Finite Group Databasebeta

← #10683228

Subgroups — #10683228

Order 768.

510 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#76480C211yesmaximal, characteristic
2#68607C211yesmaximal, characteristic, complemented
3#68602C211yesmaximal, characteristic, complemented
4C3×#1272C211yesmaximal, characteristic, complemented
5#67642C211yesmaximal, characteristic, complemented
6#67479C211yesmaximal, characteristic
7#67479C211yesmaximal, characteristic
8#7584—31nomaximal, Sylow
9C2×#947C411yescharacteristic
10C2×Q8×Dic3C411yescharacteristic
11C2×#893C411yescharacteristic
12C3×#391C2211yescharacteristic
13C2×Dic3⋊C8—21no—
14C2×C8×Dic3—21no—
15#4977C411yescharacteristic
16#4939C2211yescharacteristic
17C42×Dic3C2211yescharacteristic
18#4670—21no—
19#4670—21no—
20#4666—21no—
21#4666—21no—
22C3×#336C2211yescharacteristic
23C3×#336C2211yescharacteristic
24#4550—21no—
25#4550—21no—
26#4524C2211yescharacteristic
27#4524C2211yescharacteristic
28C4×#124—31no—
29#1839—31no—
30C4×C4⋊C8—31no—
31#1272—31no—
32#1272D311yescharacteristic, complemented
33#1270—31no—
34#1270—31no—
35C22×C3×Q8C2×C411yescharacteristic
36C22×D12C2×C411yescharacteristic
37C22×C4×D3—21no—
38C22×C3×C8—21no—
39C22×C3×C8—21no—
40C2×C3×C4⋊C4C2×C411yescharacteristic
41C2×C3×C42C2311yescharacteristic
42#947—22no—
43#947C814yescomplemented
44Q8×Dic3—42no—
45C2×(C2×D3)⋊C4—21no—
46C2×(C2×D3)⋊C4—21no—
47C2×(C2×D3)⋊C4—21no—
48C2×(C2×D3)⋊C4—21no—
49C2×(C2×D3)⋊C4—21no—
50C2×C4⋊Dic3—21no—
51C2×C4⋊Dic3C2×C411yescharacteristic
52C2×C4×Dic3—21no—
53C2×C4×Dic3C2×C411yescharacteristic
54C2×C4×Dic3C2×C411yescharacteristic
55C2×C4×Dic3D411yescharacteristic
56C2×C4×Dic3D411yescharacteristic
57C2×C4×Dic3D411yescharacteristic
58C2×C4×Dic3D411yescharacteristic
59C22×C3⋊C8—21no—
60C22×C3⋊C8—21no—
61#893—42no—
62C3×#95—21no—
63C3×#95—21no—
64C3⋊#95—21no—
65C3⋊#95—21no—
66Dic3⋊C8—24no—
67C8×Dic3—42no—
68C2×#124—31no—
69C2×C4×Q8—31no—
70C2×C4×D4—31no—
71C2×C4⋊C8—61no—
72C2×C4×C8—61no—
73#391C2×D311yescharacteristic
74#390—31no—
75#388—61no—
76#388—61no—
77#382—31no—
78C4×C22⋊C4—61no—
79C4×C22⋊C4—61no—
80C43—31no—
81#336—31no—
82#336—31no—
83#336—61no—
84#336—61no—
85#336C2×D311yescharacteristic
86#336C2×D311yescharacteristic
87C23×D3—21no—
88C2×C3×Q8—21no—
89C2×C3×Q8—41no—
90C2×C3×Q8—41no—
91C2×C3×Q8C2×C812yes—
92C22×C3×C4—21no—
93C22×C3×C4—21no—
94C22×C3×C4C22×C411yescharacteristic
95C22×C3×C4C2×D411yescharacteristic
96C22×C3×C4C2×D411yescharacteristic
97C22×Dic3—21no—
98C22×Dic3C22⋊C411yescharacteristic
99C22×Dic3C22⋊C411yescharacteristic
100C22×Dic3C22⋊C411yescharacteristic
101C22×Dic3C22⋊C411yescharacteristic
102C2×D12—21no—
103C2×D12—41no—
104C2×D12—41no—
105C2×D12C2×C812yes—
106C2×C4×D3—42no—
107C2×C4×D3—42no—
108C2×C4×D3—42no—
109C2×C3×C8—22no—
110C2×C3×C8—41no—
111C2×C3×C8—41no—
112C2×C3×C8—41no—
113C2×C3×C8—41no—
114C2×C3×C8—41no—
115C3×C4⋊C4—41no—
116C3×C42—41no—
117(C2×D3)⋊C4—24no—
118(C2×D3)⋊C4—24no—
119(C2×D3)⋊C4—41no—
120(C2×D3)⋊C4—41no—
121(C2×D3)⋊C4—41no—
122(C2×D3)⋊C4—41no—
123(C2×D3)⋊C4—42no—
124C4⋊Dic3—42no—
125C4⋊Dic3—41no—
126C4×Dic3—21no—
127C4×Dic3—21no—
128C4×Dic3—21no—
129C4×Dic3—21no—
130C4×Dic3—22no—
131C4×Dic3—22no—
132C4×Dic3—41no—
133C4×Dic3—42no—
134C4×Dic3C2×C812yes—
135C4×Dic3M1612yes—
136C2×C3⋊C8—22no—
137C2×C3⋊C8—41no—
138C2×C3⋊C8—41no—
139C2×C3⋊C8—41no—
140C2×C3⋊C8—41no—
141C2×C3⋊C8—41no—
142C22×Q8C4×D311yescharacteristic
143C22×D4—31no—
144C23×C4—61no—
145C22×C8—21no—
146C22×C8—21no—
147C22×C8—61no—
148C22×C8—61no—
149#124—34no—
150#124—62no—
151C4×Q8—122no—
152C4×D4—122no—
153C2×C4⋊C4—31no—
154C2×C4⋊C4—61no—
155C2×C4⋊C4C4×D311yescharacteristic
156C2×C22⋊C4—61no—
157C2×C22⋊C4—61no—
158C2×C22⋊C4—61no—
159C2×C22⋊C4—61no—
160C2×C22⋊C4—61no—
161C2×C42—31no—
162C2×C42—31no—
163C2×C42—31no—
164C2×C42—31no—
165C2×C42—31no—
166C2×C42—31no—
167C2×C42—61no—
168C2×C42C22×D311yescharacteristic
169C4⋊C8—64no—
170C4×C8—122no—
171#95—21no—
172#95—21no—
173#95—61no—
174#95—61no—
175C23×C3C2×C22⋊C411yescharacteristic
176C22×D3—22no—
177C22×D3—41no—
178C22×D3—41no—
179C22×D3—41no—
180C22×D3—41no—
181C22×D3—41no—
182C22×D3—41no—
183C3×Q8—42no—
184C3×Q8—42no—
185C2×C3×C4—21no—
186C2×C3×C4—21no—
187C2×C3×C4—21no—
188C2×C3×C4—21no—
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—22no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4C22×C811yescharacteristic
198C2×C3×C4C2×M1611yescharacteristic
199C2×C3×C4#13511yescharacteristic
200C2×C3×C4#13511yescharacteristic
201C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
202C2×C3×C4#13711yescharacteristic
203C2×Dic3—21no—
204C2×Dic3—21no—
205C2×Dic3—21no—
206C2×Dic3—21no—
207C2×Dic3—21no—
208C2×Dic3—21no—
209C2×Dic3—21no—
210C2×Dic3—21no—
211C2×Dic3—41no—
212C2×Dic3—41no—
213C2×Dic3—41no—
214C2×Dic3C22⋊C812yes—
215C2×Dic3C22⋊C812yes—
216C2×Dic3#10012yes—
217C2×Dic3#10412yes—
218D12—42no—
219D12—42no—
220C4×D3—84no—
221C3×C8—42no—
222C3×C8—81no—
223C3⋊C8—42no—
224C3⋊C8—81no—
225C24—61no—
226C2×Q8—21no—
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8D3×C812yes—
230C2×D4—32no—
231C2×D4—61no—
232C2×D4—121no—
233C2×D4—121no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—31no—
237C22×C4—31no—
238C22×C4—31no—
239C22×C4—31no—
240C22×C4—61no—
241C22×C4—122no—
242C22×C4—122no—
243C22×C4—122no—
244C22×C4C2×C4×D311yescharacteristic
245C22×C4D3×D411yescharacteristic
246C22×C4D3×D411yescharacteristic
247C2×C8—22no—
248C2×C8—41no—
249C2×C8—41no—
250C2×C8—41no—
251C2×C8—41no—
252C2×C8—41no—
253C2×C8—62no—
254C2×C8—121no—
255C2×C8—121no—
256C2×C8—121no—
257C2×C8—121no—
258C2×C8—121no—
259C4⋊C4—41no—
260C4⋊C4—121no—
261C4⋊C4—122no—
262C22⋊C4—64no—
263C22⋊C4—64no—
264C22⋊C4—122no—
265C22⋊C4—121no—
266C22⋊C4—121no—
267C22⋊C4—121no—
268C22⋊C4—121no—
269C42—32no—
270C42—32no—
271C42—41no—
272C42—62no—
273C42—62no—
274C42—61no—
275C42—61no—
276C42—61no—
277C42—61no—
278C42—121no—
279C42—122no—
280C22×C3C2×C22⋊C811yescharacteristic
281C22×C3#41011yescharacteristic
282C22×C3C2×#10011yescharacteristic
283C22×C3#41611yescharacteristic
284C22×C3#41711yescharacteristic
285C22×C3C2×#10411yescharacteristic
286C22×C3#42111yescharacteristic
287C2×D3—42no—
288C2×D3—42no—
289C2×D3—42no—
290C2×D3—81no—
291C2×D3—81no—
292C2×D3—81no—
293C2×D3—81no—
294C3×C4—21no—
295C3×C4—21no—
296C3×C4—21no—
297C3×C4—21no—
298C3×C4—21no—
299C3×C4—21no—
300C3×C4—42no—
301C3×C4—81no—
302Dic3—22no—
303Dic3—22no—
304Dic3—22no—
305Dic3—22no—
306Dic3—81no—
307C23—62no—
308C23—121no—
309C23—121no—
310C23—121no—
311C23—121no—
312C23—121no—
313C23—121no—
314C23D3×C22⋊C411yescharacteristic
315Q8—42no—
316Q8—42no—
317D4—122no—
318D4—122no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—22no—
326C2×C4—32no—
327C2×C4—32no—
328C2×C4—32no—
329C2×C4—32no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—61no—
336C2×C4—61no—
337C2×C4—61no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—61no—
341C2×C4—61no—
342C2×C4—61no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4—244no—
347C2×C4C2×D3×C811yescharacteristic
348C2×C4D3×M1611yescharacteristic
349C2×C4#91411yescharacteristic
350C2×C4#91611yescharacteristic
351C2×C4#91811yescharacteristic
352C2×C4#91911yescharacteristic
353C8—42no—
354C8—81no—
355C8—122no—
356C8—241no—
357C2×C3#146011yescharacteristic
358C2×C3#147611yescharacteristic
359C2×C3#147711yescharacteristic
360C2×C3#150911yescharacteristic
361C2×C3#152111yescharacteristic
362C2×C3#152711yescharacteristic
363C2×C3#155111yescharacteristic
364D3—82no—
365C22—122no—
366C22—122no—
367C22—122no—
368C22—241no—
369C22—241no—
370C22—241no—
371C22—241no—
372C22D3×C22⋊C811yescharacteristic
373C22#474011yescharacteristic
374C22D3×#10011yescharacteristic
375C22#480311yescharacteristic
376C22#480411yescharacteristic
377C22D3×#10411yescharacteristic
378C22#482111yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—21no—
385C4—42no—
386C4—62no—
387C4—62no—
388C4—62no—
389C4—62no—
390C4—81no—
391C4—241no—
392C3#758411yescharacteristic, complemented, Sylow
393C2—242no—
394C2#6810111yescharacteristic
395C2#6822911yescharacteristic
396C2#6823011yescharacteristic
397C2#6831111yescharacteristic
398C2#6831511yescharacteristic
399C2#6834311yescharacteristic
400C2#6848511yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC2×C3×C4C22×C8
FrattiniC2×C42C22×D3
FittingC3×#1272C2
socleC23×C3C2×C22⋊C4
prime radicalC22×D12C2×C4
cosocleC2×C3×C42C23
hypercenter#1272D3
hypocenterC3#7584
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1272

Complements

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

NH ≅ G/Ntype
2. #68607C2semidirect
3. #68602C2semidirect
4. C3×#1272C2semidirect
5. #67642C2semidirect
32. #1272D3semidirect
43. #947C8semidirect
392. C3#7584semidirect

Maximal subgroups

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

subgroupindex
#764802
#686072
#686022
C3×#12722
#676422
#674792
#75843