharacteristic
345C2×C3#302511yescharacteristic
346C22—21no—
347C22—21no—
348C22—24no—
349C22—44no—
350C22—42no—
351C22—42no—
352C22—82no—
353C22C2×#88211yescharacteristic
354C22C2×#90912yes—
355C4—22no—
356C4—22no—
357C4—21no—
358C4—42no—
359C4—122no—
360C4—124no—
361C4—124no—
362C4—242no—
363C4—241no—
364C4#474612yes—
365C3#1818111yescharacteristic, complemented, Sylow
366C2—22no—
367C2—44no—
368C2#7864511yescharacteristic
369C2#7869011yescharacteristic
370C2C2×#474611yescharacteristic
subgroupG/H
centerC22C2×#882
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1466C2
socleC22×C3C2×#120
prime radicalC22×C3×D4D4
hypercenter#1466D3
hypocenterC3#18181
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1466

Complements

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

NH ≅ G/Ntype
3. #79356C2semidirect
5. #70836C2semidirect
6. #68116C2semidirect
7. #68115C2semidirect
365. C3#18181semidirect

Maximal subgroups

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

subgroupindex
#857662
#823482
#793562
C3×#14662
#708362
#681162
#681152
#181813