Finite Group Databasebeta

← #10691459

Subgroups — #10691459

Order 768.

604 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C42×Q8C211yesmaximal, characteristic, complemented
2#82442C211yesmaximal, characteristic
3#71622C211yesmaximal, characteristic, complemented
4#69469C211yesmaximal, characteristic
5#69395C211yesmaximal, characteristic
6#67620C211yesmaximal, characteristic, complemented
7#67611C211yesmaximal, characteristic, complemented
8#9234—31nomaximal, Sylow
9C2×C3×C4×Q8—21no—
10C2×C3×C4×Q8C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×C4×C4⋊C4—21no—
13C3×C4×C4⋊C4—21no—
14C3×C4×C4⋊C4—22no—
15C3×C4×C4⋊C4C2211yescharacteristic
16C3×C43—21no—
17C3×C43C2211yescharacteristic
18#5227—21no—
19#5027C412yescomplemented
20C4×#215—22no—
21#5020C412yescomplemented
22#4968—21no—
23C2×#805—21no—
24C2×#805—21no—
25#4925C2211yescharacteristic
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4535—22no—
29#4451—22no—
30#3103—31no—
31C42×Q8D311yescharacteristic, complemented
32#1962—31no—
33#1502—31no—
34#1487—31no—
35#1303—31no—
36#1276—31no—
37C22×C3×Q8D411yescharacteristic
38C2×#235—21no—
39C3×C4×Q8—28no—
40C3×C4×Q8—21no—
41C3×C4×Q8—21no—
42C3×C4×Q8C2×C412yes—
43C3×C4×Q8D412yes—
44C2×C3×C4⋊C4—22no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C42—22no—
49C2×C3×C42—21no—
50C2×C3×C42—21no—
51C2×C3×C42—22no—
52C2×C3×C42—21no—
53C2×C3×C42—21no—
54C2×C3×C42D411yescharacteristic
55C2×C3×C42D411yescharacteristic
56C2×C3×C42C2311yescharacteristic
57C2×#215—22no—
58#930—21no—
59#930—21no—
60C2×#208—21no—
61#875—21no—
62#874C2×C411yescharacteristic
63C4×D12—21no—
64C4×D12—21no—
65#869C2×C411yescharacteristic
66C3×#95—22no—
67C3×#95D412yes—
68#835—22no—
69#830—21no—
70#830—21no—
71#809—22no—
72#808—22no—
73#805—42no—
74#805—42no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)C2×C412yes—
79C4×C3⋊C8—21no—
80C4×C3⋊C8—21no—
81#532—31no—
82C2×C4×Q8—21no—
83C2×C4×Q8C2×D311yescharacteristic
84#481—61no—
85#475—32no—
86#460—32no—
87#448—61no—
88C4×SD16—62no—
89#431—31no—
90#431—31no—
91#423—31no—
92C2×#104—61no—
93C2×#104—61no—
94C4×C4⋊C4—21no—
95C4×C4⋊C4—21no—
96C4×C4⋊C4—22no—
97C4×C4⋊C4C2×D311yescharacteristic
98C43—21no—
99C43C2×D311yescharacteristic
100#327—62no—
101Q8⋊C8—62no—
102C2×C3×Q8—24no—
103C2×C3×Q8—21no—
104C2×C3×Q8C22⋊C412yes—
105C22×C3×C4—21no—
106C22×C3×C4—21no—
107C22×C3×C4—21no—
108C22×C3×C4—21no—
109C22×C3×C4—22no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C2×#67—41no—
114#235—41no—
115#235—41no—
116#235—41no—
117#235—41no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12—21no—
121C2×C4×D3—41no—
122C2×Dic6—21no—
123C3×C4⋊C4—28no—
124C3×C4⋊C4—24no—
125C3×C4⋊C4—22no—
126C3×C4⋊C4—22no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4C22⋊C412yes—
129C3×C42—28no—
130C3×C42—24no—
131C3×C42—22no—
132C3×C42—22no—
133C3×C42—21no—
134C3×C42—21no—
135C3×C42—21no—
136C3×C42—42no—
137C3×C42—44no—
138C3×C42—41no—
139C3×C42—41no—
140C3×C42SD1612yescomplemented
141C3×C42C22×C411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144C3×C42#4111yescharacteristic
145#215—44no—
146(C2×D3)⋊C4—41no—
147(C2×D3)⋊C4—41no—
148C4⋊Dic3—21no—
149C4⋊Dic3—21no—
150#208—41no—
151#208—41no—
152C2×C3⋊C8—22no—
153C2×C3⋊C8—41no—
154C2×C3⋊C8—41no—
155C2×#41—61no—
156C22×Q8#6711yescharacteristic
157C2×SD16—62no—
158C2×M16—61no—
159C4⋊Q8—31no—
160#127—31no—
161#124—61no—
162#121—61no—
163#121—61no—
164C4×Q8—28no—
165C4×Q8—21no—
166C4×Q8—21no—
167C4×Q8C4×D312yes—
168C4×Q8D1212yes—
169C4×D4—61no—
170C4×D4—61no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4—21no—
173C2×C4⋊C4—21no—
174C2×C4⋊C4#6711yescharacteristic
175C2×C42—22no—
176C2×C42—21no—
177C2×C42—21no—
178C2×C42—22no—
179C2×C42—21no—
180C2×C42—21no—
181C2×C42#6711yescharacteristic
182C2×C42#6711yescharacteristic
183C2×C42C22×D311yescharacteristic
184#106—62no—
185C4⋊C8—32no—
186C4⋊C8—61no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189#104—122no—
190#104—122no—
191#103—62no—
192#102—62no—
193C22⋊C8—61no—
194C22⋊C8—61no—
195C4×C8—61no—
196C4×C8—61no—
197#95—22no—
198#95#6712yes—
199C23×C3#12011yescharacteristic
200C22×D3—41no—
201C3×Q8—24no—
202C3×Q8—22no—
203C3×Q8#10414yescomplemented
204C2×C3×C4—22no—
205C2×C3×C4—24no—
206C2×C3×C4—22no—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—24no—
212C2×C3×C4—22no—
213C2×C3×C4—41no—
214C2×C3×C4—41no—
215C2×C3×C4—42no—
216C2×C3×C4—42no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4C2×C22⋊C411yescharacteristic
220C2×C3×C4C4×D411yescharacteristic
221C2×C3×C4C4×D411yescharacteristic
222C2×C3×C4#12111yescharacteristic
223C2×C3×C4#12111yescharacteristic
224C2×C3×C4#12311yescharacteristic
225C2×C3×C4C2×SD1611yescharacteristic
226C2×C3×C4#13511yescharacteristic
227C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
228C2×C3×C4#13711yescharacteristic
229#67—81no—
230#67—81no—
231C2×Dic3—41no—
232D12—42no—
233D12—41no—
234D12—41no—
235D12—81no—
236C4×D3—81no—
237C4×D3—81no—
238Dic6—41no—
239Dic6—41no—
240C3⋊C8—42no—
241C3⋊C8—81no—
242C3⋊C8—81no—
243#41—121no—
244#41—121no—
245#41—121no—
246#41—121no—
247C2×Q8—24no—
248C2×Q8—21no—
249C2×Q8—61no—
250C2×Q8(C2×D3)⋊C412yes—
251C2×D4—61no—
252C2×D4—61no—
253C2×D4—61no—
254C2×D4—121no—
255C22×C4—21no—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—22no—
260C22×C4—121no—
261C22×C4C2×#6711yescharacteristic
262C22×C4C2×#6711yescharacteristic
263C22×C4C2×#6711yescharacteristic
264SD16—124no—
265M16—121no—
266M16—121no—
267C2×C8—62no—
268C2×C8—121no—
269C2×C8—121no—
270C4⋊C4—28no—
271C4⋊C4—24no—
272C4⋊C4—22no—
273C4⋊C4—21no—
274C4⋊C4—22no—
275C4⋊C4—61no—
276C4⋊C4—61no—
277C4⋊C4(C2×D3)⋊C412yes—
278C22⋊C4—121no—
279C22⋊C4—121no—
280C42—28no—
281C42—24no—
282C42—22no—
283C42—22no—
284C42—21no—
285C42—21no—
286C42—21no—
287C42—42no—
288C42—44no—
289C42—41no—
290C42—41no—
291C42#21512yescomplemented
292C42C2×C4×D311yescharacteristic
293C42C2×D1211yescharacteristic
294C42#23511yescharacteristic
295C42#23511yescharacteristic
296C22×C3—21no—
297C22×C3—21no—
298C22×C3#38611yescharacteristic
299C22×C3#44811yescharacteristic
300C22×C3#44911yescharacteristic
301C2×D3—81no—
302C2×D3—81no—
303C2×D3—81no—
304C3×C4—24no—
305C3×C4—24no—
306C3×C4—21no—
307C3×C4—21no—
308C3×C4—21no—
309C3×C4—42no—
310C3×C4—44no—
311C3×C4—42no—
312C3×C4C2×#10412yes—
313C3×C4C4×SD1611yescharacteristic
314C3×C4#44011yescharacteristic
315C3×C4#46011yescharacteristic
316C3×C4#46411yescharacteristic
317Dic3—81no—
318C23—121no—
319C23#95311yescharacteristic
320Q8—24no—
321Q8—22no—
322Q8—121no—
323Q8—121no—
324Q8#80514yescomplemented
325D4—122no—
326D4—121no—
327D4—121no—
328D4—241no—
329D4—241no—
330D4—241no—
331C2×C4—22no—
332C2×C4—24no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—24no—
339C2×C4—22no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—42no—
346C2×C4—121no—
347C2×C4—241no—
348C2×C4—241no—
349C2×C4C2×(C2×D3)⋊C411yescharacteristic
350C2×C4C4×#6711yescharacteristic
351C2×C4C4×#6711yescharacteristic
352C2×C4#92911yescharacteristic
353C2×C4#93011yescharacteristic
354C2×C4#93011yescharacteristic
355C2×C4C2×#21511yescharacteristic
356C2×C4#94211yescharacteristic
357C2×C4#94911yescharacteristic
358C2×C4#95011yescharacteristic
359C8—122no—
360C8—241no—
361C8—241no—
362C2×C3—21no—
363C2×C3—21no—
364C2×C3#188011yescharacteristic
365C2×C3#189611yescharacteristic
366C2×C3#190611yescharacteristic
367D3—161no—
368C22—21no—
369C22—21no—
370C22—241no—
371C22—241no—
372C22—241no—
373C22#521311yescharacteristic
374C22#522711yescharacteristic
375C22#523811yescharacteristic
376C4—24no—
377C4—24no—
378C4—21no—
379C4—21no—
380C4—21no—
381C4—42no—
382C4—44no—
383C4—42no—
384C4—241no—
385C4C2×#80512yes—
386C4C4×#21511yescharacteristic
387C4#502611yescharacteristic
388C4#502711yescharacteristic
389C4#502811yescharacteristic
390C3#923411yescharacteristic, complemented, Sylow
391C2—21no—
392C2—21no—
393C2—481no—
394C2#7153211yescharacteristic
395C2#7163311yescharacteristic
396C2#7164511yescharacteristic
subgroupG/H
centerC2×C4#930
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×C42×Q8C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenterC42×Q8D3
hypocenterC3#9234
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C42×Q8

Complements

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

NH ≅ G/Ntype
1. C3×C42×Q8C2semidirect
3. #71622C2semidirect
6. #67620C2semidirect
7. #67611C2semidirect
19. #5027C4semidirect
21. #5020C4semidirect
31. C42×Q8D3semidirect
140. C3×C42SD16semidirect
203. C3×Q8#104semidirect
291. C42#215semidirect
324. Q8#805semidirect
390. C3#9234semidirect

Maximal subgroups

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

subgroupindex
C3×C42×Q82
#824422
#716222
#694692
#693952
#676202
#676112
#92343