Finite Group Databasebeta

← #10691077

Subgroups — #10691077

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#83165C211yesmaximal, characteristic
2#79049C211yesmaximal, characteristic, complemented
3C3×#1545C211yesmaximal, characteristic, complemented
4C2×#4544C211yesmaximal, characteristic, complemented
5#70755C211yesmaximal, characteristic, complemented
6#67745C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#9173—31nomaximal, Sylow
9C2×C3×#127C2211yescharacteristic
10C2×#932—21no—
11C2×#907—21no—
12C4×#235C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15#5215—21no—
16#5119—21no—
17#5075—21no—
18#5075C412yes—
19#5072—21no—
20#5072C412yes—
21#4924C2211yescharacteristic
22C2×C4⋊(C3⋊C8)C2211yescharacteristic
23C2×C4⋊(C3⋊C8)C2211yescharacteristic
24#4839—22no—
25#4837—21no—
26D3×C4⋊C8—21no—
27C3×#331—22no—
28#4544—21no—
29#4544—21no—
30#4459—21no—
31#4456—21no—
32#3038—31no—
33#2906—31no—
34#1545D311yescharacteristic, complemented
35C2×#331—31no—
36#1477—31no—
37#1301—31no—
38#1292—31no—
39C22×C3×D4—21no—
40C22×C3×D4—21no—
41C22×C3×D4—21no—
42C2×#235D411yescharacteristic
43C2×C3×M16—21no—
44C3×#127—21no—
45C3×#127—21no—
46C3×#127—21no—
47C3×#127C2×C412yes—
48C2×C3×C42C2311yescharacteristic
49C2×#214—22no—
50#932—42no—
51#932—42no—
52C2×#213—22no—
53C4×#67—21no—
54C4×#67—21no—
55#926D411yescharacteristic
56C2×#208—21no—
57C22×C3⋊C8—21no—
58C22×C3⋊C8—21no—
59#907—42no—
60#907—42no—
61C2×ΓL(1,𝔽25)—22no—
62C2×D3×C8—22no—
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⋊C8D412yes—
72C3×C22⋊C8—21no—
73C3×C4×C8—21no—
74#832—22no—
75#832—22no—
76Dic3⋊C8—22no—
77Dic3⋊C8—22no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)—21no—
82C4⋊(C3⋊C8)C2×C412yes—
83C4⋊(C3⋊C8)D411yescharacteristic
84C4⋊(C3⋊C8)D411yescharacteristic
85C2×AGL(1,ℤ/8)—61no—
86C2×#131—61no—
87C2×#127C2×D311yescharacteristic
88C4×#41—31no—
89#460—32no—
90#460—61no—
91#459—32no—
92#459—61no—
93#431C2×D311yescharacteristic
94#424—61no—
95#424—62no—
96#423—31no—
97#423—61no—
98#423C2×D311yescharacteristic
99C2×C4⋊C8—31no—
100C2×C4⋊C8—31no—
101C2×C4⋊C8—61no—
102#418—61no—
103#331—22no—
104#331—61no—
105#331—61no—
106Q8⋊C8—61no—
107D4⋊C8—61no—
108C24×C3—41no—
109C2×C3×D4—22no—
110C2×C3×D4—22no—
111C2×C3×D4—22no—
112C2×C3×D4—41no—
113C2×C3×D4—41no—
114C2×C3×D4—41no—
115C2×C3×D4—41no—
116C2×C3×D4—41no—
117C2×C3×D4—41no—
118C2×C3×D4—41no—
119C2×C3×D4—81no—
120C2×C3×D4—81no—
121C2×C3×D4—82no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C22×C3×C4#4111yescharacteristic
125C2×#67—21no—
126#235—21no—
127#235—21no—
128#235D812yes—
129#235SD1612yes—
130C2×D12C22⋊C411yescharacteristic
131C2×C4×D3—21no—
132C2×C4×D3—21no—
133C2×C4×D3—21no—
134C2×Dic6C22⋊C411yescharacteristic
135C3×M16—41no—
136C3×M16—41no—
137C2×C3×C8—22no—
138C2×C3×C8—41no—
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—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—21no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—41no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—81no—
164ΓL(1,𝔽25)—44no—
165D3×C8—44no—
166C2×#41—31no—
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
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—62no—
178C2×M16—61no—
179C22×C8—62no—
180C22×C8—61no—
181C22×C8—61no—
182#127—21no—
183#127—21no—
184#127—21no—
185#127C4×D312yes—
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—32no—
199C4⋊C8—31no—
200C4⋊C8—31no—
201C4⋊C8—62no—
202C4⋊C8—62no—
203C4⋊C8—61no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8#6712yes—
208#102—62no—
209#102—62no—
210C22⋊C8—21no—
211C4×C8—21no—
212C23×C3—42no—
213C23×C3—82no—
214C23×C3—82no—
215C23×C3—81no—
216C23×C3—81no—
217C23×C3#12311yescharacteristic
218C22×D3—21no—
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—41no—
231C2×C3×C4C2×C22⋊C411yescharacteristic
232C2×C3×C4C4×D411yescharacteristic
233C2×C3×C4C4×D411yescharacteristic
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12111yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4C2×D811yescharacteristic
238C2×C3×C4C2×SD1611yescharacteristic
239C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
240C2×C3×C4#13711yescharacteristic
241#67—22no—
242#67—22no—
243C2×Dic3—21no—
244C2×Dic3—21no—
245C2×Dic3—21no—
246D12—21no—
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—81no—
257C3⋊C8—42no—
258C3⋊C8—41no—
259C3⋊C8—41no—
260C3⋊C8—81no—
261C24—41no—
262#41—32no—
263#41—32no—
264#41—61no—
265#41—61no—
266C2×Q8—31no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—31no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—61no—
279C2×D4—81no—
280C2×D4—81no—
281C2×D4—82no—
282C22×C4—61no—
283C22×C4—61no—
284C22×C4—61no—
285C22×C4#23511yescharacteristic
286C22×C4D3×D411yescharacteristic
287C22×C4#23711yescharacteristic
288SD16—124no—
289D8—124no—
290M16—41no—
291M16—41no—
292M16—124no—
293M16—121no—
294M16—121no—
295C2×C8—22no—
296C2×C8—41no—
297C2×C8—62no—
298C2×C8—61no—
299C2×C8—61no—
300C2×C8—124no—
301C2×C8—121no—
302C2×C8—121no—
303C2×C8—121no—
304C2×C8—241no—
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
316C42D3×D411yescharacteristic
317C42D3×D411yescharacteristic
318C42C2×#6711yescharacteristic
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—82no—
322C22×C3—82no—
323C22×C3—82no—
324C22×C3—162no—
325C22×C3—162no—
326C22×C3#38611yescharacteristic
327C22×C3#48011yescharacteristic
328C22×C3#48111yescharacteristic
329C2×D3—22no—
330C2×D3—41no—
331C2×D3—41no—
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—42no—
345C23—61no—
346C23—82no—
347C23—82no—
348C23—81no—
349C23—81no—
350C23#88311yescharacteristic
351Q8—32no—
352Q8—61no—
353D4—32no—
354D4—42no—
355D4—42no—
356D4—62no—
357D4—62no—
358D4—61no—
359D4—82no—
360D4—82no—
361D4—82no—
362D4—81no—
363D4—81no—
364D4—81no—
365D4—81no—
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×C4D3×C22⋊C411yescharacteristic
378C2×C4#67⋊C411yescharacteristic
379C2×C4#88411yescharacteristic
380C2×C4D3×D811yescharacteristic
381C2×C4#91111yescharacteristic
382C2×C4D3×SD1611yescharacteristic
383C2×C4#91511yescharacteristic
384C2×C4C4×#6711yescharacteristic
385C2×C4#93711yescharacteristic
386C2×C4#93811yescharacteristic
387C8—42no—
388C8—81no—
389C8—121no—
390C8—121no—
391C8—122no—
392C8—241no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—162no—
396C2×C3#187411yescharacteristic
397C2×C3#190311yescharacteristic
398C2×C3#190711yescharacteristic
399D3—42no—
400C22—21no—
401C22—21no—
402C22—62no—
403C22—82no—
404C22—82no—
405C22—82no—
406C22—121no—
407C22—121no—
408C22—162no—
409C22—162no—
410C22#477411yescharacteristic
411C22#477711yescharacteristic
412C22#495611yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—62no—
417C4—121no—
418C4—121no—
419C4#474811yescharacteristic
420C4#474911yescharacteristic
421C4D3×#10211yescharacteristic
422C4#477111yescharacteristic
423C4#515611yescharacteristic
424C4#517011yescharacteristic
425C3#917311yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—122no—
429C2—162no—
430C2#7008811yescharacteristic
431C2#7020011yescharacteristic
432C2#7032911yescharacteristic
subgroupG/H
centerC22#4956
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1545C2
socleC22×C3#386
prime radical#83165C2
cosocleC2×C3×C42C23
hypercenter#1545D3
hypocenterC3#9173
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1545

Complements

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

NH ≅ G/Ntype
2. #79049C2semidirect
3. C3×#1545C2semidirect
4. C2×#4544C2semidirect
5. #70755C2semidirect
6. #67745C2semidirect
7. #67711C2semidirect
34. #1545D3semidirect
425. C3#9173semidirect

Maximal subgroups

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

subgroupindex
#831652
#790492
C3×#15452
C2×#45442
#707552
#677452
#677112
#91733