Finite Group Databasebeta

← #10690906

Subgroups — #10690906

Order 768.

738 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#74000C211yesmaximal, characteristic
2C2×#4649C211yesmaximal, characteristic
3C2×#4467C211yesmaximal, characteristic
4#69910C211yesmaximal, characteristic
5C2×#4452C211yesmaximal, characteristic
6C3×#1294C211yesmaximal, characteristic
7#67624C211yesmaximal, characteristic
8#9144—31nomaximal, Sylow
9C2×#924C2211yescharacteristic
10C22×Dic3⋊C4—21no—
11C22×C4×Dic3—21no—
12C22×C4×Dic3C2211yescharacteristic
13C2×C3×C22⋊C8C2211yescharacteristic
14C3×#380C2211yescharacteristic
15C2×C3×#95C2211yescharacteristic
16C2×C3⋊#95C2211yescharacteristic
17C2×#830C2211yescharacteristic
18#4984C412yescomplemented
19#4973C412yescomplemented
20#4950—21no—
21#4943—21no—
22#4653—22no—
23#4649—22no—
24#4649—22no—
25#4550—22no—
26#4467—22no—
27#4452—22no—
28#2855—31no—
29C2×C4×C4⋊C4—31no—
30#1817—31no—
31C2×#328—31no—
32C2×#328—31no—
33#1294—31no—
34#1294D311yescharacteristic
35C23×C3×C4—21no—
36C23×C3×C4C2311yescharacteristic
37C23×Dic3—21no—
38C23×Dic3D411yescharacteristic
39C23×Dic3D411yescharacteristic
40C22×Dic6C2×C411yescharacteristic
41C22×C3×C8—21no—
42C2×C3×C4⋊C4C2×C412yes—
43C2×C3×C22⋊C4—21no—
44C2×#217—21no—
45C2×C4⋊Dic3—22no—
46C2×C4⋊Dic3C2×C411yescharacteristic
47#924—21no—
48#924—21no—
49#924—21no—
50#924—21no—
51C2×Dic3⋊C4—24no—
52C2×Dic3⋊C4—28no—
53C2×Dic3⋊C4—21no—
54C2×C4×Dic3—21no—
55C2×C4×Dic3—21no—
56C2×C4×Dic3—21no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3—24no—
59C2×C4×Dic3—42no—
60C2×C4×Dic3—42no—
61C2×C4×Dic3C2×C412yes—
62C2×C4×Dic3D412yes—
63C22×C3⋊C8—21no—
64C3×C22⋊C8—21no—
65C3×C22⋊C8—21no—
66C3×#95—21no—
67C3×#95—21no—
68C3×#95D412yes—
69C3⋊#95—21no—
70C3⋊#95—22no—
71C3⋊#95—22no—
72C3⋊#95—22no—
73C3⋊#95D412yes—
74#830—21no—
75#830—21no—
76C2×C22⋊Q8—31no—
77C22×C4⋊C4—61no—
78C22×C42—31no—
79C22×C42—61no—
80C2×C22⋊C8—31no—
81C2×C22⋊C8C2×D311yescharacteristic
82#391—32no—
83#389—32no—
84#387—62no—
85#385—61no—
86#381—61no—
87#380C2×D311yescharacteristic
88C4×C4⋊C4—62no—
89C4×C4⋊C4—62no—
90C2×#95—31no—
91C2×#95C2×D311yescharacteristic
92#336—62no—
93#328—62no—
94#328—62no—
95C24×C3C2×D411yescharacteristic
96C22×C3×C4—24no—
97C22×C3×C4—22no—
98C22×C3×C4—22no—
99C22×C3×C4—42no—
100C22×C3×C4—42no—
101C22×C3×C4—41no—
102C22×C3×C4—41no—
103C22×C3×C4C22×C411yescharacteristic
104C22×C3×C4C2×D411yescharacteristic
105C22×C3×C4C2×D411yescharacteristic
106C22×C3×C4C2×D411yescharacteristic
107C22×C3×C4#4111yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×Dic3—21no—
110C22×Dic3—21no—
111C22×Dic3—21no—
112C22×Dic3—21no—
113C22×Dic3—21no—
114C22×Dic3—21no—
115C22×Dic3—24no—
116C22×Dic3—24no—
117C22×Dic3—22no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—42no—
121C22×Dic3—41no—
122C22×Dic3—41no—
123C22×Dic3C22⋊C412yes—
124C22×Dic3C22⋊C412yes—
125C22×Dic3SD1612yes—
126C22×Dic3Q1612yes—
127C2×Dic6—21no—
128C2×Dic6—21no—
129C2×Dic6—42no—
130C2×C3×C8—42no—
131C2×C3×C8—41no—
132C3×C4⋊C4—42no—
133C3×C22⋊C4—41no—
134C3×C22⋊C4—41no—
135#217—41no—
136#217—41no—
137C4⋊Dic3—21no—
138C4⋊Dic3—21no—
139C4⋊Dic3—44no—
140Dic3⋊C4—216no—
141Dic3⋊C4—41no—
142Dic3⋊C4—41no—
143C4×Dic3—22no—
144C4×Dic3—22no—
145C4×Dic3—22no—
146C4×Dic3—22no—
147C4×Dic3—48no—
148C2×C3⋊C8—42no—
149C2×C3⋊C8—41no—
150C22×Q8—31no—
151C23×C4—21no—
152C23×C4—31no—
153C23×C4—31no—
154C23×C4—61no—
155C23×C4C22×D311yescharacteristic
156C22×C8—21no—
157C22×C8—61no—
158C22⋊Q8—61no—
159C22⋊Q8—61no—
160C22⋊Q8—61no—
161C22⋊Q8—61no—
162C2×C4⋊C4—31no—
163C2×C4⋊C4—64no—
164C2×C4⋊C4—68no—
165C2×C4⋊C4—62no—
166C2×C4⋊C4—61no—
167C2×C4⋊C4C4×D312yes—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—61no—
170C2×C42—32no—
171C2×C42—32no—
172C2×C42—61no—
173C2×C42—61no—
174C2×C42—61no—
175C2×C42—61no—
176C2×C42—64no—
177C2×C42—122no—
178C2×C42—122no—
179C22⋊C8—21no—
180C22⋊C8—21no—
181C22⋊C8—61no—
182C22⋊C8—61no—
183#95—21no—
184#95—21no—
185#95—32no—
186#95—62no—
187#95—62no—
188#95—62no—
189#95—61no—
190#95D1212yes—
191C23×C3—22no—
192C23×C3—22no—
193C23×C3C2×C22⋊C411yescharacteristic
194C23×C3#12011yescharacteristic
195C23×C3#12311yescharacteristic
196C23×C3C2×SD1611yescharacteristic
197C23×C3C2×Q1611yescharacteristic
198C23×C3AGL(1,ℤ/8)11yescharacteristic
199C23×C3#13711yescharacteristic
200C2×C3×C4—22no—
201C2×C3×C4—22no—
202C2×C3×C4—24no—
203C2×C3×C4—41no—
204C2×C3×C4—41no—
205C2×C3×C4—41no—
206C2×C3×C4—41no—
207C2×C3×C4—42no—
208C2×C3×C4—44no—
209C2×C3×C4—44no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—42no—
213C2×C3×C4—42no—
214C2×C3×C4C4×D411yescharacteristic
215C2×C3×C4C4×D411yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×Dic3—24no—
219C2×Dic3—24no—
220C2×Dic3—22no—
221C2×Dic3—22no—
222C2×Dic3—22no—
223C2×Dic3—22no—
224C2×Dic3—22no—
225C2×Dic3—22no—
226C2×Dic3—22no—
227C2×Dic3—22no—
228C2×Dic3—24no—
229C2×Dic3—42no—
230C2×Dic3—44no—
231C2×Dic3—44no—
232C2×Dic3—42no—
233C2×Dic3—41no—
234C2×Dic3—41no—
235C2×Dic3—42no—
236C2×Dic3—42no—
237C2×Dic3#10314yes—
238C2×Dic3#10414yes—
239Dic6—81no—
240Dic6—81no—
241C3×C8—81no—
242C3⋊C8—81no—
243C24D3×D411yescharacteristic
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—122no—
247C22×C4—24no—
248C22×C4—22no—
249C22×C4—22no—
250C22×C4—32no—
251C22×C4—32no—
252C22×C4—32no—
253C22×C4—32no—
254C22×C4—42no—
255C22×C4—42no—
256C22×C4—41no—
257C22×C4—41no—
258C22×C4—64no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4—61no—
262C22×C4—61no—
263C22×C4—61no—
264C22×C4—61no—
265C22×C4—64no—
266C22×C4—62no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—122no—
270C22×C4—121no—
271C22×C4—121no—
272C22×C4C2×C4×D311yescharacteristic
273C22×C4C2×D1211yescharacteristic
274C22×C4#23511yescharacteristic
275C22×C4D3×D411yescharacteristic
276C22×C4D3×D411yescharacteristic
277C22×C4#23911yescharacteristic
278C2×C8—42no—
279C2×C8—41no—
280C2×C8—122no—
281C2×C8—121no—
282C4⋊C4—42no—
283C4⋊C4—616no—
284C4⋊C4—61no—
285C4⋊C4—61no—
286C4⋊C4—124no—
287C4⋊C4—121no—
288C4⋊C4—121no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—121no—
292C22⋊C4—121no—
293C42—62no—
294C42—62no—
295C42—62no—
296C42—62no—
297C42—128no—
298C22×C3—21no—
299C22×C3—21no—
300C22×C3—21no—
301C22×C3—21no—
302C22×C3—22no—
303C22×C3—22no—
304C22×C3—22no—
305C22×C3—22no—
306C22×C3#38611yescharacteristic
307C22×C3#41012yes—
308C22×C3C2×#10311yescharacteristic
309C22×C3#41711yescharacteristic
310C22×C3C2×#10412yes—
311C22×C3#45011yescharacteristic
312C22×C3#45111yescharacteristic
313C22×C3#48311yescharacteristic
314C22×C3#48411yescharacteristic
315C3×C4—21no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—44no—
320C3×C4—82no—
321Dic3—24no—
322Dic3—24no—
323Dic3—44no—
324Dic3—81no—
325Dic3—81no—
326C23—22no—
327C23—22no—
328C23D3×C22⋊C411yescharacteristic
329C23#88211yescharacteristic
330C23#89411yescharacteristic
331C23#91111yescharacteristic
332C23D3×SD1611yescharacteristic
333C23#91511yescharacteristic
334C23D3×Q1611yescharacteristic
335Q8—241no—
336Q8—241no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—34no—
341C2×C4—34no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—42no—
347C2×C4—44no—
348C2×C4—44no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—64no—
354C2×C4—64no—
355C2×C4—62no—
356C2×C4—62no—
357C2×C4—62no—
358C2×C4—62no—
359C2×C4—62no—
360C2×C4—62no—
361C2×C4—62no—
362C2×C4—62no—
363C2×C4—64no—
364C2×C4—122no—
365C2×C4—124no—
366C2×C4—124no—
367C2×C4—122no—
368C2×C4—121no—
369C2×C4—121no—
370C2×C4—122no—
371C2×C4—122no—
372C2×C4C4×D1211yescharacteristic
373C2×C4#88411yescharacteristic
374C2×C4#89311yescharacteristic
375C2×C4#89511yescharacteristic
376C8—81no—
377C8—241no—
378C2×C3—21no—
379C2×C3—21no—
380C2×C3—21no—
381C2×C3—21no—
382C2×C3#150312yes—
383C2×C3#150912yes—
384C2×C3#187211yescharacteristic
385C2×C3#189611yescharacteristic
386C2×C3#189911yescharacteristic
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—22no—
392C22—22no—
393C22—22no—
394C22—22no—
395C22#466511yescharacteristic
396C22#474012yes—
397C22#477111yescharacteristic
398C22#477611yescharacteristic
399C22D3×#10311yescharacteristic
400C22#480911yescharacteristic
401C22D3×#10412yes—
402C22#486211yescharacteristic
403C22#488411yescharacteristic
404C4—21no—
405C4—21no—
406C4—21no—
407C4—21no—
408C4—44no—
409C4—64no—
410C4—64no—
411C4—82no—
412C4—124no—
413C4—241no—
414C4—241no—
415C3#914411yescharacteristic, complemented, Sylow
416C2—21no—
417C2—21no—
418C2—21no—
419C2—21no—
420C2#6827912yes—
421C2#6830912yes—
422C2#6875611yescharacteristic
423C2#6887811yescharacteristic
424C2#6895411yescharacteristic
subgroupG/H
centerC23D3×C22⋊C4
commutatorC22×C3×C4C22×C4
FrattiniC23×C4C22×D3
FittingC3×#1294C2
socleC23×C3C2×C22⋊C4
prime radicalC24×C3C2×D4
cosocleC23×C3×C4C23
hypercenter#1294D3
hypocenterC3#9144
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1294

Complements

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

NH ≅ G/Ntype
18. #4984C4semidirect
19. #4973C4semidirect
415. C3#9144semidirect

Maximal subgroups

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

subgroupindex
#740002
C2×#46492
C2×#44672
#699102
C2×#44522
C3×#12942
#676242
#91443