Finite Group Databasebeta

← #10691129

Subgroups — #10691129

Order 768.

544 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#2351C211yesmaximal, characteristic, complemented
2#82441C211yesmaximal, characteristic
3#71621C211yesmaximal, characteristic, complemented
4#69545C211yesmaximal, characteristic
5#69469C211yesmaximal, characteristic
6#67631C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#9180—31nomaximal, Sylow
9C2×C3×C4×Q8—21no—
10C2×C3×C4×Q8C2211yescharacteristic
11C2×#942—21no—
12#5477C2211yescharacteristic
13C3×#389—21no—
14C3×#389—22no—
15C3×#382—21no—
16C3×#382C2211yescharacteristic
17C3×C4×C4⋊C4—21no—
18C3×C4×C4⋊C4C2211yescharacteristic
19#5031C412yescomplemented
20#5028—22no—
21#5027C412yescomplemented
22#4966—21no—
23#4925C2211yescharacteristic
24#4925C2211yescharacteristic
25#4924C2211yescharacteristic
26#4535—22no—
27#4451—21no—
28#4451—21no—
29#4451—22no—
30#3037—31no—
31#2351D311yescharacteristic, complemented
32#1517—31no—
33#1502—31no—
34#1479—31no—
35#1303—31no—
36#1286—31no—
37C22×C3×Q8D411yescharacteristic
38C2×#235—21no—
39C3×C4×Q8—21no—
40C3×C4×Q8—21no—
41C3×C4×Q8—42no—
42C3×C4×Q8—42no—
43C3×C4×Q8C2×C412yes—
44C3×C4×Q8D412yes—
45C2×C3×C4⋊C4—21no—
46C2×C3×C4⋊C4—22no—
47C2×C3×C4⋊C4—22no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C42—21no—
52C2×C3×C42—22no—
53C2×C3×C42—21no—
54C2×C3×C42C2311yescharacteristic
55C2×#216—22no—
56#942—42no—
57#942—42no—
58C2×#215—22no—
59#930—21no—
60#930—21no—
61C2×#208—21no—
62#875—21no—
63#874C2×C411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#869C2×C411yescharacteristic
67C3×#95—22no—
68C3×#95—21no—
69C3×#95D411yescharacteristic
70C3×#95D411yescharacteristic
71C3×#95D412yes—
72#830—21no—
73#830—21no—
74#810—22no—
75#809—22no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)C2×C412yes—
80C4×C3⋊C8—21no—
81C4×C3⋊C8—21no—
82C2×#137—61no—
83#532—31no—
84C2×C4×Q8—21no—
85C2×C4×Q8C2×D311yescharacteristic
86#464—62no—
87#462—32no—
88#460—32no—
89#431—31no—
90#431—31no—
91#423—31no—
92#417—61no—
93#389—21no—
94#389—22no—
95#382—21no—
96#382C2×D311yescharacteristic
97C4×C4⋊C4—21no—
98C4×C4⋊C4C2×D311yescharacteristic
99#327—61no—
100#327—61no—
101#327—62no—
102Q8⋊C8—62no—
103C2×C3×Q8—21no—
104C2×C3×Q8—21no—
105C2×C3×Q8—21no—
106C2×C3×Q8C22⋊C412yes—
107C2×C3×Q8D812yes—
108C2×C3×Q8SD1612yes—
109C22×C3×C4—21no—
110C22×C3×C4—21no—
111C22×C3×C4—21no—
112C22×C3×C4—21no—
113C22×C3×C4—22no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C2×#67—41no—
118#235—41no—
119#235—41no—
120#235—41no—
121#235—41no—
122C2×D12—21no—
123C2×D12—21no—
124C2×D12—21no—
125C2×C4×D3—41no—
126C2×Dic6—21no—
127C3×C4⋊C4—22no—
128C3×C4⋊C4—22no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—42no—
131C3×C4⋊C4—44no—
132C3×C4⋊C4—44no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C4⋊C4C22⋊C412yes—
136C3×C42—22no—
137C3×C42—22no—
138C3×C42—44no—
139C3×C42—41no—
140C3×C42—41no—
141C3×C42C22×C411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42#4111yescharacteristic
144C3×C42#4111yescharacteristic
145#216—44no—
146#215—44no—
147(C2×D3)⋊C4—41no—
148(C2×D3)⋊C4—41no—
149C4⋊Dic3—21no—
150C4⋊Dic3—21no—
151#208—41no—
152#208—41no—
153C2×C3⋊C8—22no—
154C2×C3⋊C8—41no—
155C2×C3⋊C8—41no—
156C2×#41—61no—
157C22×Q8#6711yescharacteristic
158#137—122no—
159#137—122no—
160C2×Q16—62no—
161C2×SD16—62no—
162C2×M16—61no—
163C4⋊Q8—31no—
164#127—31no—
165#124—61no—
166#121—61no—
167#121—61no—
168C4×Q8—21no—
169C4×Q8—21no—
170C4×Q8—42no—
171C4×Q8—42no—
172C4×Q8C4×D312yes—
173C4×Q8D1212yes—
174C4×D4—61no—
175C4×D4—61no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—22no—
178C2×C4⋊C4—22no—
179C2×C4⋊C4—21no—
180C2×C4⋊C4—21no—
181C2×C4⋊C4#6711yescharacteristic
182C2×C42—21no—
183C2×C42—22no—
184C2×C42—21no—
185C2×C42C22×D311yescharacteristic
186C4⋊C8—32no—
187C4⋊C8—61no—
188C4⋊C8—61no—
189C4⋊C8—61no—
190#103—62no—
191#102—62no—
192C22⋊C8—61no—
193C22⋊C8—61no—
194C4×C8—61no—
195C4×C8—61no—
196#95—22no—
197#95—21no—
198#95#6711yescharacteristic
199#95#6711yescharacteristic
200#95#6712yes—
201C23×C3#12011yescharacteristic
202C22×D3—41no—
203C3×Q8—22no—
204C3×Q8—22no—
205C3×Q8—22no—
206C3×Q8#10214yes—
207C2×C3×C4—22no—
208C2×C3×C4—22no—
209C2×C3×C4—22no—
210C2×C3×C4—22no—
211C2×C3×C4—22no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4—41no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—42no—
219C2×C3×C4—41no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—41no—
223C2×C3×C4C2×C22⋊C411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4C4×D411yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12311yescharacteristic
229C2×C3×C4C2×D811yescharacteristic
230C2×C3×C4C2×SD1611yescharacteristic
231C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
232C2×C3×C4#13711yescharacteristic
233#67—81no—
234#67—81no—
235C2×Dic3—41no—
236D12—41no—
237D12—41no—
238D12—81no—
239D12—81no—
240C4×D3—81no—
241C4×D3—81no—
242Dic6—41no—
243Dic6—41no—
244C3⋊C8—42no—
245C3⋊C8—81no—
246C3⋊C8—81no—
247#41—121no—
248#41—121no—
249#41—121no—
250#41—121no—
251C2×Q8—21no—
252C2×Q8—21no—
253C2×Q8—21no—
254C2×Q8—61no—
255C2×Q8(C2×D3)⋊C412yes—
256C2×Q8#21312yes—
257C2×Q8#21512yes—
258C2×D4—61no—
259C2×D4—61no—
260C2×D4—61no—
261C2×D4—121no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—22no—
267C22×C4—121no—
268C22×C4C2×#6711yescharacteristic
269C22×C4C2×#6711yescharacteristic
270C22×C4C2×#6711yescharacteristic
271Q16—124no—
272SD16—124no—
273M16—121no—
274M16—121no—
275C2×C8—62no—
276C2×C8—121no—
277C2×C8—121no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—44no—
282C4⋊C4—42no—
283C4⋊C4—44no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—61no—
287C4⋊C4—61no—
288C4⋊C4(C2×D3)⋊C412yes—
289C22⋊C4—121no—
290C22⋊C4—121no—
291C42—22no—
292C42—22no—
293C42—44no—
294C42—41no—
295C42—41no—
296C42C2×C4×D311yescharacteristic
297C42C2×D1211yescharacteristic
298C42#23511yescharacteristic
299C42#23511yescharacteristic
300C22×C3—21no—
301C22×C3—21no—
302C22×C3#38611yescharacteristic
303C22×C3#44711yescharacteristic
304C22×C3#44811yescharacteristic
305C2×D3—81no—
306C2×D3—81no—
307C2×D3—81no—
308C3×C4—24no—
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—42no—
314C3×C4—42no—
315C3×C4—82no—
316C3×C4—81no—
317C3×C4C2×#10211yescharacteristic
318C3×C4#41711yescharacteristic
319C3×C4#48011yescharacteristic
320C3×C4#48111yescharacteristic
321Dic3—81no—
322C23—121no—
323C23#95311yescharacteristic
324Q8—22no—
325Q8—22no—
326Q8—22no—
327Q8—121no—
328Q8—121no—
329Q8#80914yes—
330D4—121no—
331D4—121no—
332D4—241no—
333D4—241no—
334D4—241no—
335D4—241no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—42no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—41no—
347C2×C4—42no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—121no—
353C2×C4—241no—
354C2×C4—241no—
355C2×C4C2×(C2×D3)⋊C411yescharacteristic
356C2×C4C4×#6711yescharacteristic
357C2×C4C4×#6711yescharacteristic
358C2×C4#92911yescharacteristic
359C2×C4#93011yescharacteristic
360C2×C4#93011yescharacteristic
361C2×C4C2×#21311yescharacteristic
362C2×C4#93211yescharacteristic
363C2×C4C2×#21511yescharacteristic
364C2×C4#94211yescharacteristic
365C8—122no—
366C8—241no—
367C8—241no—
368C2×C3—21no—
369C2×C3—21no—
370C2×C3#187411yescharacteristic
371C2×C3#190511yescharacteristic
372C2×C3#190611yescharacteristic
373D3—161no—
374C22—21no—
375C22—21no—
376C22—241no—
377C22—241no—
378C22—241no—
379C22#521311yescharacteristic
380C22#521711yescharacteristic
381C22#522711yescharacteristic
382C4—24no—
383C4—21no—
384C4—21no—
385C4—21no—
386C4—21no—
387C4—42no—
388C4—42no—
389C4—82no—
390C4—81no—
391C4—241no—
392C4C2×#80911yescharacteristic
393C4#496611yescharacteristic
394C4#496711yescharacteristic
395C4#496811yescharacteristic
396C3#918011yescharacteristic, complemented, Sylow
397C2—21no—
398C2—21no—
399C2—481no—
400C2#7156011yescharacteristic
401C2#7164511yescharacteristic
402C2#7166211yescharacteristic
subgroupG/H
centerC22#5213
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#2351C2
socleC22×C3#386
prime radical#5477C22
cosocleC2×C3×C42C23
hypercenter#2351D3
hypocenterC3#9180
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2351

Complements

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

NH ≅ G/Ntype
1. C3×#2351C2semidirect
3. #71621C2semidirect
6. #67631C2semidirect
7. #67620C2semidirect
19. #5031C4semidirect
21. #5027C4semidirect
31. #2351D3semidirect
396. C3#9180semidirect

Maximal subgroups

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

subgroupindex
C3×#23512
#824412
#716212
#695452
#694692
#676312
#676202
#91803