Finite Group Databasebeta

← #10691089

Subgroups — #10691089

Order 768.

516 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
1C3×#3038C211yesmaximal, characteristic
2#82515C211yesmaximal, characteristic, complemented
3#71759C211yesmaximal, characteristic, complemented
4C2×#4544C211yesmaximal, characteristic, complemented
5#71639C211yesmaximal, characteristic, complemented
6#67714C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#9174—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C2×C3×#127C2211yescharacteristic
11C3×C4×#41C2211yescharacteristic
12C3×#460—21no—
13C3×#460C412yes—
14C3×#459—21no—
15C3×#459C412yes—
16C3×#423C2211yescharacteristic
17C3×#418—21no—
18#5234—21no—
19#5000—22no—
20#4997—21no—
21#4926—21no—
22#4925—21no—
23#4925C2211yescharacteristic
24#4924C2211yescharacteristic
25C2×C4⋊(C3⋊C8)C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27#4544—21no—
28#4544—21no—
29#4544—22no—
30#4535—21no—
31D4⋊(C3⋊C8)—21no—
32#3038D311yescharacteristic
33#2925—31no—
34#1545—31no—
35C2×#331—31no—
36#1477—31no—
37#1301—31no—
38#1292—31no—
39C2×C3×#41D411yescharacteristic
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C22×C3×D4—21no—
43C3×AGL(1,ℤ/8)—42no—
44C3×AGL(1,ℤ/8)—42no—
45C2×C3×SD16—22no—
46C2×C3×D8—22no—
47C2×C3×M16—21no—
48C3×#127—21no—
49C3×#127—21no—
50C3×#127—21no—
51C3×#127C2×C412yes—
52C3×C4×Q8C2×C411yescharacteristic
53C3×C4×Q8D411yescharacteristic
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4C2×C411yescharacteristic
57C3×C4×D4D411yescharacteristic
58C3×#117—21no—
59C3×#117D411yescharacteristic
60C2×C3×C42—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#208—21no—
63C22×C3⋊C8—21no—
64C22×C3⋊C8—21no—
65C3×C4⋊C8—21no—
66C3×C4⋊C8C2×C412yes—
67C3×#102—22no—
68C3×#102—22no—
69#830—22no—
70#830—22no—
71#830—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4⋊(C3⋊C8)—21no—
75C4⋊(C3⋊C8)—21no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)D411yescharacteristic
78C4⋊(C3⋊C8)D411yescharacteristic
79C4⋊(C3⋊C8)D412yes—
80#803—22no—
81C4×C3⋊C8—22no—
82C4×C3⋊C8—21no—
83C2×AGL(1,ℤ/8)—21no—
84C2×#127C2×D311yescharacteristic
85C4×#41C2×D311yescharacteristic
86#460—21no—
87#460Dic312yes—
88#459—21no—
89#459Dic312yes—
90#433—62no—
91#432—61no—
92#431—31no—
93#431—61no—
94#423—31no—
95#423C2×D311yescharacteristic
96C2×C4⋊C8—31no—
97C2×C4⋊C8—31no—
98#418—21no—
99#408—61no—
100#405—61no—
101#331—61no—
102#331—61no—
103#331—62no—
104Q8⋊C8—61no—
105D4⋊C8—61no—
106C24×C3—41no—
107C3×#41—21no—
108C3×#41—21no—
109C3×#41—21no—
110C3×#41—21no—
111C2×C3×Q8C22⋊C411yescharacteristic
112C2×C3×D4—21no—
113C2×C3×D4—22no—
114C2×C3×D4—22no—
115C2×C3×D4—22no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—81no—
124C2×C3×D4—81no—
125C2×C3×D4—82no—
126C2×C3×D4C22⋊C411yescharacteristic
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4#4111yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C3×SD16—44no—
134C3×D8—44no—
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—22no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4C22⋊C411yescharacteristic
141C3×C4⋊C4C22⋊C411yescharacteristic
142C3×C22⋊C4—21no—
143C3×C22⋊C4—21no—
144C3×C22⋊C4—21no—
145C3×C42—21no—
146C3×C42—21no—
147C3×C42—21no—
148C3×C42C22×C411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152#208—41no—
153#208—41no—
154C2×C3⋊C8—21no—
155C2×C3⋊C8—22no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—81no—
162C2×#41#6711yescharacteristic
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166AGL(1,ℤ/8)—42no—
167AGL(1,ℤ/8)—42no—
168C2×SD16—22no—
169C2×D8—22no—
170C2×M16—21no—
171C2×M16—61no—
172C22×C8—61no—
173C22×C8—61no—
174#127—21no—
175#127—21no—
176#127—21no—
177#127C2×Dic312yes—
178C4×Q8C2×Dic311yescharacteristic
179C4×Q8#6711yescharacteristic
180C4×D4—21no—
181C4×D4—21no—
182C4×D4C2×Dic311yescharacteristic
183C4×D4#6711yescharacteristic
184#117—21no—
185#117#6711yescharacteristic
186C2×C42—21no—
187C2×C42C22×D311yescharacteristic
188C4⋊C8—21no—
189C4⋊C8—31no—
190C4⋊C8—31no—
191C4⋊C8—32no—
192C4⋊C8—61no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8—61no—
197C4⋊C8C2×Dic312yes—
198#102—22no—
199#102—22no—
200C22⋊C8—62no—
201C22⋊C8—62no—
202C22⋊C8—61no—
203#97—62no—
204C4×C8—62no—
205C4×C8—61no—
206C23×C3—21no—
207C23×C3—42no—
208C23×C3—82no—
209C23×C3—82no—
210C23×C3—81no—
211C23×C3—81no—
212C23×C3#12311yescharacteristic
213C3×Q8—21no—
214C3×Q8—21no—
215C3×D4—21no—
216C3×D4—21no—
217C3×D4—41no—
218C3×D4—41no—
219C3×D4—42no—
220C3×D4—42no—
221C3×D4—82no—
222C3×D4—82no—
223C3×D4—82no—
224C3×D4—81no—
225C3×D4—81no—
226C3×D4—81no—
227C3×D4—81no—
228C2×C3×C4—21no—
229C2×C3×C4—21no—
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—81no—
239C2×C3×C4C2×C22⋊C411yescharacteristic
240C2×C3×C4C4×D411yescharacteristic
241C2×C3×C4C4×D411yescharacteristic
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#13511yescharacteristic
246C2×C3×C4#13511yescharacteristic
247C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
248C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
249C3×C8—42no—
250C3⋊C8—41no—
251C3⋊C8—41no—
252C3⋊C8—42no—
253C3⋊C8—81no—
254C3⋊C8—81no—
255C24—41no—
256#41—21no—
257#41—21no—
258#41—21no—
259#41—21no—
260C2×Q8#21711yescharacteristic
261C2×D4—21no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—81no—
273C2×D4—81no—
274C2×D4—82no—
275C2×D4#21711yescharacteristic
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4#23711yescharacteristic
280C22×C4#23711yescharacteristic
281C22×C4C2×#6711yescharacteristic
282SD16—44no—
283D8—44no—
284M16—41no—
285M16—41no—
286M16—121no—
287M16—121no—
288C2×C8—22no—
289C2×C8—61no—
290C2×C8—62no—
291C2×C8—61no—
292C2×C8—121no—
293C2×C8—121no—
294C2×C8—121no—
295C2×C8—121no—
296C2×C8—241no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4#21711yescharacteristic
300C4⋊C4#21711yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42D3×D411yescharacteristic
308C42D3×D411yescharacteristic
309C42C22×Dic311yescharacteristic
310C42C2×#6711yescharacteristic
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—41no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—82no—
317C22×C3—82no—
318C22×C3—82no—
319C22×C3—162no—
320C22×C3—162no—
321C22×C3#38611yescharacteristic
322C22×C3#48211yescharacteristic
323C22×C3#48211yescharacteristic
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—41no—
328C3×C4—41no—
329C3×C4—41no—
330C3×C4#41611yescharacteristic
331C3×C4#41811yescharacteristic
332C3×C4#44911yescharacteristic
333C3×C4#44911yescharacteristic
334C3×C4#47111yescharacteristic
335C3×C4#47211yescharacteristic
336C23—21no—
337C23—42no—
338C23—82no—
339C23—82no—
340C23—81no—
341C23—81no—
342C23#93511yescharacteristic
343Q8—21no—
344Q8—21no—
345D4—21no—
346D4—21no—
347D4—41no—
348D4—41no—
349D4—42no—
350D4—42no—
351D4—82no—
352D4—82no—
353D4—82no—
354D4—81no—
355D4—81no—
356D4—81no—
357D4—81no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—81no—
369C2×C4#93211yescharacteristic
370C2×C4#93211yescharacteristic
371C2×C4D4×Dic311yescharacteristic
372C2×C4D4×Dic311yescharacteristic
373C2×C4#93711yescharacteristic
374C2×C4#93911yescharacteristic
375C2×C4#93911yescharacteristic
376C2×C4#95011yescharacteristic
377C2×C4#95011yescharacteristic
378C2×C4C2×#21711yescharacteristic
379C8—42no—
380C8—121no—
381C8—121no—
382C8—122no—
383C8—241no—
384C8—241no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—81no—
388C2×C3—162no—
389C2×C3#187511yescharacteristic
390C2×C3#190311yescharacteristic
391C2×C3#190711yescharacteristic
392C22—21no—
393C22—21no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—82no—
398C22—82no—
399C22—82no—
400C22—162no—
401C22—162no—
402C22#503511yescharacteristic
403C22#503511yescharacteristic
404C22#522011yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4#503711yescharacteristic
412C4#503711yescharacteristic
413C4#519911yescharacteristic
414C4#520011yescharacteristic
415C4#521511yescharacteristic
416C4#523311yescharacteristic
417C3#917411yescharacteristic, complemented, Sylow
418C2—21no—
419C2—21no—
420C2—81no—
421C2—162no—
422C2#7167811yescharacteristic
423C2#7209011yescharacteristic
424C2#7209111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3038C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3038D3
hypocenterC3#9174
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3038

Complements

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

NH ≅ G/Ntype
2. #82515C2semidirect
3. #71759C2semidirect
4. C2×#4544C2semidirect
5. #71639C2semidirect
6. #67714C2semidirect
7. #67711C2semidirect
417. C3#9174semidirect

Maximal subgroups

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

subgroupindex
C3×#30382
#825152
#717592
C2×#45442
#716392
#677142
#677112
#91743