Finite Group Databasebeta

← #10691088

Subgroups — #10691088

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#79056C211yesmaximal, characteristic
2#79040C211yesmaximal, characteristic, complemented
3#70502C211yesmaximal, characteristic, complemented
4C2×#4485C211yesmaximal, characteristic, complemented
5#70425C211yesmaximal, characteristic, complemented
6C3×#1292C211yesmaximal, characteristic, complemented
7#67714C211yesmaximal, characteristic, complemented
8#9174—31nomaximal, Sylow
9C2×#908—21no—
10C2×#874C2211yescharacteristic
11C4×#235C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C2×C3×C4⋊C8C2211yescharacteristic
15#5131—21no—
16#5129—21no—
17#5122—21no—
18#4925C2211yescharacteristic
19C2×C4⋊(C3⋊C8)C2211yescharacteristic
20#4848—21no—
21#4848C412yes—
22#4843—21no—
23#4843C412yes—
24#4840—22no—
25#4835—21no—
26#4833—21no—
27#4485—22no—
28#4485—21no—
29#4485—21no—
30#4484—21no—
31#4483—21no—
32#3038—31no—
33#2925—31no—
34#1545—31no—
35C2×#331—31no—
36#1477—31no—
37#1301—31no—
38#1292D311yescharacteristic, complemented
39C2×#235D411yescharacteristic
40C22×D12—21no—
41C22×D12—21no—
42C22×D12—21no—
43C2×C3×M16—21no—
44C2×C3×M16—21no—
45C22×C3×C8—21no—
46C2×C3×C42C2311yescharacteristic
47C4×#67—21no—
48C4×#67—21no—
49#926D411yescharacteristic
50C22×C3⋊C8—21no—
51#908—42no—
52#908—42no—
53C2×D24—22no—
54C2×#204—22no—
55#874—21no—
56#874—21no—
57#874—21no—
58#874C2×C412yes—
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⋊C8—21no—
69C3×C4⋊C8C2×C412yes—
70C3×C4⋊C8D412yes—
71#830—21no—
72#821—22no—
73#821—22no—
74(C2×D3)⋊C8—22no—
75(C2×D3)⋊C8—22no—
76#815—22no—
77C8×Dic3—22no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D411yescharacteristic
81C4⋊(C3⋊C8)D411yescharacteristic
82C4×C3⋊C8—21no—
83C2×AGL(1,ℤ/8)—61no—
84C2×#127—31no—
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#423C2×D311yescharacteristic
95#423C2×D311yescharacteristic
96C2×C4⋊C8—31no—
97C2×C4⋊C8C2×D311yescharacteristic
98#418—61no—
99#408—61no—
100#405—61no—
101#331—61no—
102#331—61no—
103#331—62no—
104Q8⋊C8—61no—
105D4⋊C8—61no—
106C23×D3—41no—
107C22×C3×C4C2×D411yescharacteristic
108C22×C3×C4#4111yescharacteristic
109C22×C3×C4#4111yescharacteristic
110C2×#67—21no—
111#235—21no—
112#235—21no—
113#235—21no—
114#235—21no—
115C2×D12—22no—
116C2×D12—22no—
117C2×D12—22no—
118C2×D12—41no—
119C2×D12—41no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—82no—
126C2×D12—81no—
127C2×D12—81no—
128C2×D12C22⋊C411yescharacteristic
129C2×C4×D3—21no—
130C2×C4×D3—21no—
131C2×C4×D3—21no—
132C2×Dic6C22⋊C411yescharacteristic
133C3×M16—41no—
134C3×M16—41no—
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—22no—
138C2×C3×C8—22no—
139C2×C3×C8—41no—
140C2×C3×C8—81no—
141C3×C42C22×C411yescharacteristic
142C3×C42C2×D411yescharacteristic
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145#217—21no—
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—
155C2×C3⋊C8—21no—
156C2×C3⋊C8—21no—
157C2×C3⋊C8—41no—
158C2×C3⋊C8—41no—
159C2×C3⋊C8—41no—
160D24—44no—
161#204—44no—
162C2×#41—31no—
163C22×D4—61no—
164C22×D4—61no—
165C22×D4—61no—
166AGL(1,ℤ/8)—122no—
167AGL(1,ℤ/8)—122no—
168C2×SD16—62no—
169C2×D8—62no—
170C2×M16—21no—
171C2×M16—21no—
172C22×C8—21no—
173C22×C8—61no—
174#127—32no—
175#127—61no—
176#127—61no—
177#127—61no—
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—21no—
192C4⋊C8—31no—
193C4⋊C8—31no—
194C4⋊C8—61no—
195C4⋊C8—61no—
196C4⋊C8C4×D312yes—
197C4⋊C8D1212yes—
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#12311yescharacteristic
207C22×D3—21no—
208C22×D3—42no—
209C22×D3—82no—
210C22×D3—82no—
211C22×D3—81no—
212C22×D3—81no—
213C2×C3×C4—21no—
214C2×C3×C4—21no—
215C2×C3×C4—41no—
216C2×C3×C4C2×C22⋊C411yescharacteristic
217C2×C3×C4C4×D411yescharacteristic
218C2×C3×C4C4×D411yescharacteristic
219C2×C3×C4#12011yescharacteristic
220C2×C3×C4#12111yescharacteristic
221C2×C3×C4#12111yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4#13511yescharacteristic
224C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
225C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
226#67—41no—
227#67—41no—
228C2×Dic3—21no—
229C2×Dic3—21no—
230C2×Dic3—21no—
231D12—21no—
232D12—21no—
233D12—42no—
234D12—42no—
235D12—82no—
236D12—82no—
237D12—82no—
238D12—81no—
239D12—81no—
240D12—81no—
241D12—81no—
242C4×D3—41no—
243C4×D3—41no—
244C4×D3—41no—
245C4×D3—41no—
246C4×D3—81no—
247Dic6—21no—
248Dic6—21no—
249C3×C8—42no—
250C3×C8—42no—
251C3×C8—81no—
252C3⋊C8—41no—
253C3⋊C8—41no—
254C3⋊C8—81no—
255C24—121no—
256#41—61no—
257#41—61no—
258#41—61no—
259#41—61no—
260C2×Q8—31no—
261C2×D4—31no—
262C2×D4—61no—
263C2×D4—62no—
264C2×D4—62no—
265C2×D4—62no—
266C2×D4—121no—
267C2×D4—121no—
268C2×D4—121no—
269C2×D4—121no—
270C2×D4—121no—
271C2×D4—121no—
272C2×D4—121no—
273C2×D4—241no—
274C2×D4—241no—
275C2×D4—242no—
276C22×C4—61no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4#23511yescharacteristic
280C22×C4D3×D411yescharacteristic
281C22×C4#23911yescharacteristic
282SD16—124no—
283D8—124no—
284M16—41no—
285M16—41no—
286M16—41no—
287M16—41no—
288C2×C8—22no—
289C2×C8—22no—
290C2×C8—41no—
291C2×C8—61no—
292C2×C8—61no—
293C2×C8—81no—
294C2×C8—121no—
295C2×C8—121no—
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
308C42C2×D1211yescharacteristic
309C42D3×D411yescharacteristic
310C42D3×D411yescharacteristic
311C22×C3—21no—
312C22×C3—21no—
313C22×C3#38611yescharacteristic
314C22×C3#48211yescharacteristic
315C22×C3#48211yescharacteristic
316C2×D3—41no—
317C2×D3—41no—
318C2×D3—41no—
319C2×D3—82no—
320C2×D3—82no—
321C2×D3—82no—
322C2×D3—162no—
323C2×D3—162no—
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—61no—
337C23—122no—
338C23—242no—
339C23—242no—
340C23—241no—
341C23—241no—
342C23#89411yescharacteristic
343Q8—61no—
344Q8—61no—
345D4—61no—
346D4—61no—
347D4—121no—
348D4—121no—
349D4—122no—
350D4—122no—
351D4—242no—
352D4—242no—
353D4—242no—
354D4—241no—
355D4—241no—
356D4—241no—
357D4—241no—
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×C4C4×D1211yescharacteristic
370C2×C4D3×C22⋊C411yescharacteristic
371C2×C4#88211yescharacteristic
372C2×C4#88411yescharacteristic
373C2×C4#89311yescharacteristic
374C2×C4#89511yescharacteristic
375C2×C4#91111yescharacteristic
376C2×C4#91411yescharacteristic
377C2×C4#91611yescharacteristic
378C2×C4#91911yescharacteristic
379C8—42no—
380C8—42no—
381C8—81no—
382C8—121no—
383C8—121no—
384C8—241no—
385C2×C3—21no—
386C2×C3—21no—
387C2×C3#187511yescharacteristic
388C2×C3#190311yescharacteristic
389C2×C3#190711yescharacteristic
390D3—81no—
391D3—162no—
392C22—21no—
393C22—21no—
394C22—121no—
395C22—121no—
396C22—121no—
397C22—242no—
398C22—242no—
399C22—242no—
400C22—482no—
401C22—482no—
402C22#466511yescharacteristic
403C22#486611yescharacteristic
404C22#488511yescharacteristic
405C4—21no—
406C4—21no—
407C4—21no—
408C4—121no—
409C4—121no—
410C4—121no—
411C4#474911yescharacteristic
412C4#475411yescharacteristic
413C4#477011yescharacteristic
414C4#478111yescharacteristic
415C4#480411yescharacteristic
416C4#481011yescharacteristic
417C3#917411yescharacteristic, complemented, Sylow
418C2—21no—
419C2—21no—
420C2—241no—
421C2—482no—
422C2#6872611yescharacteristic
423C2#6888111yescharacteristic
424C2#6900611yescharacteristic
subgroupG/H
centerC22#4665
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1292C2
socleC22×C3#386
prime radical#79056C2
cosocleC2×C3×C42C23
hypercenter#1292D3
hypocenterC3#9174
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1292

Complements

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

NH ≅ G/Ntype
2. #79040C2semidirect
3. #70502C2semidirect
4. C2×#4485C2semidirect
5. #70425C2semidirect
6. C3×#1292C2semidirect
7. #67714C2semidirect
38. #1292D3semidirect
417. C3#9174semidirect

Maximal subgroups

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

subgroupindex
#790562
#790402
#705022
C2×#44852
#704252
C3×#12922
#677142
#91743