Finite Group Databasebeta

← #10691218

Subgroups — #10691218

Order 768.

624 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 624 conjugacy classes of subgroups — too many to draw. Only its 127 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#82286C211yesmaximal, characteristic
2#82204C211yesmaximal, characteristic, complemented
3C3×#1458C211yesmaximal, characteristic, complemented
4#70774C212yesmaximal, complemented
5#67738C212yesmaximal, complemented
6#9197—31nomaximal, Sylow
7C2×C3×#117C2211yescharacteristic
8C2×#930—21no—
9C22×#208—21no—
10C2×#874C2211yescharacteristic
11C3×#431C2212yes—
12#4999—24no—
13#4991—21no—
14#4991C412yes—
15#4960—21no—
16#4960C412yes—
17#4939—21no—
18#4936C2211yescharacteristic
19#4924—21no—
20#4924C2212yes—
21C3×#324—21no—
22C3×#323—21no—
23#4485—24no—
24#2905—31no—
25#2619—31no—
26#1545—32no—
27#1458D311yescharacteristic, complemented
28#1286—32no—
29C23×C3×C4D411yescharacteristic
30C22×#67—22no—
31C22×D12—21no—
32C22×D12—21no—
33C22×D12C2×C412yes—
34C3×#117—22no—
35C3×#117—22no—
36C2×C3×C4⋊C4C2×C411yescharacteristic
37C2×C3×C4⋊C4D411yescharacteristic
38C2×C3×C22⋊C4C2×C411yescharacteristic
39C2×C3×C22⋊C4D411yescharacteristic
40C2×C3×C42D411yescharacteristic
41C2×C3×C42C2311yescharacteristic
42#930—42no—
43#930—42no—
44C2×(C2×D3)⋊C4—22no—
45C2×(C2×D3)⋊C4—22no—
46C2×(C2×D3)⋊C4—22no—
47C2×C4⋊Dic3—21no—
48C2×#208—22no—
49C2×#208—22no—
50C2×#208—42no—
51C2×#208—42no—
52C22×C3⋊C8—22no—
53#874—22no—
54#874—22no—
55C3×C4⋊C8—22no—
56C3×C22⋊C8—22no—
57C3×C4×C8—22no—
58C3⋊#95—21no—
59C3⋊#95C2×C412yes—
60C4⋊(C3⋊C8)—24no—
61C4⋊(C3⋊C8)—22no—
62C4⋊(C3⋊C8)D414yescomplemented
63C22×M16—61no—
64C2×#127—31no—
65C2×#121—61no—
66C2×#117C2×D311yescharacteristic
67#431C2×D312yes—
68#424—64no—
69#423—32no—
70#423—61no—
71#394—32no—
72#394—61no—
73#392—32no—
74#392—61no—
75#390—61no—
76#382—31no—
77#331—64no—
78#324—21no—
79#323—21no—
80C24×C3C22⋊C411yescharacteristic
81C23×D3—22no—
82C22×C3×C4—21no—
83C22×C3×C4—21no—
84C22×C3×C4—21no—
85C22×C3×C4—21no—
86C22×C3×C4C22⋊C411yescharacteristic
87C22×C3×C4C22⋊C411yescharacteristic
88C22×C3×C4C22⋊C411yescharacteristic
89C22×C3×C4D812yes—
90C22×C3×C4SD1612yes—
91C22×C3×C4C22×C411yescharacteristic
92C22×C3×C4C2×D411yescharacteristic
93C22×C3×C4C2×D411yescharacteristic
94C2×#67—44no—
95C2×#67—82no—
96C2×#67—82no—
97C22×Dic3—22no—
98C2×D12—24no—
99C2×D12—41no—
100C2×D12—42no—
101C2×D12—41no—
102C2×D12—41no—
103C2×D12—41no—
104C2×D12—41no—
105C2×D12—41no—
106C2×D12—82no—
107C2×D12—82no—
108C2×C3×C8—42no—
109C3×C4⋊C4—22no—
110C3×C4⋊C4—22no—
111C3×C22⋊C4—22no—
112C3×C22⋊C4—22no—
113C3×C42—21no—
114C3×C42—21no—
115C3×C42C2×D412yes—
116C3×C42#4112yes—
117(C2×D3)⋊C4—44no—
118(C2×D3)⋊C4—82no—
119(C2×D3)⋊C4—82no—
120C4⋊Dic3—41no—
121C4⋊Dic3—41no—
122#208—44no—
123#208—42no—
124#208—42no—
125C2×C3⋊C8—24no—
126C2×C3⋊C8—44no—
127C22×D4—32no—
128C22×D4—62no—
129C22×D4—61no—
130C22×D4—61no—
131C23×C4D1211yescharacteristic
132C2×M16—62no—
133C2×M16—62no—
134C2×M16—122no—
135C2×M16—122no—
136C22×C8—62no—
137#127—62no—
138#127—62no—
139#121—122no—
140#121—122no—
141#117—22no—
142#117—22no—
143C2×C4⋊C4—61no—
144C2×C4⋊C4C4×D311yescharacteristic
145C2×C4⋊C4#6711yescharacteristic
146C2×C22⋊C4—62no—
147C2×C22⋊C4—62no—
148C2×C22⋊C4—62no—
149C2×C22⋊C4C4×D311yescharacteristic
150C2×C22⋊C4#6711yescharacteristic
151C2×C42D1211yescharacteristic
152C2×C42C22×D311yescharacteristic
153C4⋊C8—22no—
154C4⋊C8—34no—
155C4⋊C8—64no—
156C4⋊C8—62no—
157C22⋊C8—22no—
158C4×C8—22no—
159#95—32no—
160#95—61no—
161C23×C3—21no—
162C23×C3—21no—
163C23×C3—41no—
164C23×C3—41no—
165C23×C3#10212yes—
166C23×C3C2×C22⋊C411yescharacteristic
167C22×D3—42no—
168C22×D3—42no—
169C22×D3—42no—
170C22×D3—42no—
171C22×D3—42no—
172C22×D3—42no—
173C22×D3—42no—
174C2×C3×C4—21no—
175C2×C3×C4—21no—
176C2×C3×C4—22no—
177C2×C3×C4—22no—
178C2×C3×C4—22no—
179C2×C3×C4—22no—
180C2×C3×C4—21no—
181C2×C3×C4—21no—
182C2×C3×C4—41no—
183C2×C3×C4—41no—
184C2×C3×C4—41no—
185C2×C3×C4—41no—
186C2×C3×C4#10212yes—
187C2×C3×C4C4×D412yes—
188C2×C3×C4#12011yescharacteristic
189C2×C3×C4#12112yes—
190C2×C3×C4#12311yescharacteristic
191C2×C3×C4C2×D811yescharacteristic
192C2×C3×C4C2×SD1611yescharacteristic
193C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
194C2×C3×C4#13711yescharacteristic
195#67—88no—
196C2×Dic3—42no—
197C2×Dic3—42no—
198C2×Dic3—42no—
199D12—84no—
200D12—84no—
201D12—81no—
202D12—81no—
203D12—81no—
204D12—81no—
205C3×C8—82no—
206C3⋊C8—44no—
207C24—62no—
208C24(C2×D3)⋊C411yescharacteristic
209C2×D4—64no—
210C2×D4—124no—
211C2×D4—121no—
212C2×D4—121no—
213C2×D4—121no—
214C2×D4—121no—
215C2×D4—121no—
216C2×D4—121no—
217C2×D4—122no—
218C2×D4—242no—
219C2×D4—242no—
220C2×D4—242no—
221C2×D4—242no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—62no—
227C22×C4#20412yes—
228C22×C4D2412yes—
229C22×C4(C2×D3)⋊C411yescharacteristic
230C22×C4(C2×D3)⋊C411yescharacteristic
231C22×C4(C2×D3)⋊C411yescharacteristic
232C22×C4C2×C4×D311yescharacteristic
233C22×C4C2×D1211yescharacteristic
234C22×C4C2×#6711yescharacteristic
235M16—124no—
236M16—122no—
237M16—122no—
238C2×C8—42no—
239C2×C8—64no—
240C2×C8—124no—
241C4⋊C4—22no—
242C4⋊C4—22no—
243C4⋊C4—121no—
244C4⋊C4—121no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—124no—
248C22⋊C4—242no—
249C22⋊C4—242no—
250C42—21no—
251C42—21no—
252C42D3×D412yes—
253C42#23712yes—
254C22×C3—21no—
255C22×C3—21no—
256C22×C3—22no—
257C22×C3—42no—
258C22×C3—41no—
259C22×C3—41no—
260C22×C3—81no—
261C22×C3#38611yescharacteristic
262C22×C3C2×#10211yescharacteristic
263C22×C3#41711yescharacteristic
264C2×D3—82no—
265C2×D3—82no—
266C2×D3—82no—
267C2×D3—82no—
268C2×D3—82no—
269C2×D3—82no—
270C2×D3—82no—
271C3×C4—22no—
272C3×C4—22no—
273C3×C4—22no—
274C3×C4—41no—
275C3×C4—41no—
276C3×C4#44711yescharacteristic
277C3×C4#44811yescharacteristic
278C3×C4#48011yescharacteristic
279C3×C4#48111yescharacteristic
280Dic3—82no—
281C23—21no—
282C23—21no—
283C23—41no—
284C23—41no—
285C23—122no—
286C23—122no—
287C23—122no—
288C23—122no—
289C23—122no—
290C23—122no—
291C23—122no—
292C23#82112yes—
293C23C2×(C2×D3)⋊C411yescharacteristic
294D4—244no—
295D4—244no—
296D4—248no—
297D4—241no—
298D4—241no—
299D4—241no—
300D4—241no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—22no—
306C2×C4—22no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—41no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—122no—
314C2×C4—122no—
315C2×C4—122no—
316C2×C4#82112yes—
317C2×C4#67⋊C412yes—
318C2×C4#88211yescharacteristic
319C2×C4#88611yescharacteristic
320C2×C4C2×#20411yescharacteristic
321C2×C4C2×D2411yescharacteristic
322C2×C4#90811yescharacteristic
323C2×C4#90911yescharacteristic
324C2×C4#93912yes—
325C8—82no—
326C8—124no—
327C2×C3—21no—
328C2×C3—21no—
329C2×C3—42no—
330C2×C3#187411yescharacteristic
331C2×C3#191011yescharacteristic
332C2×C3#191411yescharacteristic
333D3—162no—
334C22—21no—
335C22—21no—
336C22—22no—
337C22—42no—
338C22—41no—
339C22—41no—
340C22—81no—
341C22—242no—
342C22—242no—
343C22—242no—
344C22—242no—
345C22—242no—
346C22—242no—
347C22—242no—
348C22#495811yescharacteristic
349C22C2×#82111yescharacteristic
350C22#513311yescharacteristic
351C4—22no—
352C4—22no—
353C4—22no—
354C4—41no—
355C4—41no—
356C4—242no—
357C4#473211yescharacteristic
358C4#473511yescharacteristic
359C4#473611yescharacteristic
360C4#473811yescharacteristic
361C3#919711yescharacteristic, complemented, Sylow
362C2—21no—
363C2—21no—
364C2—42no—
365C2—482no—
366C2#6993211yescharacteristic
367C2#7004711yescharacteristic
368C2#7013711yescharacteristic
subgroupG/H
centerC22#4958
commutatorC22×C3×C4C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1458C2
socleC22×C3#386
prime radical#82286C2
cosocleC2×C3×C42C23
hypercenter#1458D3
hypocenterC3#9197
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1458

Complements

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

NH ≅ G/Ntype
2. #82204C2semidirect
3. C3×#1458C2semidirect
4. #70774C2semidirect
5. #67738C2semidirect
27. #1458D3semidirect
62. C4⋊(C3⋊C8)D4semidirect
361. C3#9197semidirect

Maximal subgroups

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

subgroupindex
#822862
#822042
C3×#14582
#707742
#677382
#91973