Finite Group Databasebeta

← #10691090

Subgroups — #10691090

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
1#83165C211yesmaximal, characteristic
2#79000C211yesmaximal, characteristic, complemented
3C2×C3×#331C211yesmaximal, characteristic, complemented
4#71759C211yesmaximal, characteristic, complemented
5#70425C211yesmaximal, characteristic, complemented
6#67707C211yesmaximal, characteristic, complemented
7#67705C211yesmaximal, characteristic, complemented
8#9174—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×#932—21no—
11C4×#235C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14#5215—21no—
15#5110—21no—
16#5101—21no—
17#5075—21no—
18#5075C412yes—
19#5072—21no—
20#5072C412yes—
21#4925C2211yescharacteristic
22#4924C2211yescharacteristic
23#4924C2211yescharacteristic
24#4840—22no—
25#4835—21no—
26#4833—21no—
27C3×#331—21no—
28C3×#331—21no—
29#4544—22no—
30#4484—21no—
31#4483—21no—
32#3038—31no—
33#2925—31no—
34#1545—31no—
35C2×#331D311yescharacteristic, complemented
36#1477—31no—
37#1301—31no—
38#1292—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C2×#235D411yescharacteristic
43C22×C3×C8—21no—
44C22×C3×C8—21no—
45C3×#127—21no—
46C3×#127—21no—
47C3×#127—21no—
48C3×#127C2×C412yes—
49C2×C3×C42C2311yescharacteristic
50C2×#214—22no—
51#932—42no—
52#932—42no—
53C2×#213—22no—
54C4×#67—21no—
55C4×#67—21no—
56#926D411yescharacteristic
57C2×#208—21no—
58C2×#208—21no—
59C4×D12C2×C411yescharacteristic
60C4×D12D411yescharacteristic
61#872—21no—
62C42×D3—21no—
63C4×Dic6C2×C411yescharacteristic
64C4×Dic6D411yescharacteristic
65C3×C4⋊C8—21no—
66C3×C4⋊C8—21no—
67C3×C4⋊C8—21no—
68C3×C4⋊C8D411yescharacteristic
69C3×C4⋊C8D411yescharacteristic
70#832—22no—
71#832—22no—
72#830—21no—
73(C2×D3)⋊C8—22no—
74(C2×D3)⋊C8—22no—
75#815—22no—
76C8×Dic3—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)C2×C412yes—
81C4⋊(C3⋊C8)D412yes—
82C4×C3⋊C8—21no—
83C2×AGL(1,ℤ/8)—61no—
84C2×#127C2×D311yescharacteristic
85C4×#41—31no—
86#460—32no—
87#460—61no—
88#459—32no—
89#459—61no—
90#433—62no—
91#432—61no—
92#431—31no—
93#431—61no—
94#423—31no—
95#423—31no—
96C2×C4⋊C8C2×D311yescharacteristic
97C2×C4⋊C8C2×D311yescharacteristic
98#418—61no—
99#408—61no—
100#405—61no—
101#331—21no—
102#331—21no—
103#331—62no—
104Q8⋊C8—61no—
105D4⋊C8—61no—
106C24×C3—41no—
107C2×C3×D4—22no—
108C2×C3×D4—22no—
109C2×C3×D4—22no—
110C2×C3×D4—41no—
111C2×C3×D4—41no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—81no—
118C2×C3×D4—81no—
119C2×C3×D4—82no—
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4#4111yescharacteristic
122C22×C3×C4#4111yescharacteristic
123C2×#67—21no—
124#235—21no—
125#235—21no—
126#235—21no—
127#235—21no—
128C2×D12C22⋊C411yescharacteristic
129C2×C4×D3—21no—
130C2×C4×D3—21no—
131C2×C4×D3—21no—
132C2×Dic6C22⋊C411yescharacteristic
133C2×C3×C8—21no—
134C2×C3×C8—21no—
135C2×C3×C8—41no—
136C2×C3×C8—41no—
137C2×C3×C8—41no—
138C2×C3×C8—81no—
139C3×C42C22×C411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42C2×D411yescharacteristic
143#217—21no—
144#214—44no—
145#213—44no—
146(C2×D3)⋊C4—21no—
147(C2×D3)⋊C4—21no—
148C4⋊Dic3C22⋊C411yescharacteristic
149C4⋊Dic3C22⋊C411yescharacteristic
150Dic3⋊C4—21no—
151Dic3⋊C4—21no—
152C4×Dic3—21no—
153C4×Dic3—21no—
154C4×Dic3—21no—
155#208—41no—
156#208—41no—
157#208—41no—
158#208—41no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—41no—
162C2×#41—31no—
163C22×D4—21no—
164C22×D4—21no—
165C22×D4—21no—
166AGL(1,ℤ/8)—122no—
167AGL(1,ℤ/8)—122no—
168C2×SD16—62no—
169C2×D8—62no—
170C2×M16—61no—
171C2×M16—61no—
172C22×C8—21no—
173C22×C8—21no—
174#127—21no—
175#127—21no—
176#127—21no—
177#127C4×D312yes—
178C4×Q8—31no—
179C4×Q8—31no—
180C4×D4—31no—
181C4×D4—31no—
182C4×D4—61no—
183C4×D4—61no—
184#117—31no—
185#117—61no—
186C2×C42—61no—
187C2×C42C22×D311yescharacteristic
188C4⋊C8—21no—
189C4⋊C8—21no—
190C4⋊C8—21no—
191C4⋊C8—32no—
192C4⋊C8—32no—
193C4⋊C8—61no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8#6711yescharacteristic
197C4⋊C8#6711yescharacteristic
198#102—62no—
199#102—62no—
200C22⋊C8—62no—
201C22⋊C8—62no—
202C22⋊C8—61no—
203#97—62no—
204C4×C8—62no—
205C4×C8—61no—
206C23×C3—42no—
207C23×C3—82no—
208C23×C3—82no—
209C23×C3—81no—
210C23×C3—81no—
211C23×C3#12311yescharacteristic
212C22×D3—21no—
213C3×D4—42no—
214C3×D4—42no—
215C3×D4—82no—
216C3×D4—82no—
217C3×D4—82no—
218C3×D4—81no—
219C3×D4—81no—
220C3×D4—81no—
221C3×D4—81no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—41no—
225C2×C3×C4C2×C22⋊C411yescharacteristic
226C2×C3×C4C4×D411yescharacteristic
227C2×C3×C4C4×D411yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12111yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#13511yescharacteristic
232C2×C3×C4#13511yescharacteristic
233C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
234C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
235#67—41no—
236#67—41no—
237C2×Dic3—21no—
238C2×Dic3—21no—
239C2×Dic3—21no—
240D12—21no—
241D12—21no—
242C4×D3—41no—
243C4×D3—41no—
244C4×D3—41no—
245C4×D3—41no—
246C4×D3—81no—
247Dic6—21no—
248Dic6—21no—
249C3×C8—41no—
250C3×C8—41no—
251C3×C8—81no—
252C3⋊C8—42no—
253C3⋊C8—42no—
254C3⋊C8—81no—
255C24—41no—
256#41—61no—
257#41—61no—
258#41—61no—
259#41—61no—
260C2×Q8—31no—
261C2×D4—22no—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—31no—
265C2×D4—41no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—61no—
273C2×D4—81no—
274C2×D4—81no—
275C2×D4—82no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4#23511yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4#23711yescharacteristic
282SD16—124no—
283D8—124no—
284M16—121no—
285M16—121no—
286M16—121no—
287M16—121no—
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—41no—
291C2×C8—41no—
292C2×C8—41no—
293C2×C8—62no—
294C2×C8—62no—
295C2×C8—81no—
296C2×C8—121no—
297C4⋊C4—31no—
298C4⋊C4—31no—
299C4⋊C4—61no—
300C4⋊C4—61no—
301C22⋊C4—61no—
302C22⋊C4—61no—
303C22⋊C4—61no—
304C42—61no—
305C42—61no—
306C42—61no—
307C42C2×C4×D311yescharacteristic
308C42D3×D411yescharacteristic
309C42D3×D411yescharacteristic
310C42C2×#6711yescharacteristic
311C22×C3—21no—
312C22×C3—21no—
313C22×C3—82no—
314C22×C3—82no—
315C22×C3—82no—
316C22×C3—162no—
317C22×C3—162no—
318C22×C3#38611yescharacteristic
319C22×C3#48211yescharacteristic
320C22×C3#48211yescharacteristic
321C2×D3—41no—
322C2×D3—41no—
323C2×D3—41no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4#41611yescharacteristic
328C3×C4#41811yescharacteristic
329C3×C4#44911yescharacteristic
330C3×C4#44911yescharacteristic
331C3×C4#47111yescharacteristic
332C3×C4#47211yescharacteristic
333Dic3—41no—
334Dic3—41no—
335Dic3—41no—
336C23—42no—
337C23—61no—
338C23—82no—
339C23—82no—
340C23—81no—
341C23—81no—
342C23#88311yescharacteristic
343Q8—61no—
344Q8—61no—
345D4—42no—
346D4—42no—
347D4—61no—
348D4—61no—
349D4—82no—
350D4—82no—
351D4—82no—
352D4—81no—
353D4—81no—
354D4—81no—
355D4—81no—
356D4—121no—
357D4—121no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—41no—
361C2×C4—61no—
362C2×C4—61no—
363C2×C4—61no—
364C2×C4—121no—
365C2×C4—121no—
366C2×C4—121no—
367C2×C4—121no—
368C2×C4—241no—
369C2×C4D3×C22⋊C411yescharacteristic
370C2×C4#67⋊C411yescharacteristic
371C2×C4#88411yescharacteristic
372C2×C4#91111yescharacteristic
373C2×C4#91211yescharacteristic
374C2×C4#91411yescharacteristic
375C2×C4#91611yescharacteristic
376C2×C4C4×#6711yescharacteristic
377C2×C4#93711yescharacteristic
378C2×C4#93811yescharacteristic
379C8—41no—
380C8—41no—
381C8—81no—
382C8—122no—
383C8—122no—
384C8—241no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3—162no—
388C2×C3#187511yescharacteristic
389C2×C3#190311yescharacteristic
390C2×C3#190711yescharacteristic
391D3—81no—
392C22—21no—
393C22—21no—
394C22—82no—
395C22—82no—
396C22—82no—
397C22—121no—
398C22—121no—
399C22—121no—
400C22—162no—
401C22—162no—
402C22#477911yescharacteristic
403C22#478411yescharacteristic
404C22#495611yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—121no—
409C4—121no—
410C4—121no—
411C4#477011yescharacteristic
412C4#477211yescharacteristic
413C4#515811yescharacteristic
414C4#516011yescharacteristic
415C4#517211yescharacteristic
416C4#517711yescharacteristic
417C3#917411yescharacteristic, complemented, Sylow
418C2—21no—
419C2—21no—
420C2—162no—
421C2—241no—
422C2#7008711yescharacteristic
423C2#7020111yescharacteristic
424C2#7021811yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC2×C3×#331C2
socleC22×C3#386
prime radical#83165C2
cosocleC2×C3×C42C23
hypercenterC2×#331D3
hypocenterC3#9174
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#331

Complements

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

NH ≅ G/Ntype
2. #79000C2semidirect
3. C2×C3×#331C2semidirect
4. #71759C2semidirect
5. #70425C2semidirect
6. #67707C2semidirect
7. #67705C2semidirect
35. C2×#331D3semidirect
417. C3#9174semidirect

Maximal subgroups

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

subgroupindex
#831652
#790002
C2×C3×#3312
#717592
#704252
#677072
#677052
#91743