Finite Group Databasebeta

← #10691082

Subgroups — #10691082

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
1C3×#3038C211yesmaximal, characteristic
2#82512C211yesmaximal, characteristic, complemented
3#71759C211yesmaximal, characteristic, complemented
4C2×#4544C211yesmaximal, characteristic, complemented
5#71639C211yesmaximal, characteristic, complemented
6#67714C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#9173—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C2×C3×#127C2211yescharacteristic
11C3×C4×#41C2211yescharacteristic
12C2×#948—21no—
13C3×#460—21no—
14C3×#460C412yes—
15C3×#459—21no—
16C3×#459C412yes—
17C3×#423C2211yescharacteristic
18C3×#418—21no—
19#4999—21no—
20#4999—22no—
21#4925C2211yescharacteristic
22#4924—21no—
23#4924C2211yescharacteristic
24C2×C4⋊(C3⋊C8)—21no—
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#2906—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
62#948—42no—
63#948—42no—
64C2×#208—22no—
65C2×#208—21no—
66C22×C3⋊C8—22no—
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C4⋊C8C2×C412yes—
71C3×#102—22no—
72C3×#102—22no—
73#830—21no—
74C4⋊(C3⋊C8)—22no—
75C4⋊(C3⋊C8)—22no—
76C4⋊(C3⋊C8)—21no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D411yescharacteristic
82C4⋊(C3⋊C8)D411yescharacteristic
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85C2×AGL(1,ℤ/8)—21no—
86C2×#131—61no—
87C2×#127C2×D311yescharacteristic
88C4×#41C2×D311yescharacteristic
89#460—21no—
90#460Dic312yes—
91#459—21no—
92#459Dic312yes—
93#431—31no—
94#424—62no—
95#424—61no—
96#423—31no—
97#423—61no—
98#423C2×D311yescharacteristic
99C2×C4⋊C8—31no—
100C2×C4⋊C8—31no—
101C2×C4⋊C8—61no—
102#418—21no—
103#331—61no—
104#331—61no—
105#331—62no—
106Q8⋊C8—61no—
107D4⋊C8—61no—
108C24×C3—41no—
109C3×#41—21no—
110C3×#41—21no—
111C3×#41D812yes—
112C3×#41SD1612yes—
113C2×C3×Q8C22⋊C411yescharacteristic
114C2×C3×D4—21no—
115C2×C3×D4—22no—
116C2×C3×D4—22no—
117C2×C3×D4—22no—
118C2×C3×D4—41no—
119C2×C3×D4—41no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—81no—
126C2×C3×D4—81no—
127C2×C3×D4—82no—
128C2×C3×D4C22⋊C411yescharacteristic
129C22×C3×C4—21no—
130C22×C3×C4—21no—
131C22×C3×C4—21no—
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4#4111yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C3×SD16—44no—
136C3×D8—44no—
137C3×M16—41no—
138C3×M16—41no—
139C2×C3×C8—22no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4C22⋊C411yescharacteristic
143C3×C4⋊C4C22⋊C411yescharacteristic
144C3×C22⋊C4—21no—
145C3×C22⋊C4—21no—
146C3×C22⋊C4—21no—
147C3×C42—21no—
148C3×C42—21no—
149C3×C42—21no—
150C3×C42C22×C411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154#208—44no—
155#208—41no—
156#208—41no—
157C2×C3⋊C8—21no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—21no—
160C2×C3⋊C8—44no—
161C2×C3⋊C8—41no—
162C2×C3⋊C8—41no—
163C2×C3⋊C8—41no—
164C2×C3⋊C8—41no—
165C2×C3⋊C8—81no—
166C2×#41#6711yescharacteristic
167C22×D4—21no—
168C22×D4—21no—
169C22×D4—21no—
170AGL(1,ℤ/8)—42no—
171AGL(1,ℤ/8)—42no—
172C2×SD16—22no—
173C2×D8—22no—
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#127C2×Dic312yes—
186C4×Q8C2×Dic311yescharacteristic
187C4×Q8#6711yescharacteristic
188C4×D4—21no—
189C4×D4—21no—
190C4×D4C2×Dic311yescharacteristic
191C4×D4#6711yescharacteristic
192#117—21no—
193#117#6711yescharacteristic
194C2×C42—21no—
195C2×C42C22×D311yescharacteristic
196C4⋊C8—21no—
197C4⋊C8—31no—
198C4⋊C8—31no—
199C4⋊C8—32no—
200C4⋊C8—62no—
201C4⋊C8—62no—
202C4⋊C8—61no—
203C4⋊C8—61no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207C4⋊C8C2×Dic312yes—
208#102—22no—
209#102—22no—
210C22⋊C8—61no—
211C4×C8—61no—
212C23×C3—21no—
213C23×C3—42no—
214C23×C3—82no—
215C23×C3—82no—
216C23×C3—81no—
217C23×C3—81no—
218C23×C3#12311yescharacteristic
219C3×Q8—21no—
220C3×Q8#10212yes—
221C3×D4—22no—
222C3×D4—22no—
223C3×D4—21no—
224C3×D4—42no—
225C3×D4—42no—
226C3×D4—82no—
227C3×D4—82no—
228C3×D4—82no—
229C3×D4—81no—
230C3×D4—81no—
231C3×D4—81no—
232C3×D4—81no—
233C3×D4#10212yes—
234C2×C3×C4—21no—
235C2×C3×C4—22no—
236C2×C3×C4—22no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—81no—
245C2×C3×C4C2×C22⋊C411yescharacteristic
246C2×C3×C4C4×D411yescharacteristic
247C2×C3×C4C4×D411yescharacteristic
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4C2×D811yescharacteristic
252C2×C3×C4C2×SD1611yescharacteristic
253C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
254C2×C3×C4#13711yescharacteristic
255C3×C8—42no—
256C3⋊C8—41no—
257C3⋊C8—42no—
258C3⋊C8—41no—
259C3⋊C8—81no—
260C3⋊C8—81no—
261C24—41no—
262#41—21no—
263#41—21no—
264#41#21312yes—
265#41#21412yes—
266C2×Q8#21711yescharacteristic
267C2×D4—21no—
268C2×D4—22no—
269C2×D4—22no—
270C2×D4—22no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—81no—
279C2×D4—81no—
280C2×D4—82no—
281C2×D4#21711yescharacteristic
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4#23711yescharacteristic
286C22×C4#23711yescharacteristic
287C22×C4C2×#6711yescharacteristic
288SD16—44no—
289D8—44no—
290M16—41no—
291M16—41no—
292M16—124no—
293M16—121no—
294M16—121no—
295C2×C8—22no—
296C2×C8—61no—
297C2×C8—61no—
298C2×C8—62no—
299C2×C8—124no—
300C2×C8—121no—
301C2×C8—121no—
302C2×C8—121no—
303C2×C8—121no—
304C2×C8—241no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4#21711yescharacteristic
308C4⋊C4#21711yescharacteristic
309C22⋊C4—21no—
310C22⋊C4—21no—
311C22⋊C4—21no—
312C42—21no—
313C42—21no—
314C42—21no—
315C42D3×D411yescharacteristic
316C42D3×D411yescharacteristic
317C42C22×Dic311yescharacteristic
318C42C2×#6711yescharacteristic
319C22×C3—21no—
320C22×C3—21no—
321C22×C3—22no—
322C22×C3—41no—
323C22×C3—41no—
324C22×C3—82no—
325C22×C3—82no—
326C22×C3—82no—
327C22×C3—162no—
328C22×C3—162no—
329C22×C3#38611yescharacteristic
330C22×C3#48011yescharacteristic
331C22×C3#48111yescharacteristic
332C3×C4—22no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—41no—
337C3×C4—41no—
338C3×C4C2×#10211yescharacteristic
339C3×C4#41711yescharacteristic
340C3×C4#44711yescharacteristic
341C3×C4#44811yescharacteristic
342C3×C4#47111yescharacteristic
343C3×C4#47211yescharacteristic
344C23—21no—
345C23—42no—
346C23—82no—
347C23—82no—
348C23—81no—
349C23—81no—
350C23#93511yescharacteristic
351Q8—21no—
352Q8#83212yes—
353D4—22no—
354D4—22no—
355D4—21no—
356D4—42no—
357D4—42no—
358D4—82no—
359D4—82no—
360D4—82no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—81no—
365D4#83212yes—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—22no—
370C2×C4—22no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—81no—
377C2×C4C2×#21311yescharacteristic
378C2×C4C2×#21411yescharacteristic
379C2×C4D4×Dic311yescharacteristic
380C2×C4D4×Dic311yescharacteristic
381C2×C4#93711yescharacteristic
382C2×C4#93911yescharacteristic
383C2×C4#93911yescharacteristic
384C2×C4#94911yescharacteristic
385C2×C4#95111yescharacteristic
386C2×C4C2×#21711yescharacteristic
387C8—42no—
388C8—121no—
389C8—122no—
390C8—121no—
391C8—241no—
392C8—241no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—42no—
396C2×C3—162no—
397C2×C3#187411yescharacteristic
398C2×C3#190311yescharacteristic
399C2×C3#190711yescharacteristic
400C22—21no—
401C22—21no—
402C22—22no—
403C22—41no—
404C22—41no—
405C22—82no—
406C22—82no—
407C22—82no—
408C22—162no—
409C22—162no—
410C22#503311yescharacteristic
411C22#503411yescharacteristic
412C22#522011yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4#503611yescharacteristic
420C4#504011yescharacteristic
421C4#519911yescharacteristic
422C4#520011yescharacteristic
423C4C2×#83211yescharacteristic
424C4#523211yescharacteristic
425C3#917311yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—42no—
429C2—162no—
430C2#7167711yescharacteristic
431C2#7209011yescharacteristic
432C2#7209111yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3038C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3038D3
hypocenterC3#9173
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. #82512C2semidirect
3. #71759C2semidirect
4. C2×#4544C2semidirect
5. #71639C2semidirect
6. #67714C2semidirect
7. #67711C2semidirect
425. C3#9173semidirect

Maximal subgroups

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

subgroupindex
C3×#30382
#825122
#717592
C2×#45442
#716392
#677142
#677112
#91733