Finite Group Databasebeta

← #11714121

Subgroups — #11714121

Order 768.

355 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 355 conjugacy classes of subgroups — too many to draw. Only its 59 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#85339C211yesmaximal, characteristic
2#85337C211yesmaximal, characteristic
3#2875⋊C3C211yesmaximal, characteristic
4#73126C212yesmaximal
5#73125C212yesmaximal
6#33223—31nomaximal, Sylow
7C2×#131⋊C3C2212yes—
8C4×#41⋊C3C2211yescharacteristic
9#5426C2212yes—
10#5426C2211yescharacteristic
11#5426C2211yescharacteristic
12#5391—21no—
13#5389—21no—
14#5389—21no—
15#5386—21no—
16#4702—41nomaximal
17C2×#501—31no—
18#3080—31no—
19#3079—31no—
20#2875D311yescharacteristic
21#1998—31no—
22#1997—31no—
23#1952—32no—
24#1951—32no—
25#1849—31no—
26#1816—32no—
27#1815—32no—
28C2×#41⋊C3C2311yescharacteristic
29#870—41no—
30#869—41no—
31#131⋊C3Q814yescomplemented
32C4×SL(2,𝔽3)D411yescharacteristic
33C4×SL(2,𝔽3)D411yescharacteristic
34#859—22no—
35#859—21no—
36#859—21no—
37C3×#97—41no—
38#818—42no—
39#817—42no—
40C2×#131C2×D312yes—
41C2×C4⋊Q8—31no—
42C2×#125—31no—
43C4×#41C2×D311yescharacteristic
44#501—62no—
45#501—61no—
46#501—61no—
47#479—61no—
48#479—61no—
49#478—61no—
50#478—61no—
51#477—61no—
52#476—61no—
53#475—61no—
54#474—61no—
55C2×#108—32no—
56C2×#108—32no—
57#428—32no—
58#428—31no—
59#428—31no—
60C2×#107—32no—
61C2×#106—32no—
62#423—31no—
63#423—31no—
64#417—32no—
65#417—31no—
66#417—31no—
67#405—32no—
68C4×M16—31no—
69C2×#97—31no—
70#343—32no—
71#343—32no—
72#343—32no—
73#343—32no—
74C2×Dic6—81no—
75#41⋊C3C2×Q812yes—
76C2×SL(2,𝔽3)C2×D411yescharacteristic
77C2×C3×C8—42no—
78C3×C42—41no—
79C4⋊Dic3—41no—
80C4⋊Dic3—42no—
81C4⋊Dic3—41no—
82C4⋊Dic3—81no—
83Dic3⋊C4—81no—
84Dic3⋊C4—81no—
85C2×#41C22×D311yescharacteristic
86C22×Q8—61no—
87#131—34no—
88#131Dic614yescomplemented
89C2×M16—32no—
90C2×M16—32no—
91C2×M16—31no—
92C2×M16—31no—
93C22×C8—32no—
94C4⋊Q8—61no—
95C4⋊Q8—61no—
96C4⋊Q8—61no—
97C4⋊Q8—61no—
98#125—61no—
99#125—61no—
100#125—61no—
101#125—61no—
102C4×Q8D1211yescharacteristic
103C4×Q8D1211yescharacteristic
104C4×D4—31no—
105C4×D4—31no—
106#117—31no—
107C2×C4⋊C4—31no—
108C2×C4⋊C4—31no—
109C2×C4⋊C4—32no—
110C2×C4⋊C4—61no—
111C2×C4⋊C4—61no—
112C2×C4⋊C4—61no—
113C2×C42—31no—
114#108—64no—
115#108—64no—
116#107—62no—
117#107—61no—
118#107—61no—
119#107—62no—
120#107—62no—
121#106—62no—
122#106—61no—
123#106—61no—
124#106—62no—
125#106—62no—
126C4⋊C8—61no—
127C4⋊C8—61no—
128C4⋊C8—61no—
129C4⋊C8—61no—
130#103—62no—
131#103—61no—
132#103—61no—
133#102—62no—
134#102—61no—
135#102—61no—
136#97—32no—
137#97—31no—
138#97S411yescharacteristic
139C4×C8—61no—
140C2×C3×C4—41no—
141C2×C3×C4—41no—
142C2×C3×C4—41no—
143C2×Dic3—81no—
144C2×Dic3—81no—
145C2×Dic3—82no—
146Dic6—162no—
147SL(2,𝔽3)C4⋊Q811yescharacteristic
148C3×C8—44no—
149#41—32no—
150#41C2×Dic612yes—
151C2×Q8—121no—
152C2×Q8—121no—
153C2×Q8—242no—
154C2×Q8C2×D1211yescharacteristic
155C2×D4—31no—
156C22×C4—31no—
157C22×C4—31no—
158C22×C4—31no—
159C22×C4—61no—
160C22×C4—61no—
161C22×C4—62no—
162M16—34no—
163M16—62no—
164M16—62no—
165M16—62no—
166M16—62no—
167M16—62no—
168C2×C8—32no—
169C2×C8—34no—
170C2×C8—34no—
171C2×C8—62no—
172C2×C8—61no—
173C2×C8—61no—
174C2×C8C2×S412yes—
175C4⋊C4—31no—
176C4⋊C4—31no—
177C4⋊C4—61no—
178C4⋊C4—61no—
179C4⋊C4—62no—
180C4⋊C4—61no—
181C4⋊C4—62no—
182C4⋊C4—61no—
183C4⋊C4—121no—
184C4⋊C4—121no—
185C4⋊C4—121no—
186C4⋊C4—121no—
187C4⋊C4—121no—
188C4⋊C4—121no—
189C22⋊C4—61no—
190C42—31no—
191C42—31no—
192C42—31no—
193C42C2×S411yescharacteristic
194C22×C3—41no—
195C3×C4—42no—
196C3×C4—81no—
197C3×C4—81no—
198Dic3—162no—
199Dic3—161no—
200Dic3—161no—
201C23—31no—
202Q8—31no—
203Q8—242no—
204Q8—242no—
205Q8#86911yescharacteristic
206D4—32no—
207D4—61no—
208C2×C4—31no—
209C2×C4—32no—
210C2×C4—32no—
211C2×C4—31no—
212C2×C4—31no—
213C2×C4—121no—
214C2×C4—121no—
215C2×C4—121no—
216C2×C4—121no—
217C2×C4—122no—
218C2×C4—241no—
219C2×C4—241no—
220C2×C4—242no—
221C2×C4C4⋊S411yescharacteristic
222C2×C4C4⋊S411yescharacteristic
223C2×C4C22×S411yescharacteristic
224C8—34no—
225C8—122no—
226C8—121no—
227C8—121no—
228C8A4⋊Q814yes—
229C2×C3—41no—
230C2×C3—41no—
231C2×C3—41no—
232C22—32no—
233C22—61no—
234C22—61no—
235C22C2×C4⋊S411yescharacteristic
236C4—21no—
237C4—21no—
238C4—31no—
239C4—31no—
240C4—61no—
241C4—61no—
242C4—242no—
243C4—241no—
244C4—241no—
245C4C2×A4⋊Q812yes—
246C3—41noSylow
247C2—62no—
248C2#8528311yescharacteristic
249C2#8545411yescharacteristic
250C2#8545511yescharacteristic
subgroupG/H
center, socleC22C2×C4⋊S4
commutator, prime radical, cosocleC2×#41⋊C3C23
FrattiniC2×C4C22×S4
Fitting#2875D3
hypercenter#97S4
hypocenterSL(2,𝔽3)C4⋊Q8
LCSG ▷ C2×#41⋊C3 ▷ SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C42 ◁ #97

Complements

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

NH ≅ G/Ntype
31. #131⋊C3Q8semidirect
88. #131Dic6semidirect

Maximal subgroups

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

subgroupindex
#853392
#853372
#2875⋊C32
#731262
#731252
#332233
#47024