Finite Group Databasebeta

← #10691921

Subgroups — #10691921

Order 768.

508 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2327C211yesmaximal, characteristic, complemented
2#83215C211yesmaximal, characteristic
3#71773C211yesmaximal, characteristic, complemented
4#69545C211yesmaximal, characteristic
5#69440C211yesmaximal, characteristic
6#67631C211yesmaximal, characteristic, complemented
7#67614C211yesmaximal, characteristic, complemented
8#9300—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×C3×C4×Q8—21no—
11#5477C2211yescharacteristic
12C3×#391—21no—
13C3×#387—22no—
14C3×#387—22no—
15C3×#382C2211yescharacteristic
16C3×#376C2211yescharacteristic
17#5088C412yescomplemented
18#5087—22no—
19#5078C412yescomplemented
20#4967—22no—
21C2×#805—21no—
22C2×#805—21no—
23#4925C2211yescharacteristic
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26#4547—22no—
27#4451—22no—
28#3226—31no—
29#2327D311yescharacteristic, complemented
30#1963—31no—
31#1546—31no—
32#1517—31no—
33#1286—31no—
34#1274—31no—
35C22×C3×Q8—21no—
36C2×#235—21no—
37C3×C4⋊Q8—21no—
38C3×C4⋊Q8—21no—
39C3×C4⋊Q8C2×C412yes—
40C3×C4⋊Q8D412yes—
41C3×C4×Q8—42no—
42C3×C4×Q8—42no—
43C2×C3×C4⋊C4—21no—
44C2×C3×C4⋊C4—22no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4D412yes—
47C2×C3×C42—22no—
48C2×C3×C42D411yescharacteristic
49C2×C3×C42D411yescharacteristic
50C2×C3×C42C2311yescharacteristic
51#930—21no—
52#930—21no—
53C2×#208—21no—
54#875—21no—
55#874C2×C411yescharacteristic
56C4×D12—21no—
57C4×D12—21no—
58#869C2×C411yescharacteristic
59C3×#95—24no—
60C3×#95—21no—
61C3×#95—21no—
62C3×#95—21no—
63C3×#95D412yes—
64#830—21no—
65#830—21no—
66#809—24no—
67#807—24no—
68#805—42no—
69#805—42no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8C2×C412yes—
76#532—31no—
77C2×C4⋊Q8C2×D311yescharacteristic
78C2×C4×Q8—21no—
79#500—32no—
80#486—32no—
81#480—62no—
82#474—62no—
83#431—31no—
84#431—31no—
85#423—31no—
86C2×#104—61no—
87C2×#104—61no—
88#391—21no—
89#387—22no—
90#387—22no—
91#382C2×D311yescharacteristic
92#376C2×D311yescharacteristic
93#332—62no—
94#327—62no—
95C2×C3×Q8—22no—
96C2×C3×Q8—22no—
97C2×C3×Q8—41no—
98C2×C3×Q8—41no—
99C2×C3×Q8—41no—
100C2×C3×Q8—41no—
101C22×C3×C4—21no—
102C22×C3×C4—22no—
103C22×C3×C4—21no—
104C22×C3×C4—22no—
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4C2×D411yescharacteristic
108C2×#67—41no—
109#235—41no—
110#235—41no—
111#235—41no—
112#235—41no—
113C2×D12—21no—
114C2×D12—21no—
115C2×D12—21no—
116C2×C4×D3—41no—
117C2×Dic6—21no—
118C3×C4⋊C4—22no—
119C3×C4⋊C4—22no—
120C3×C4⋊C4—22no—
121C3×C4⋊C4—44no—
122C3×C4⋊C4—41no—
123C3×C4⋊C4—41no—
124C3×C4⋊C4C22⋊C414yes—
125C3×C42—21no—
126C3×C42—21no—
127C3×C42—21no—
128C3×C42—44no—
129C3×C42D812yes—
130C3×C42C22×C411yescharacteristic
131C3×C42C2×D411yescharacteristic
132C3×C42#4111yescharacteristic
133C3×C42#4111yescharacteristic
134(C2×D3)⋊C4—41no—
135(C2×D3)⋊C4—41no—
136C4⋊Dic3—21no—
137C4⋊Dic3—21no—
138#208—41no—
139#208—41no—
140C2×C3⋊C8—22no—
141C2×C3⋊C8—41no—
142C2×C3⋊C8—41no—
143C2×#41—61no—
144C22×Q8—21no—
145C2×M16—61no—
146C4⋊Q8—21no—
147C4⋊Q8—21no—
148C4⋊Q8—31no—
149C4⋊Q8C4×D312yes—
150C4⋊Q8D1212yes—
151#127—31no—
152#124—61no—
153#121—61no—
154#121—61no—
155C4×Q8—42no—
156C4×Q8—42no—
157C4×D4—61no—
158C4×D4—61no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—22no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4#6712yes—
163C2×C42—22no—
164C2×C42#6711yescharacteristic
165C2×C42#6711yescharacteristic
166C2×C42C22×D311yescharacteristic
167#107—64no—
168C4⋊C8—61no—
169C4⋊C8—61no—
170C4⋊C8—61no—
171C4⋊C8—61no—
172#104—122no—
173#104—122no—
174#102—64no—
175C22⋊C8—61no—
176C22⋊C8—61no—
177C4×C8—32no—
178C4×C8—61no—
179#95—24no—
180#95—21no—
181#95—21no—
182#95—21no—
183#95#6712yes—
184C23×C3#12011yescharacteristic
185C22×D3—41no—
186C3×Q8—42no—
187C3×Q8—42no—
188C3×Q8—42no—
189C3×Q8—42no—
190C2×C3×C4—24no—
191C2×C3×C4—22no—
192C2×C3×C4—41no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—42no—
197C2×C3×C4—42no—
198C2×C3×C4—41no—
199C2×C3×C4—42no—
200C2×C3×C4—41no—
201C2×C3×C4—42no—
202C2×C3×C4—42no—
203C2×C3×C4—41no—
204C2×C3×C4C2×C22⋊C411yescharacteristic
205C2×C3×C4C4×D411yescharacteristic
206C2×C3×C4C4×D411yescharacteristic
207C2×C3×C4#12111yescharacteristic
208C2×C3×C4#12111yescharacteristic
209C2×C3×C4#12311yescharacteristic
210C2×C3×C4C2×D811yescharacteristic
211C2×C3×C4#13511yescharacteristic
212C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
213C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
214#67—81no—
215#67—81no—
216C2×Dic3—41no—
217D12—42no—
218D12—41no—
219D12—41no—
220D12—81no—
221C4×D3—81no—
222C4×D3—81no—
223Dic6—41no—
224Dic6—41no—
225C3⋊C8—42no—
226C3⋊C8—81no—
227C3⋊C8—81no—
228#41—121no—
229#41—121no—
230#41—121no—
231#41—121no—
232C2×Q8—22no—
233C2×Q8—22no—
234C2×Q8—41no—
235C2×Q8—41no—
236C2×Q8—41no—
237C2×Q8—41no—
238C2×Q8—61no—
239C2×D4—61no—
240C2×D4—61no—
241C2×D4—61no—
242C2×D4—121no—
243C22×C4—21no—
244C22×C4—22no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—121no—
248C22×C4C2×#6711yescharacteristic
249C22×C4C2×#6711yescharacteristic
250C22×C4C2×#6711yescharacteristic
251M16—121no—
252M16—121no—
253C2×C8—62no—
254C2×C8—121no—
255C2×C8—121no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—22no—
259C4⋊C4—44no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—61no—
263C4⋊C4—61no—
264C4⋊C4(C2×D3)⋊C414yes—
265C22⋊C4—121no—
266C22⋊C4—121no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—44no—
271C42#21312yes—
272C42C2×C4×D311yescharacteristic
273C42C2×D1211yescharacteristic
274C42#23511yescharacteristic
275C42#23511yescharacteristic
276C22×C3—21no—
277C22×C3—21no—
278C22×C3#38611yescharacteristic
279C22×C3#44711yescharacteristic
280C22×C3#44911yescharacteristic
281C2×D3—81no—
282C2×D3—81no—
283C2×D3—81no—
284C3×C4—21no—
285C3×C4—21no—
286C3×C4—21no—
287C3×C4—21no—
288C3×C4—42no—
289C3×C4—44no—
290C3×C4—82no—
291C3×C4—81no—
292C3×C4C4×D811yescharacteristic
293C3×C4#44211yescharacteristic
294C3×C4#45911yescharacteristic
295C3×C4#46311yescharacteristic
296Dic3—81no—
297C23—121no—
298C23#95311yescharacteristic
299Q8—42no—
300Q8—42no—
301Q8—42no—
302Q8—42no—
303Q8—121no—
304Q8—121no—
305D4—122no—
306D4—121no—
307D4—121no—
308D4—241no—
309D4—241no—
310D4—241no—
311C2×C4—24no—
312C2×C4—22no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—41no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—41no—
320C2×C4—42no—
321C2×C4—41no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—41no—
325C2×C4—121no—
326C2×C4—241no—
327C2×C4—241no—
328C2×C4C2×(C2×D3)⋊C411yescharacteristic
329C2×C4C4×#6711yescharacteristic
330C2×C4C4×#6711yescharacteristic
331C2×C4#92911yescharacteristic
332C2×C4#93011yescharacteristic
333C2×C4#93011yescharacteristic
334C2×C4C2×#21311yescharacteristic
335C2×C4#93211yescharacteristic
336C2×C4#94911yescharacteristic
337C2×C4#95011yescharacteristic
338C8—122no—
339C8—241no—
340C8—241no—
341C2×C3—21no—
342C2×C3—21no—
343C2×C3#187911yescharacteristic
344C2×C3#190511yescharacteristic
345C2×C3#191211yescharacteristic
346D3—161no—
347C22—21no—
348C22—21no—
349C22—241no—
350C22—241no—
351C22—241no—
352C22#521311yescharacteristic
353C22#521711yescharacteristic
354C22#523811yescharacteristic
355C4—21no—
356C4—21no—
357C4—21no—
358C4—21no—
359C4—42no—
360C4—44no—
361C4—82no—
362C4—81no—
363C4—241no—
364C4C4×#21311yescharacteristic
365C4#501411yescharacteristic
366C4#501511yescharacteristic
367C4#501611yescharacteristic
368C3#930011yescharacteristic, complemented, Sylow
369C2—21no—
370C2—21no—
371C2—481no—
372C2#7161311yescharacteristic
373C2#7166211yescharacteristic
374C2#7166311yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2327C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#2327D3
hypocenterC3#9300
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2327

Complements

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

NH ≅ G/Ntype
1. C3×#2327C2semidirect
3. #71773C2semidirect
6. #67631C2semidirect
7. #67614C2semidirect
17. #5088C4semidirect
19. #5078C4semidirect
29. #2327D3semidirect
368. C3#9300semidirect

Maximal subgroups

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

subgroupindex
C3×#23272
#832152
#717732
#695452
#694402
#676312
#676142
#93003