Finite Group Databasebeta

← #10691081

Subgroups — #10691081

Order 768.

562 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79056C211yesmaximal, characteristic
2#79049C211yesmaximal, characteristic, complemented
3#70502C211yesmaximal, characteristic, complemented
4C2×#4485C211yesmaximal, characteristic, complemented
5#70425C211yesmaximal, characteristic, complemented
6C3×#1292C211yesmaximal, characteristic, complemented
7#67714C211yesmaximal, characteristic, complemented
8#9173—31nomaximal, Sylow
9C2×#908—21no—
10C2×#907—21no—
11C2×#874C2211yescharacteristic
12C4×#235C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C2×C3×C4⋊C8C2211yescharacteristic
16#5131—21no—
17#5119—21no—
18#4925C2211yescharacteristic
19C2×C4⋊(C3⋊C8)C2211yescharacteristic
20#4848—21no—
21#4848C412yes—
22#4843—21no—
23#4843C412yes—
24#4839—22no—
25#4837—21no—
26D3×C4⋊C8—21no—
27#4485—22no—
28#4485—21no—
29#4485—21no—
30#4484—21no—
31#4483—21no—
32#3038—31no—
33#2906—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—
53#907—42no—
54#907—42no—
55C2×D24—22no—
56C2×#204—22no—
57C2×ΓL(1,𝔽25)—22no—
58C2×D3×C8—22no—
59#874—21no—
60#874—21no—
61#874—21no—
62#874C2×C412yes—
63C4×D12C2×C411yescharacteristic
64C4×D12D411yescharacteristic
65#872—21no—
66C42×D3—21no—
67C4×Dic6C2×C411yescharacteristic
68C4×Dic6D411yescharacteristic
69C3×C4⋊C8—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8C2×C412yes—
74C3×C4⋊C8D412yes—
75#830—21no—
76#821—22no—
77#821—22no—
78Dic3⋊C8—22no—
79Dic3⋊C8—22no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)D411yescharacteristic
83C4⋊(C3⋊C8)D411yescharacteristic
84C4×C3⋊C8—21no—
85C2×AGL(1,ℤ/8)—61no—
86C2×#131—61no—
87C2×#127—31no—
88C4×#41—31no—
89#460—32no—
90#460—61no—
91#459—32no—
92#459—61no—
93#431—31no—
94#424—61no—
95#424—62no—
96#423—61no—
97#423C2×D311yescharacteristic
98#423C2×D311yescharacteristic
99C2×C4⋊C8—31no—
100C2×C4⋊C8—61no—
101C2×C4⋊C8C2×D311yescharacteristic
102#418—61no—
103#331—61no—
104#331—61no—
105#331—62no—
106Q8⋊C8—61no—
107D4⋊C8—61no—
108C23×D3—41no—
109C22×C3×C4C2×D411yescharacteristic
110C22×C3×C4#4111yescharacteristic
111C22×C3×C4#4111yescharacteristic
112C2×#67—21no—
113#235—21no—
114#235—21no—
115#235D812yes—
116#235SD1612yes—
117C2×D12—22no—
118C2×D12—22no—
119C2×D12—22no—
120C2×D12—41no—
121C2×D12—41no—
122C2×D12—41no—
123C2×D12—41no—
124C2×D12—41no—
125C2×D12—41no—
126C2×D12—41no—
127C2×D12—82no—
128C2×D12—81no—
129C2×D12—81no—
130C2×D12C22⋊C411yescharacteristic
131C2×C4×D3—21no—
132C2×C4×D3—21no—
133C2×C4×D3—21no—
134C2×Dic6C22⋊C411yescharacteristic
135C3×M16—41no—
136C3×M16—41no—
137C3×M16—41no—
138C3×M16—41no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—41no—
142C2×C3×C8—81no—
143C3×C42C22×C411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147#217—21no—
148(C2×D3)⋊C4—21no—
149(C2×D3)⋊C4—21no—
150C4⋊Dic3C22⋊C411yescharacteristic
151C4⋊Dic3C22⋊C411yescharacteristic
152Dic3⋊C4—21no—
153Dic3⋊C4—21no—
154C4×Dic3—21no—
155C4×Dic3—21no—
156C4×Dic3—21no—
157C2×C3⋊C8—21no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—41no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162D24—44no—
163#204—44no—
164ΓL(1,𝔽25)—44no—
165D3×C8—44no—
166C2×#41—31no—
167C22×D4—61no—
168C22×D4—61no—
169C22×D4—61no—
170AGL(1,ℤ/8)—122no—
171AGL(1,ℤ/8)—122no—
172C2×SD16—62no—
173C2×D8—62no—
174#131—122no—
175#131—122no—
176C2×M16—21no—
177C2×M16—21no—
178C2×M16—62no—
179C22×C8—21no—
180C22×C8—62no—
181C22×C8—61no—
182#127—32no—
183#127—61no—
184#127—61no—
185#127—61no—
186C4×Q8—31no—
187C4×Q8—31no—
188C4×D4—31no—
189C4×D4—31no—
190C4×D4—61no—
191C4×D4—61no—
192#117—31no—
193#117—61no—
194C2×C42—61no—
195C2×C42C22×D311yescharacteristic
196C4⋊C8—21no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—31no—
201C4⋊C8—31no—
202C4⋊C8—62no—
203C4⋊C8—62no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206C4⋊C8C4×D312yes—
207C4⋊C8D1212yes—
208#102—62no—
209#102—62no—
210C22⋊C8—61no—
211C4×C8—61no—
212C23×C3#12311yescharacteristic
213C22×D3—21no—
214C22×D3—42no—
215C22×D3—82no—
216C22×D3—82no—
217C22×D3—81no—
218C22×D3—81no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—41no—
222C2×C3×C4C2×C22⋊C411yescharacteristic
223C2×C3×C4C4×D411yescharacteristic
224C2×C3×C4C4×D411yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4C2×D811yescharacteristic
229C2×C3×C4C2×SD1611yescharacteristic
230C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
231C2×C3×C4#13711yescharacteristic
232#67—22no—
233#67—22no—
234C2×Dic3—21no—
235C2×Dic3—21no—
236C2×Dic3—21no—
237D12—21no—
238D12—42no—
239D12—42no—
240D12—82no—
241D12—82no—
242D12—82no—
243D12—81no—
244D12—81no—
245D12—81no—
246D12—81no—
247D12#10212yes—
248C4×D3—22no—
249C4×D3—22no—
250C4×D3—41no—
251C4×D3—41no—
252C4×D3—81no—
253Dic6—21no—
254Dic6#10212yes—
255C3×C8—42no—
256C3×C8—42no—
257C3×C8—81no—
258C3⋊C8—41no—
259C3⋊C8—41no—
260C3⋊C8—81no—
261C24—121no—
262#41—32no—
263#41—32no—
264#41—61no—
265#41—61no—
266C2×Q8—31no—
267C2×D4—31no—
268C2×D4—61no—
269C2×D4—62no—
270C2×D4—62no—
271C2×D4—62no—
272C2×D4—121no—
273C2×D4—121no—
274C2×D4—121no—
275C2×D4—121no—
276C2×D4—121no—
277C2×D4—121no—
278C2×D4—121no—
279C2×D4—241no—
280C2×D4—241no—
281C2×D4—242no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4#23511yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4#23911yescharacteristic
288SD16—124no—
289D8—124no—
290M16—41no—
291M16—41no—
292M16—41no—
293M16—41no—
294M16—124no—
295C2×C8—22no—
296C2×C8—22no—
297C2×C8—41no—
298C2×C8—61no—
299C2×C8—61no—
300C2×C8—81no—
301C2×C8—124no—
302C2×C8—121no—
303C2×C8—121no—
304C2×C8—121no—
305C4⋊C4—31no—
306C4⋊C4—31no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C22⋊C4—61no—
310C22⋊C4—61no—
311C22⋊C4—61no—
312C42—61no—
313C42—61no—
314C42—61no—
315C42C2×C4×D311yescharacteristic
316C42C2×D1211yescharacteristic
317C42D3×D411yescharacteristic
318C42D3×D411yescharacteristic
319C22×C3—21no—
320C22×C3—21no—
321C22×C3#38611yescharacteristic
322C22×C3#48011yescharacteristic
323C22×C3#48111yescharacteristic
324C2×D3—22no—
325C2×D3—41no—
326C2×D3—41no—
327C2×D3—82no—
328C2×D3—82no—
329C2×D3—82no—
330C2×D3—162no—
331C2×D3—162no—
332C3×C4—21no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4C2×#10211yescharacteristic
336C3×C4#41711yescharacteristic
337C3×C4#44711yescharacteristic
338C3×C4#44811yescharacteristic
339C3×C4#47111yescharacteristic
340C3×C4#47211yescharacteristic
341Dic3—22no—
342Dic3—41no—
343Dic3—41no—
344C23—61no—
345C23—122no—
346C23—242no—
347C23—242no—
348C23—241no—
349C23—241no—
350C23#89411yescharacteristic
351Q8—32no—
352Q8—61no—
353D4—32no—
354D4—62no—
355D4—62no—
356D4—61no—
357D4—122no—
358D4—122no—
359D4—242no—
360D4—242no—
361D4—242no—
362D4—241no—
363D4—241no—
364D4—241no—
365D4—241no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—41no—
369C2×C4—61no—
370C2×C4—62no—
371C2×C4—62no—
372C2×C4—61no—
373C2×C4—61no—
374C2×C4—121no—
375C2×C4—121no—
376C2×C4—241no—
377C2×C4C4×D1211yescharacteristic
378C2×C4D3×C22⋊C411yescharacteristic
379C2×C4#88211yescharacteristic
380C2×C4#88411yescharacteristic
381C2×C4#89311yescharacteristic
382C2×C4#89511yescharacteristic
383C2×C4D3×D811yescharacteristic
384C2×C4D3×SD1611yescharacteristic
385C2×C4#91411yescharacteristic
386C2×C4#91811yescharacteristic
387C8—42no—
388C8—42no—
389C8—81no—
390C8—121no—
391C8—121no—
392C8—241no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3#187411yescharacteristic
396C2×C3#190311yescharacteristic
397C2×C3#190711yescharacteristic
398D3—42no—
399D3—162no—
400C22—21no—
401C22—21no—
402C22—62no—
403C22—121no—
404C22—121no—
405C22—242no—
406C22—242no—
407C22—242no—
408C22—482no—
409C22—482no—
410C22#466511yescharacteristic
411C22#486311yescharacteristic
412C22#488211yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—62no—
417C4—121no—
418C4—121no—
419C4#474911yescharacteristic
420C4#475411yescharacteristic
421C4D3×#10211yescharacteristic
422C4#477311yescharacteristic
423C4#480311yescharacteristic
424C4#480611yescharacteristic
425C3#917311yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—122no—
429C2—482no—
430C2#6872611yescharacteristic
431C2#6888011yescharacteristic
432C2#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#9173
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. #79049C2semidirect
3. #70502C2semidirect
4. C2×#4485C2semidirect
5. #70425C2semidirect
6. C3×#1292C2semidirect
7. #67714C2semidirect
38. #1292D3semidirect
425. C3#9173semidirect

Maximal subgroups

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

subgroupindex
#790562
#790492
#705022
C2×#44852
#704252
C3×#12922
#677142
#91733