Finite Group Databasebeta

← #10711069

Subgroups — #10711069

Order 768.

740 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 740 conjugacy classes of subgroups — too many to draw. Only its 107 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
1C2×#5672C211yesmaximal, characteristic
2C2×C3×#355C211yesmaximal, characteristic
3#71724C211yesmaximal, characteristic
4C2×#4478C212yesmaximal, complemented
5#70103C212yesmaximal, complemented
6#12775—31nomaximal, Sylow
7C2×C3×#124C2211yescharacteristic
8#5812C2211yescharacteristic
9C2×#944C2212yes—
10C2×#935—21no—
11#5672—22no—
12#5672—24no—
13#5672C414yescomplemented
14C2×#879—22no—
15C2×#878—22no—
16C2×#870—21no—
17C2×C3×#101C2212yes—
18C2×#836C2211yescharacteristic
19#5215—21no—
20#5120—22no—
21C3×#355—22no—
22#4555—22no—
23#4523—21no—
24#4478—24no—
25C2×#563—31no—
26C2×#356—32no—
27C2×#355D311yescharacteristic
28#1888—31no—
29#1886—32no—
30C22×C3×Q8C2311yescharacteristic
31C22×C3×D4D411yescharacteristic
32C23×Dic3—21no—
33C22×Dic6D412yes—
34C2×C3×M16—22no—
35C3×#124—22no—
36C3×#124C2×C412yes—
37C3×#124D412yes—
38C2×C3×C22⋊C4—22no—
39C2×C3×C42D411yescharacteristic
40C2×#217—22no—
41C2×#217—21no—
42Q8×Dic3—42no—
43#944—22no—
44#944—22no—
45#944C2×C414yes—
46#944D414yes—
47#935—24no—
48#935—24no—
49#926—22no—
50#926—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C2×Dic3⋊C4—22no—
54C2×Dic3⋊C4—22no—
55C2×Dic3⋊C4—22no—
56C2×C4×Dic3D412yes—
57C2×#208—21no—
58#879—28no—
59#879—28no—
60#878—28no—
61#878—28no—
62#870—24no—
63#870—24no—
64C3×#101—22no—
65C3×#101—22no—
66#836—22no—
67#832—42no—
68#816—44no—
69#563—34no—
70#563—64no—
71#563—62no—
72C2×C4⋊Q8—32no—
73C2×#126—62no—
74C2×#125—61no—
75C2×#124C2×D311yescharacteristic
76C2×#123—61no—
77C2×C22⋊Q8—62no—
78#519—31no—
79#419—62no—
80#418—61no—
81C2×#101—31no—
82C2×#101C2×D312yes—
83#356—64no—
84#355—22no—
85#339—62no—
86#339—61no—
87C24×C3—21no—
88C2×C3×Q8—41no—
89C2×C3×Q8—41no—
90C2×C3×Q8C22×C411yescharacteristic
91C2×C3×Q8C2×D412yes—
92C2×C3×Q8C2×D411yescharacteristic
93C2×C3×D4—21no—
94C2×C3×D4—41no—
95C2×C3×D4—41no—
96C2×C3×D4C22⋊C412yes—
97C22×C3×C4—21no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D412yes—
100C22×Dic3—22no—
101C22×Dic3—22no—
102C22×Dic3—24no—
103C22×Dic3—81no—
104C22×Dic3—81no—
105C2×Dic6—22no—
106C2×Dic6—42no—
107C2×Dic6—42no—
108C2×Dic6C22⋊C414yes—
109C3×M16—42no—
110C3×M16—42no—
111C2×C3×C8—42no—
112C3×C22⋊C4—24no—
113C3×C22⋊C4—24no—
114C3×C42—21no—
115C3×C42C22⋊C412yes—
116#217—28no—
117#217—24no—
118#217—41no—
119C4⋊Dic3—24no—
120C4⋊Dic3—24no—
121C4⋊Dic3—24no—
122C4⋊Dic3—22no—
123Dic3⋊C4—28no—
124Dic3⋊C4—24no—
125Dic3⋊C4—24no—
126Dic3⋊C4—24no—
127Dic3⋊C4—24no—
128C4×Dic3—22no—
129C4×Dic3—41no—
130C4×Dic3—42no—
131C4×Dic3C22⋊C414yes—
132#208—41no—
133#208—41no—
134C2×C3⋊C8—41no—
135C22×Q8—32no—
136C22×Q8C22×D311yescharacteristic
137C22×D4D1211yescharacteristic
138C23×C4—61no—
139C2×M16—22no—
140C2×M16—61no—
141C4⋊Q8—34no—
142C4⋊Q8—34no—
143C4⋊Q8—62no—
144C4⋊Q8—62no—
145#126—68no—
146#126—68no—
147#125—64no—
148#125—64no—
149#124—22no—
150#124C4×D312yes—
151#124#6712yes—
152#123—64no—
153#123—64no—
154C22⋊Q8—68no—
155C22⋊Q8—68no—
156C4×Q8—122no—
157#117—61no—
158#117—62no—
159C2×C4⋊C4—61no—
160C2×C4⋊C4—62no—
161C2×C4⋊C4—61no—
162C2×C4⋊C4—62no—
163C2×C4⋊C4—62no—
164C2×C22⋊C4—22no—
165C2×C22⋊C4—62no—
166C2×C22⋊C4—61no—
167C2×C42—32no—
168C2×C42D1211yescharacteristic
169#103—124no—
170#102—122no—
171#101—22no—
172#101—22no—
173#101—62no—
174C23×C3—22no—
175C23×C3—44no—
176C23×C3—41no—
177C23×C3—41no—
178C23×C3#12011yescharacteristic
179C3×Q8—41no—
180C3×Q8—42no—
181C3×Q8—41no—
182C3×D4—42no—
183C3×D4—81no—
184C2×C3×C4—21no—
185C2×C3×C4—22no—
186C2×C3×C4—21no—
187C2×C3×C4—22no—
188C2×C3×C4—22no—
189C2×C3×C4—42no—
190C2×C3×C4C2×C22⋊C412yes—
191C2×C3×C4C2×C22⋊C411yescharacteristic
192C2×C3×C4#12011yescharacteristic
193C2×C3×C4#12012yes—
194C2×Dic3—24no—
195C2×Dic3—24no—
196C2×Dic3—44no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—84no—
201Dic6—44no—
202Dic6—82no—
203C3×C8—82no—
204C3⋊C8—81no—
205C24—21no—
206C2×Q8—34no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—62no—
210C2×Q8—122no—
211C2×Q8—122no—
212C2×Q8C2×C4×D311yescharacteristic
213C2×Q8D3×D412yes—
214C2×Q8C2×#6711yescharacteristic
215C2×D4—21no—
216C2×D4—41no—
217C2×D4—41no—
218C2×D4(C2×D3)⋊C412yes—
219C22×C4—21no—
220C22×C4—62no—
221C22×C4—62no—
222C22×C4—64no—
223C22×C4—241no—
224C22×C4—241no—
225C22×C4C2×D1211yescharacteristic
226C22×C4D3×D412yes—
227M16—42no—
228M16—42no—
229M16—121no—
230M16—121no—
231C2×C8—42no—
232C2×C8—121no—
233C4⋊C4—64no—
234C4⋊C4—68no—
235C4⋊C4—64no—
236C4⋊C4—62no—
237C4⋊C4—64no—
238C4⋊C4—64no—
239C4⋊C4—64no—
240C4⋊C4—64no—
241C4⋊C4—64no—
242C22⋊C4—24no—
243C22⋊C4—24no—
244C22⋊C4—68no—
245C22⋊C4—64no—
246C22⋊C4—121no—
247C42—21no—
248C42—34no—
249C42—62no—
250C42—122no—
251C42—121no—
252C42(C2×D3)⋊C412yes—
253C22×C3—22no—
254C22×C3—44no—
255C22×C3—42no—
256C22×C3—82no—
257C22×C3—82no—
258C22×C3#37911yescharacteristic
259C22×C3B(4,𝔽2)11yescharacteristic
260C22×C3#45811yescharacteristic
261C3×C4—21no—
262C3×C4—22no—
263C3×C4—22no—
264C3×C4—21no—
265C3×C4—42no—
266Dic3—44no—
267Dic3—44no—
268Dic3—82no—
269C23—22no—
270C23—44no—
271C23—41no—
272C23—41no—
273C23#88211yescharacteristic
274Q8—41no—
275Q8—42no—
276Q8—41no—
277Q8—124no—
278Q8—242no—
279D4—42no—
280D4—81no—
281C2×C4—22no—
282C2×C4—21no—
283C2×C4—22no—
284C2×C4—21no—
285C2×C4—22no—
286C2×C4—42no—
287C2×C4—64no—
288C2×C4—64no—
289C2×C4—124no—
290C2×C4—122no—
291C2×C4—122no—
292C2×C4—122no—
293C2×C4—244no—
294C2×C4D3×C22⋊C412yes—
295C2×C4#88211yescharacteristic
296C2×C4C2×(C2×D3)⋊C411yescharacteristic
297C2×C4#93712yes—
298C8—82no—
299C8—241no—
300C2×C3—21no—
301C2×C3—21no—
302C2×C3—82no—
303C2×C3#179011yescharacteristic
304C2×C3#220312yes—
305C22—22no—
306C22—44no—
307C22—42no—
308C22—82no—
309C22—82no—
310C22#474111yescharacteristic
311C22#474211yescharacteristic
312C22#495511yescharacteristic
313C4—22no—
314C4—21no—
315C4—22no—
316C4—21no—
317C4—42no—
318C4—124no—
319C4—124no—
320C4—242no—
321C3#1277511yescharacteristic, complemented, Sylow
322C2—21no—
323C2—21no—
324C2—82no—
325C2#6916212yes—
326C2#7003311yescharacteristic
subgroupG/H
centerC22#4741
commutatorC2×C3×Q8C22×C4
FrattiniC22×Q8C22×D3
FittingC2×C3×#355C2
socleC22×C3B(4,𝔽2)
prime radicalC22×C3×D4D4
cosocleC22×C3×Q8C23
hypercenterC2×#355D3
hypocenterC3#12775
LCSG ▷ C2×C3×Q8 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ C2×#355

Complements

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

NH ≅ G/Ntype
4. C2×#4478C2semidirect
5. #70103C2semidirect
13. #5672C4semidirect
321. C3#12775semidirect

Maximal subgroups

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

subgroupindex
C2×#56722
C2×C3×#3552
#717242
C2×#44782
#701032
#127753