Finite Group Databasebeta

← #10698007

Subgroups — #10698007

Order 768.

554 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#5093C211yesmaximal, characteristic
2C2×#5079C211yesmaximal, characteristic
3C2×#4682C211yesmaximal, characteristic
4C3×#1990C211yesmaximal, characteristic
5#69582C211yesmaximal, characteristic
6#69579C211yesmaximal, characteristic
7C3⋊#1273C211yesmaximal, characteristic
8#10313—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#869C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13C3×#384C2211yescharacteristic
14C2×#835—22no—
15C2×#818—22no—
16#5093—22no—
17#5093—22no—
18#5079—22no—
19#5079—22no—
20C2×#810—22no—
21C2×#810—22no—
22C2×C4×C3⋊C8C2211yescharacteristic
23C2×C4×C3⋊C8C2211yescharacteristic
24#4682—22no—
25#4682—22no—
26C3×#340—22no—
27#4529—22no—
28C2×#500—31no—
29C2×#487—31no—
30C2×#487—31no—
31#1990—31no—
32#1990D311yescharacteristic
33#1956—31no—
34#1273—31no—
35C22×C3×Q8—21no—
36C22×Dic6—21no—
37C22×Dic6—21no—
38C22×C3×C8D412yes—
39C3×C4⋊Q8—21no—
40C3×C4⋊Q8—21no—
41C3×C4⋊Q8—21no—
42C3×C4⋊Q8D412yes—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4—21no—
48C2×C3×C42C2311yescharacteristic
49C2×C4⋊Dic3—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3—21no—
52C2×C4⋊Dic3—21no—
53C22×C3⋊C8D412yes—
54C22×C3⋊C8D412yes—
55#869—21no—
56#869—21no—
57#869—21no—
58#869—21no—
59#869—21no—
60#869—21no—
61#869D412yes—
62#869D412yes—
63C3×C4×C8—21no—
64C3×C4×C8—21no—
65#835—28no—
66#818—28no—
67#810—28no—
68#810—44no—
69C4×C3⋊C8—21no—
70C4×C3⋊C8—21no—
71C4×C3⋊C8—21no—
72C4×C3⋊C8—21no—
73C2×C4⋊Q8—31no—
74C2×C4⋊Q8—31no—
75C2×C4⋊Q8C2×D311yescharacteristic
76#500—62no—
77#500—62no—
78#487—62no—
79#487—62no—
80#487—62no—
81#487—62no—
82C2×#107—62no—
83C2×#103—62no—
84C2×#103—62no—
85C2×#103—62no—
86C2×C4×C8—31no—
87C2×C4×C8—31no—
88C2×C4×C8C2×D311yescharacteristic
89#384C2×D311yescharacteristic
90#340—22no—
91#340—62no—
92C2×C3×Q8—22no—
93C2×C3×Q8—41no—
94C2×C3×Q8—42no—
95C2×C3×Q8—42no—
96C22×C3×C4—41no—
97C22×C3×C4—41no—
98C22×C3×C4C2×D411yescharacteristic
99C22×C3×C4C2×D411yescharacteristic
100C22×C3×C4C2×D411yescharacteristic
101C22×Dic3—41no—
102C22×Dic3—41no—
103C2×Dic6—22no—
104C2×Dic6—22no—
105C2×Dic6—42no—
106C2×Dic6—42no—
107C2×Dic6—42no—
108C2×Dic6—42no—
109C2×Dic6—41no—
110C2×Dic6—41no—
111C2×C3×C8—22no—
112C2×C3×C8—22no—
113C2×C3×C8Q1614yes—
114C3×C4⋊C4—22no—
115C3×C4⋊C4—22no—
116C3×C4⋊C4—41no—
117C3×C4⋊C4—41no—
118C3×C4⋊C4—42no—
119C3×C4⋊C4—42no—
120C3×C4⋊C4—42no—
121C3×C42C2×D411yescharacteristic
122C3×C42C2×D411yescharacteristic
123C3×C42C2×D411yescharacteristic
124C3×C42#4111yescharacteristic
125C4⋊Dic3—22no—
126C4⋊Dic3—22no—
127C4⋊Dic3—22no—
128C4⋊Dic3—22no—
129C4⋊Dic3—41no—
130C4⋊Dic3—41no—
131C4⋊Dic3—41no—
132C4⋊Dic3—41no—
133C2×C3⋊C8—22no—
134C2×C3⋊C8—22no—
135C2×C3⋊C8—22no—
136C2×C3⋊C8—22no—
137C2×C3⋊C8—22no—
138C2×C3⋊C8—22no—
139C22×Q8—21no—
140C22×Q8—61no—
141C22×Q8—61no—
142C22×C8—32no—
143C22×C8—32no—
144C22×C8#6712yes—
145C4⋊Q8—21no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—32no—
149C4⋊Q8—32no—
150C4⋊Q8—61no—
151C4⋊Q8—61no—
152C4⋊Q8—61no—
153C4⋊Q8—61no—
154C4⋊Q8—61no—
155C4⋊Q8—61no—
156C4⋊Q8D1212yes—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—21no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—61no—
163C2×C4⋊C4—61no—
164C2×C4⋊C4—61no—
165C2×C4⋊C4—61no—
166C2×C42C22×D311yescharacteristic
167#107—68no—
168#103—68no—
169#103—68no—
170#103—124no—
171C4×C8—21no—
172C4×C8—21no—
173C4×C8—61no—
174C4×C8—61no—
175C4×C8—61no—
176C4×C8—61no—
177C23×C3#12711yescharacteristic
178C3×Q8—44no—
179C3×Q8—82no—
180C2×C3×C4—42no—
181C2×C3×C4—81no—
182C2×C3×C4—81no—
183C2×C3×C4—81no—
184C2×C3×C4—81no—
185C2×C3×C4—81no—
186C2×C3×C4#12011yescharacteristic
187C2×C3×C4#12011yescharacteristic
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12111yescharacteristic
190C2×C3×C4#12111yescharacteristic
191C2×C3×C4#12111yescharacteristic
192C2×C3×C4C2×Q1612yes—
193C2×C3×C4#13512yes—
194C2×C3×C4#13512yes—
195C2×C3×C4AGL(1,ℤ/8)12yes—
196C2×C3×C4#13712yes—
197C2×C3×C4#13712yes—
198C2×Dic3—42no—
199C2×Dic3—42no—
200C2×Dic3—81no—
201C2×Dic3—81no—
202C2×Dic3—81no—
203C2×Dic3—81no—
204Dic6—44no—
205Dic6—44no—
206Dic6—82no—
207Dic6—82no—
208C3×C8—24no—
209C3⋊C8—42no—
210C3⋊C8—42no—
211C2×Q8—22no—
212C2×Q8—41no—
213C2×Q8—42no—
214C2×Q8—42no—
215C2×Q8—62no—
216C2×Q8—62no—
217C2×Q8—121no—
218C2×Q8—121no—
219C2×Q8—122no—
220C2×Q8—122no—
221C2×Q8—122no—
222C2×Q8—122no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—121no—
226C22×C4—121no—
227C22×C4D3×D411yescharacteristic
228C22×C4D3×D411yescharacteristic
229C22×C4C2×#6711yescharacteristic
230C2×C8—22no—
231C2×C8—22no—
232C2×C8—62no—
233C2×C8—62no—
234C2×C8—62no—
235C2×C8—62no—
236C2×C8—62no—
237C2×C8—62no—
238C2×C8#21614yes—
239C4⋊C4—22no—
240C4⋊C4—22no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—62no—
247C4⋊C4—62no—
248C4⋊C4—62no—
249C4⋊C4—62no—
250C4⋊C4—121no—
251C4⋊C4—121no—
252C4⋊C4—121no—
253C4⋊C4—121no—
254C42C2×D1211yescharacteristic
255C42#23511yescharacteristic
256C42D3×D411yescharacteristic
257C42D3×D411yescharacteristic
258C22×C3#39211yescharacteristic
259C22×C3#49411yescharacteristic
260C22×C3#49511yescharacteristic
261C22×C3#49511yescharacteristic
262C22×C3#49611yescharacteristic
263C22×C3#49711yescharacteristic
264C22×C3#49711yescharacteristic
265C3×C4—82no—
266C3×C4—161no—
267C3×C4#44912yes—
268C3×C4#45112yes—
269C3×C4#45212yes—
270C3×C4#46212yes—
271C3×C4#46312yes—
272C3×C4#46412yes—
273Dic3—82no—
274Dic3—82no—
275C23#94011yescharacteristic
276Q8—44no—
277Q8—82no—
278Q8—124no—
279Q8—124no—
280Q8—242no—
281Q8—242no—
282C2×C4—42no—
283C2×C4—81no—
284C2×C4—81no—
285C2×C4—81no—
286C2×C4—81no—
287C2×C4—81no—
288C2×C4—122no—
289C2×C4—122no—
290C2×C4—241no—
291C2×C4—241no—
292C2×C4—241no—
293C2×C4—241no—
294C2×C4#88211yescharacteristic
295C2×C4#88211yescharacteristic
296C2×C4#88411yescharacteristic
297C2×C4#88411yescharacteristic
298C2×C4#91212yes—
299C2×C4#91412yes—
300C2×C4#91512yes—
301C2×C4#91912yes—
302C2×C4#93011yescharacteristic
303C2×C4#93711yescharacteristic
304C2×C4C2×#21612yes—
305C2×C4#95112yes—
306C8—24no—
307C8—122no—
308C8—122no—
309C2×C3#171011yescharacteristic
310C2×C3#171811yescharacteristic
311C2×C3#171911yescharacteristic
312C2×C3#172311yescharacteristic
313C2×C3#201211yescharacteristic
314C2×C3#201511yescharacteristic
315C2×C3#201611yescharacteristic
316C22#496011yescharacteristic
317C22#506011yescharacteristic
318C22#509211yescharacteristic
319C22#515511yescharacteristic
320C22#516711yescharacteristic
321C22#517611yescharacteristic
322C22#519111yescharacteristic
323C4—82no—
324C4—161no—
325C4—242no—
326C4—242no—
327C4#476712yes—
328C4#478312yes—
329C4#481012yes—
330C4#481912yes—
331C4#503112yes—
332C4#505012yes—
333C3#1031311yescharacteristic, complemented, Sylow
334C2#6969111yescharacteristic
335C2#7021511yescharacteristic
336C2#7027711yescharacteristic
337C2#7119011yescharacteristic
338C2#7122211yescharacteristic
339C2#7134111yescharacteristic
340C2#7136111yescharacteristic
subgroupG/H
centerC23#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1990C2
socle, prime radicalC23×C3#127
hypercenter#1990D3
hypocenterC3#10313
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
333. C3#10313semidirect

Maximal subgroups

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

subgroupindex
C2×#50932
C2×#50792
C2×#46822
C3×#19902
#695822
#695792
C3⋊#12732
#103133