Finite Group Databasebeta

← #10767979

Subgroups — #10767979

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
2C3×#2398C211yesmaximal, characteristic
3#82448C212yesmaximal, complemented
4#71603C211yesmaximal, characteristic
5#69710C214yesmaximal, complemented
6#69701C214yesmaximal, complemented
7#69433C212yesmaximal, complemented
8#18181—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11#5798—21no—
12C2×#924—22no—
13#5543—22no—
14#5539—24no—
15D4×Dic6—22no—
16#5500—21no—
17#5483C2214yes—
18#5470C2212yes—
19C3×#400—21no—
20C3×#397C2212yes—
21C3×#394—21no—
22C3×#393C2212yes—
23C3×C4×C22⋊C4—22no—
24C3×#376C2211yescharacteristic
25#5228C2214yes—
26#5031—22no—
27#5030—24no—
28#5020—22no—
29#4971C2214yes—
30#4926C2212yes—
31#4525C2214yes—
32#4452C2218yes—
33#3501—31no—
34#3113—32no—
35#2398D311yescharacteristic
36#1608—34no—
37#1602—34no—
38#1506—32no—
39#1466—31no—
40C22×C3×Q8C2311yescharacteristic
41C22×C3×D4D411yescharacteristic
42C23×C3×C4D412yes—
43C22×Dic6—22no—
44C3×C4×Q8—22no—
45C3×C4×Q8—22no—
46C3×C4×D4—22no—
47C3×C4×D4—22no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4D411yescharacteristic
50C2×C3×C4⋊C4C2311yescharacteristic
51C2×C3×C22⋊C4—22no—
52C2×C3×C22⋊C4—22no—
53C2×C3×C22⋊C4C2312yes—
54C2×C3×C42—21no—
55C2×C3×C42—21no—
56C2×C3×C42C2311yescharacteristic
57C2×#217—24no—
58C2×#216—24no—
59D4×Dic3—24no—
60D4×Dic3—24no—
61#926—22no—
62C2×C4⋊Dic3—22no—
63#924—24no—
64#924—24no—
65#924—22no—
66#924—24no—
67#924C2314yes—
68C2×Dic3⋊C4—24no—
69#890—24no—
70#888—24no—
71#878—28no—
72#878—28no—
73#870—22no—
74#869D412yes—
75#869D412yes—
76C4×Dic6—22no—
77C4×Dic6D414yes—
78C3×#95—21no—
79C3×#95—22no—
80C3×#95—22no—
81C3×#95C2312yes—
82#835—24no—
83#830C2314yes—
84#810—24no—
85#808—24no—
86C4⋊(C3⋊C8)—22no—
87#803—22no—
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×Q8C2×D311yescharacteristic
97C2×C4×D4C2×D311yescharacteristic
98#483—34no—
99#475—62no—
100#462—62no—
101#451—34no—
102#441—64no—
103#432—32no—
104#400—21no—
105#397C2×D312yes—
106#394—21no—
107#393C2×D312yes—
108C4×C22⋊C4—22no—
109#376C2×D311yescharacteristic
110#328—38no—
111#323—34no—
112C24×C3C2×D412yes—
113C2×C3×Q8—22no—
114C2×C3×Q8—42no—
115C2×C3×D4—24no—
116C2×C3×D4—22no—
117C22×C3×C4—21no—
118C22×C3×C4—22no—
119C22×C3×C4—22no—
120C22×C3×C4—22no—
121C22×C3×C4—21no—
122C22×C3×C4—44no—
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D412yes—
126C22×C3×C4C2411yescharacteristic
127C22×Dic3—24no—
128C22×Dic3—24no—
129C2×Dic6—22no—
130C2×Dic6—24no—
131C2×Dic6—84no—
132C2×Dic6C2×D414yes—
133C3×C4⋊C4—22no—
134C3×C4⋊C4—22no—
135C3×C4⋊C4—42no—
136C3×C22⋊C4—24no—
137C3×C22⋊C4—44no—
138C3×C22⋊C4—44no—
139C3×C42—42no—
140C3×C42—41no—
141C3×C42—41no—
142C3×C42C2×D412yes—
143C3×C42C2×D412yes—
144#217—28no—
145#217—24no—
146#216—48no—
147C4⋊Dic3—24no—
148C4⋊Dic3—22no—
149C4⋊Dic3—22no—
150C4⋊Dic3—22no—
151C4⋊Dic3C2×D414yes—
152Dic3⋊C4—28no—
153Dic3⋊C4—24no—
154Dic3⋊C4—24no—
155C4×Dic3—24no—
156C2×C3⋊C8—24no—
157C22×Q8—62no—
158C22×Q8C22×D311yescharacteristic
159C22×D4#6711yescharacteristic
160C23×C4#6712yes—
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—22no—
175C4×Q8—22no—
176C4×Q8—34no—
177C4×Q8—62no—
178C4×D4—22no—
179C4×D4—22no—
180C4×D4—64no—
181C4×D4—64no—
182#117—62no—
183C2×C4⋊C4—21no—
184C2×C4⋊C4—64no—
185C2×C4⋊C4—62no—
186C2×C4⋊C4#6711yescharacteristic
187C2×C4⋊C4C22×D311yescharacteristic
188C2×C22⋊C4—22no—
189C2×C22⋊C4—22no—
190C2×C22⋊C4—64no—
191C2×C22⋊C4C22×D312yes—
192C2×C42—21no—
193C2×C42—21no—
194C2×C42C22×D311yescharacteristic
195#106—64no—
196C4⋊C8—62no—
197#103—64no—
198#103—64no—
199C22⋊C8—34no—
200#97—62no—
201#95—21no—
202#95—22no—
203#95—22no—
204#95C22×D312yes—
205C23×C3—22no—
206C23×C3—22no—
207C23×C3—42no—
208C23×C3—42no—
209C23×C3#13714yes—
210C23×C3C22×D411yescharacteristic
211C3×Q8—24no—
212C3×Q8—42no—
213C3×D4—28no—
214C2×C3×C4—22no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—22no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—41no—
221C2×C3×C4—44no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—42no—
228C2×C3×C4—82no—
229C2×C3×C4#12012yes—
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4C22×D412yes—
233C2×Dic3—24no—
234C2×Dic3—24no—
235C2×Dic3—84no—
236C2×Dic3—84no—
237Dic6—44no—
238Dic6—44no—
239Dic6—84no—
240C3⋊C8—44no—
241C24C2×#6712yes—
242C2×Q8—22no—
243C2×Q8—34no—
244C2×Q8—42no—
245C2×Q8—64no—
246C2×Q8—62no—
247C2×Q8—244no—
248C2×D4—24no—
249C2×D4—22no—
250C22×C4—21no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—21no—
255C22×C4—44no—
256C22×C4—64no—
257C22×C4—64no—
258C22×C4C2×#6711yescharacteristic
259C22×C4C2×#6711yescharacteristic
260C22×C4C2×#6712yes—
261C22×C4C23×D311yescharacteristic
262Q16—128no—
263C2×C8—64no—
264C4⋊C4—22no—
265C4⋊C4—22no—
266C4⋊C4—34no—
267C4⋊C4—42no—
268C4⋊C4—64no—
269C4⋊C4—68no—
270C4⋊C4—64no—
271C4⋊C4—62no—
272C4⋊C4—62no—
273C4⋊C4—62no—
274C4⋊C4—64no—
275C22⋊C4—24no—
276C22⋊C4—44no—
277C22⋊C4—44no—
278C22⋊C4—68no—
279C22⋊C4—64no—
280C42—42no—
281C42—41no—
282C42—41no—
283C42—64no—
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—42no—
300C3×C4—82no—
301C3×C4—81no—
302C3×C4#45512yes—
303Dic3—44no—
304Dic3—44no—
305C23—22no—
306C23—22no—
307C23—42no—
308C23—42no—
309C23#95114yes—
310C23C22×#6711yescharacteristic
311Q8—24no—
312Q8—42no—
313Q8—124no—
314Q8—124no—
315Q8—244no—
316D4—28no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—44no—
325C2×C4—42no—
326C2×C4—42no—
327C2×C4—42no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—64no—
332C2×C4—64no—
333C2×C4—82no—
334C2×C4—244no—
335C2×C4—244no—
336C2×C4#93711yescharacteristic
337C2×C4#93712yes—
338C2×C4#93711yescharacteristic
339C2×C4C2×D3×D412yes—
340C8—124no—
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×#93711yescharacteristic
354C22C2×#95112yes—
355C4—21no—
356C4—22no—
357C4—22no—
358C4—42no—
359C4—42no—
360C4—82no—
361C4—81no—
362C4—124no—
363C4—124no—
364C4#506112yes—
365C3#1818111yescharacteristic, complemented, Sylow
366C2—22no—
367C2—44no—
368C2#8293311yescharacteristic
369C2#8293611yescharacteristic
370C2C2×#506111yescharacteristic
subgroupG/H
centerC22C2×#937
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#2398C2
socleC22×C3C2×#120
prime radicalC22×C3×D4D4
hypercenter#2398D3
hypocenterC3#18181
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2398

Complements

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

NH ≅ G/Ntype
3. #82448C2semidirect
5. #69710C2semidirect
6. #69701C2semidirect
7. #69433C2semidirect
365. C3#18181semidirect

Maximal subgroups

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

subgroupindex
#857662
C3×#23982
#824482
#716032
#697102
#697012
#694332
#181813