Finite Group Databasebeta

← #10696933

Subgroups — #10696933

Order 768.

690 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 113 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×#3151C211yesmaximal, characteristic
2#83317C211yesmaximal, characteristic, complemented
3C2×#5076C211yesmaximal, characteristic, complemented
4#71802C211yesmaximal, characteristic, complemented
5#71759C211yesmaximal, characteristic, complemented
6#69879C211yesmaximal, characteristic, complemented
7#67667C211yesmaximal, characteristic, complemented
8#10146—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—22no—
10C3×#532C2211yescharacteristic
11C2×C3×#127C2211yescharacteristic
12C22×#214—22no—
13C2×#869C2211yescharacteristic
14C3×#496—22no—
15C3×#493—21no—
16C3×#492—21no—
17C3×C4×M16C2211yescharacteristic
18#5093—21no—
19#5076—21no—
20#5076—22no—
21#5076—22no—
22#5040—22no—
23#5019—22no—
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26C2×C4×C3⋊C8C2211yescharacteristic
27#4554—22no—
28#4544—22no—
29D4⋊(C3⋊C8)—22no—
30#3206—31no—
31#3151D311yescharacteristic
32C2×#492—31no—
33#1997—31no—
34#1545—31no—
35#1492—31no—
36#1278—31no—
37C2×C3×#41—21no—
38C22×C3×D4—21no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×Dic6D412yes—
42C3×AGL(1,ℤ/8)—44no—
43C3×AGL(1,ℤ/8)—44no—
44C2×C3×SD16—24no—
45C2×C3×D8—24no—
46C2×C3×M16D412yes—
47C3×C4⋊Q8D411yescharacteristic
48C3×#127—21no—
49C3×#127—21no—
50C3×#127—21no—
51C3×#127D412yes—
52C3×#127D411yescharacteristic
53C3×#124—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#214—28no—
60C2×#214—24no—
61C2×#214—48no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3—21no—
64C2×#208—21no—
65C22×C3⋊C8—21no—
66#869—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70C3×#97—21no—
71C3×C4×C8—21no—
72#830—21no—
73#810—24no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)D412yes—
77C4×C3⋊C8—21no—
78C4×C3⋊C8—21no—
79C4×C3⋊C8D412yes—
80C2×AGL(1,ℤ/8)—22no—
81C22×SD16—62no—
82#532C2×D311yescharacteristic
83C2×C4⋊Q8—31no—
84C2×#127C2×D311yescharacteristic
85#496—22no—
86#493—21no—
87#492—21no—
88#492—62no—
89#492—62no—
90#492—61no—
91#487—61no—
92#461—62no—
93#448—62no—
94#431—31no—
95#423—31no—
96C4×M16C2×D311yescharacteristic
97C2×C4×C8—31no—
98#343—62no—
99#331—62no—
100D4⋊C8—62no—
101C24×C3—41no—
102C3×#41—42no—
103C3×#41—42no—
104C2×C3×Q8—21no—
105C2×C3×D4—21no—
106C2×C3×D4—22no—
107C2×C3×D4—21no—
108C2×C3×D4—22no—
109C2×C3×D4—21no—
110C2×C3×D4—22no—
111C2×C3×D4—44no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—42no—
117C2×C3×D4—42no—
118C2×C3×D4—82no—
119C22×C3×C4—41no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4#4111yescharacteristic
123C22×Dic3—22no—
124C2×Dic6—22no—
125C2×Dic6—22no—
126C2×Dic6—44no—
127C3×SD16—48no—
128C3×D8—48no—
129C3×M16—22no—
130C3×M16SD1614yescomplemented
131C2×C3×C8—22no—
132C3×C4⋊C4—21no—
133C3×C4⋊C4—21no—
134C3×C22⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42C2×D411yescharacteristic
137C3×C42C2×D411yescharacteristic
138C3×C42C2×D411yescharacteristic
139C3×C42C2×D411yescharacteristic
140#214—416no—
141#214—84no—
142C4⋊Dic3—41no—
143C4⋊Dic3—41no—
144C4⋊Dic3—41no—
145C4⋊Dic3—41no—
146#208—42no—
147C2×C3⋊C8—24no—
148C2×C3⋊C8—22no—
149C2×C3⋊C8—22no—
150C2×C3⋊C8—41no—
151C2×#41—21no—
152C22×Q8—32no—
153C22×D4—21no—
154C22×D4—21no—
155C22×D4—21no—
156AGL(1,ℤ/8)—44no—
157AGL(1,ℤ/8)—44no—
158C2×SD16—24no—
159C2×SD16—68no—
160C2×SD16—64no—
161C2×SD16—128no—
162C2×D8—24no—
163C2×M16—61no—
164C2×M16#6712yes—
165C22×C8—61no—
166C4⋊Q8—61no—
167C4⋊Q8—61no—
168C4⋊Q8—61no—
169C4⋊Q8—61no—
170C4⋊Q8#6711yescharacteristic
171#127—21no—
172#127—21no—
173#127—21no—
174#127#6712yes—
175#127#6711yescharacteristic
176#124—21no—
177#121—21no—
178#121—21no—
179C4×D4—21no—
180C4×D4—21no—
181C2×C4⋊C4—61no—
182C2×C4⋊C4—61no—
183C2×C42C22×D311yescharacteristic
184C4⋊C8—32no—
185C4⋊C8—61no—
186C4⋊C8—61no—
187#103—64no—
188C22⋊C8—61no—
189#97—21no—
190C4×C8—21no—
191C4×C8—32no—
192C4×C8—61no—
193C4×C8—61no—
194C23×C3—41no—
195C23×C3—42no—
196C23×C3—82no—
197C23×C3—82no—
198C23×C3—82no—
199C23×C3#12111yescharacteristic
200C3×Q8—42no—
201C3×D4—44no—
202C3×D4—42no—
203C3×D4—44no—
204C3×D4—42no—
205C3×D4—42no—
206C3×D4—82no—
207C3×D4—82no—
208C3×D4—82no—
209C3×D4—82no—
210C3×D4—81no—
211C3×D4—81no—
212C2×C3×C4—22no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—81no—
216C2×C3×C4—81no—
217C2×C3×C4#12011yescharacteristic
218C2×C3×C4#12011yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#12711yescharacteristic
223C2×C3×C4C2×SD1612yes—
224C2×C3×C4AGL(1,ℤ/8)12yes—
225C2×Dic3—42no—
226C2×Dic3—82no—
227Dic6—44no—
228Dic6—44no—
229C3×C8—24no—
230C3⋊C8—24no—
231C3⋊C8—42no—
232C3⋊C8—81no—
233C24—41no—
234#41—42no—
235#41—42no—
236C2×Q8—21no—
237C2×Q8—62no—
238C2×Q8—62no—
239C2×Q8—124no—
240C2×D4—21no—
241C2×D4—22no—
242C2×D4—21no—
243C2×D4—22no—
244C2×D4—21no—
245C2×D4—22no—
246C2×D4—44no—
247C2×D4—41no—
248C2×D4—42no—
249C2×D4—42no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—82no—
254C22×C4—41no—
255C22×C4—62no—
256C22×C4D3×D411yescharacteristic
257C22×C4#23711yescharacteristic
258C22×C4C2×#6711yescharacteristic
259SD16—48no—
260SD16—1216no—
261SD16—244no—
262D8—48no—
263M16—22no—
264M16—122no—
265M16#21414yescomplemented
266C2×C8—22no—
267C2×C8—64no—
268C2×C8—62no—
269C2×C8—62no—
270C2×C8—121no—
271C4⋊C4—21no—
272C4⋊C4—21no—
273C4⋊C4—121no—
274C4⋊C4—121no—
275C4⋊C4—121no—
276C4⋊C4—121no—
277C22⋊C4—41no—
278C22⋊C4—41no—
279C42D3×D411yescharacteristic
280C42D3×D411yescharacteristic
281C42C2×#6711yescharacteristic
282C42C2×#6711yescharacteristic
283C22×C3—21no—
284C22×C3—21no—
285C22×C3—82no—
286C22×C3—81no—
287C22×C3—81no—
288C22×C3—82no—
289C22×C3—81no—
290C22×C3—82no—
291C22×C3—164no—
292C22×C3#39211yescharacteristic
293C22×C3#46012yes—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—81no—
297C3×C4#45012yes—
298C3×C4#45312yes—
299C3×C4#47112yes—
300C3×C4#49211yescharacteristic
301C3×C4#49611yescharacteristic
302Dic3—82no—
303C23—41no—
304C23—42no—
305C23—82no—
306C23—82no—
307C23—82no—
308C23#93811yescharacteristic
309Q8—42no—
310Q8—124no—
311Q8—124no—
312D4—44no—
313D4—42no—
314D4—42no—
315D4—42no—
316D4—44no—
317D4—82no—
318D4—82no—
319D4—82no—
320D4—82no—
321D4—81no—
322D4—81no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—41no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4—122no—
329C2×C4—242no—
330C2×C4#93212yes—
331C2×C4C2×#21412yes—
332C2×C4#93711yescharacteristic
333C2×C4#93711yescharacteristic
334C2×C4#93911yescharacteristic
335C2×C4#93911yescharacteristic
336C2×C4#94011yescharacteristic
337C2×C4#95311yescharacteristic
338C8—24no—
339C8—64no—
340C8—122no—
341C8—241no—
342C2×C3—22no—
343C2×C3—161no—
344C2×C3—162no—
345C2×C3#200211yescharacteristic
346C2×C3#201411yescharacteristic
347C2×C3#201911yescharacteristic
348C22—21no—
349C22—21no—
350C22—81no—
351C22—82no—
352C22—82no—
353C22—81no—
354C22—81no—
355C22—82no—
356C22—164no—
357C22#507512yes—
358C22#522311yescharacteristic
359C4—21no—
360C4—21no—
361C4—81no—
362C4—242no—
363C4#507411yescharacteristic
364C4#507611yescharacteristic
365C4#519812yes—
366C4#520012yes—
367C4#521812yes—
368C3#1014611yescharacteristic, complemented, Sylow
369C2—22no—
370C2—162no—
371C2—161no—
372C2#7176811yescharacteristic
373C2#7209411yescharacteristic
374C2#7209711yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3151C2
socleC22×C3#392
hypercenter#3151D3
hypocenterC3#10146
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3151

Complements

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

NH ≅ G/Ntype
2. #83317C2semidirect
3. C2×#5076C2semidirect
4. #71802C2semidirect
5. #71759C2semidirect
6. #69879C2semidirect
7. #67667C2semidirect
130. C3×M16SD16semidirect
265. M16#214semidirect
368. C3#10146semidirect

Maximal subgroups

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

subgroupindex
C3×#31512
#833172
C2×#50762
#718022
#717592
#698792
#676672
#101463