Finite Group Databasebeta

← #10692077

Subgroups — #10692077

Order 768.

558 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3038C211yesmaximal, characteristic
2#82521C211yesmaximal, characteristic, complemented
3#71761C211yesmaximal, characteristic, complemented
4#71759C211yesmaximal, characteristic, complemented
5#71646C211yesmaximal, characteristic, complemented
6#67715C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#9322—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C2×C3×#127C2211yescharacteristic
11C3×C4×#41C2211yescharacteristic
12C3×#460—21no—
13C3×#460C412yescomplemented
14C3×#459—21no—
15C3×#459C412yescomplemented
16C3×#423C2211yescharacteristic
17C3×#418—21no—
18C2×#837—21no—
19C2×#832—22no—
20#5024—21no—
21#5012—21no—
22#5011—22no—
23C4×C4⋊Dic3C2211yescharacteristic
24#4926—21no—
25#4925C2211yescharacteristic
26#4924—21no—
27#4924C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29#4544—21no—
30#4544—21no—
31#4532—21no—
32#3038D311yescharacteristic
33#2989—31no—
34#1982—31no—
35#1806—31no—
36#1545—31no—
37#1303—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
62C2×C4⋊Dic3—21no—
63C2×C4×Dic3—21no—
64C2×#208—22no—
65C2×#208—21no—
66C22×C3⋊C8D412yes—
67C3×C4⋊C8—21no—
68C3×C4⋊C8C2×C412yes—
69C3×#102—22no—
70C3×#102—22no—
71#837—42no—
72#832—44no—
73C3⋊#95—21no—
74#830—22no—
75#830—21no—
76C4⋊(C3⋊C8)—22no—
77C4⋊(C3⋊C8)—21no—
78C4⋊(C3⋊C8)—21no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)—21no—
81C4⋊(C3⋊C8)D411yescharacteristic
82C4⋊(C3⋊C8)D411yescharacteristic
83#803—22no—
84C4×C3⋊C8—21no—
85C2×AGL(1,ℤ/8)—21no—
86C2×#127C2×D311yescharacteristic
87C4×#41C2×D311yescharacteristic
88#460—21no—
89#460Dic312yescomplemented
90#459—21no—
91#459Dic312yescomplemented
92#446—61no—
93#436—61no—
94#435—62no—
95#432—61no—
96#431—31no—
97#423—31no—
98#423—61no—
99#423C2×D311yescharacteristic
100C2×C4⋊C8—31no—
101C2×#104—61no—
102#418—21no—
103C2×#102—62no—
104C4×C4⋊C4—31no—
105#339—61no—
106#331—61no—
107#331—61no—
108C24×C3—41no—
109C3×#41—21no—
110C3×#41—21no—
111C3×#41—21no—
112C3×#41—21no—
113C2×C3×Q8C22⋊C411yescharacteristic
114C2×C3×D4—21no—
115C2×C3×D4—22no—
116C2×C3×D4—22no—
117C2×C3×D4—22no—
118C2×C3×D4—42no—
119C2×C3×D4—42no—
120C2×C3×D4—41no—
121C2×C3×D4—41no—
122C2×C3×D4—41no—
123C2×C3×D4—41no—
124C2×C3×D4—41no—
125C2×C3×D4—41no—
126C2×C3×D4—41no—
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×C4C2×D411yescharacteristic
134C22×C3×C4#4111yescharacteristic
135C22×Dic3—41no—
136C3×SD16—44no—
137C3×D8—44no—
138C3×M16—41no—
139C3×M16—41no—
140C2×C3×C8—22no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4C22⋊C411yescharacteristic
144C3×C4⋊C4C22⋊C411yescharacteristic
145C3×C22⋊C4—21no—
146C3×C22⋊C4—21no—
147C3×C22⋊C4—21no—
148C3×C42—21no—
149C3×C42—21no—
150C3×C42—21no—
151C3×C42C22×C411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42#4111yescharacteristic
155C4⋊Dic3—22no—
156C4⋊Dic3—41no—
157C4×Dic3—41no—
158C4×Dic3—41no—
159#208—41no—
160#208—41no—
161#208—82no—
162C2×C3⋊C8—21no—
163C2×C3⋊C8—21no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—42no—
166C2×C3⋊C8—41no—
167C2×#41#6711yescharacteristic
168C22×D4—21no—
169C22×D4—21no—
170C22×D4—21no—
171AGL(1,ℤ/8)—42no—
172AGL(1,ℤ/8)—42no—
173C2×SD16—22no—
174C2×D8—22no—
175C2×M16—21no—
176C2×M16—62no—
177C2×M16—61no—
178C22×C8—32no—
179#127—21no—
180#127—21no—
181#127—21no—
182#127C2×Dic312yes—
183C4×Q8C2×Dic311yescharacteristic
184C4×Q8#6711yescharacteristic
185C4×D4—21no—
186C4×D4—21no—
187C4×D4C2×Dic311yescharacteristic
188C4×D4#6711yescharacteristic
189#117—21no—
190#117#6711yescharacteristic
191C2×C4⋊C4—61no—
192C2×C42—21no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195C4⋊C8—21no—
196C4⋊C8—31no—
197C4⋊C8—31no—
198C4⋊C8—62no—
199C4⋊C8—61no—
200C4⋊C8—61no—
201C4⋊C8—61no—
202C4⋊C8—61no—
203C4⋊C8C2×Dic312yes—
204#104—122no—
205#102—22no—
206#102—22no—
207#102—124no—
208C22⋊C8—62no—
209C22⋊C8—61no—
210#97—62no—
211C4×C8—61no—
212#95—61no—
213C23×C3—21no—
214C23×C3—42no—
215C23×C3—82no—
216C23×C3—82no—
217C23×C3—81no—
218C23×C3—81no—
219C23×C3#12111yescharacteristic
220C3×Q8—21no—
221C3×Q8—21no—
222C3×D4—21no—
223C3×D4—21no—
224C3×D4—44no—
225C3×D4—41no—
226C3×D4—41no—
227C3×D4—42no—
228C3×D4—42no—
229C3×D4—82no—
230C3×D4—82no—
231C3×D4—81no—
232C3×D4—81no—
233C3×D4—81no—
234C3×D4—81no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—21no—
238C2×C3×C4—21no—
239C2×C3×C4—21no—
240C2×C3×C4—41no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—81no—
246C2×C3×C4C2×C22⋊C411yescharacteristic
247C2×C3×C4C4×D411yescharacteristic
248C2×C3×C4C4×D411yescharacteristic
249C2×C3×C4#12011yescharacteristic
250C2×C3×C4#12111yescharacteristic
251C2×C3×C4#12311yescharacteristic
252C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
253C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
254C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
255C2×C3×C4#13711yescharacteristic
256C2×Dic3—42no—
257C2×Dic3—81no—
258C2×Dic3—81no—
259C3×C8—42no—
260C3⋊C8—42no—
261C3⋊C8—41no—
262C3⋊C8—41no—
263C3⋊C8—81no—
264C24—41no—
265#41—21no—
266#41—21no—
267#41—21no—
268#41—21no—
269C2×Q8#21711yescharacteristic
270C2×D4—21no—
271C2×D4—22no—
272C2×D4—22no—
273C2×D4—22no—
274C2×D4—42no—
275C2×D4—42no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—41no—
281C2×D4—41no—
282C2×D4—41no—
283C2×D4—82no—
284C2×D4#21711yescharacteristic
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—21no—
288C22×C4—121no—
289C22×C4D3×D411yescharacteristic
290C22×C4#23711yescharacteristic
291C22×C4C2×#6711yescharacteristic
292SD16—44no—
293D8—44no—
294M16—41no—
295M16—41no—
296M16—121no—
297M16—121no—
298M16—242no—
299C2×C8—22no—
300C2×C8—61no—
301C2×C8—61no—
302C2×C8—62no—
303C2×C8—122no—
304C2×C8—121no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—62no—
308C4⋊C4—121no—
309C4⋊C4#21711yescharacteristic
310C4⋊C4#21711yescharacteristic
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C42—21no—
315C42—21no—
316C42—21no—
317C42—121no—
318C42—121no—
319C42D3×D411yescharacteristic
320C42#23711yescharacteristic
321C42C22×Dic311yescharacteristic
322C42C2×#6711yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—82no—
329C22×C3—82no—
330C22×C3—82no—
331C22×C3—162no—
332C22×C3—162no—
333C22×C3#38611yescharacteristic
334C22×C3#46811yescharacteristic
335C22×C3#46911yescharacteristic
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—41no—
340C3×C4—41no—
341C3×C4—41no—
342C3×C4#41711yescharacteristic
343C3×C4#41811yescharacteristic
344C3×C4#45311yescharacteristic
345C3×C4#45411yescharacteristic
346C3×C4#46811yescharacteristic
347C3×C4#46911yescharacteristic
348Dic3—82no—
349C23—21no—
350C23—42no—
351C23—82no—
352C23—82no—
353C23—81no—
354C23—81no—
355C23#93911yescharacteristic
356Q8—21no—
357Q8—21no—
358D4—21no—
359D4—21no—
360D4—44no—
361D4—41no—
362D4—41no—
363D4—42no—
364D4—42no—
365D4—82no—
366D4—82no—
367D4—81no—
368D4—81no—
369D4—81no—
370D4—81no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—81no—
382C2×C4—122no—
383C2×C4—241no—
384C2×C4—241no—
385C2×C4#93211yescharacteristic
386C2×C4#93211yescharacteristic
387C2×C4D4×Dic311yescharacteristic
388C2×C4D4×Dic311yescharacteristic
389C2×C4#93511yescharacteristic
390C2×C4#93711yescharacteristic
391C2×C4#93911yescharacteristic
392C2×C4#94911yescharacteristic
393C2×C4#95111yescharacteristic
394C2×C4C2×#21711yescharacteristic
395C8—42no—
396C8—122no—
397C8—121no—
398C8—121no—
399C8—241no—
400C2×C3—21no—
401C2×C3—21no—
402C2×C3—81no—
403C2×C3—162no—
404C2×C3#188311yescharacteristic
405C2×C3#189511yescharacteristic
406C2×C3#190311yescharacteristic
407C22—21no—
408C22—21no—
409C22—41no—
410C22—41no—
411C22—41no—
412C22—82no—
413C22—82no—
414C22—82no—
415C22—162no—
416C22—162no—
417C22#503911yescharacteristic
418C22#504211yescharacteristic
419C22#522011yescharacteristic
420C4—21no—
421C4—21no—
422C4—21no—
423C4—41no—
424C4—41no—
425C4—41no—
426C4—242no—
427C4#503911yescharacteristic
428C4#504211yescharacteristic
429C4#519811yescharacteristic
430C4#520111yescharacteristic
431C4#521511yescharacteristic
432C4#523211yescharacteristic
433C3#932211yescharacteristic, complemented, Sylow
434C2—21no—
435C2—21no—
436C2—81no—
437C2—162no—
438C2#7168211yescharacteristic
439C2#7208811yescharacteristic
440C2#7209011yescharacteristic
subgroupG/H
centerC22#5220
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3038C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3038D3
hypocenterC3#9322
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. #82521C2semidirect
3. #71761C2semidirect
4. #71759C2semidirect
5. #71646C2semidirect
6. #67715C2semidirect
7. #67711C2semidirect
13. C3×#460C4semidirect
15. C3×#459C4semidirect
89. #460Dic3semidirect
91. #459Dic3semidirect
433. C3#9322semidirect

Maximal subgroups

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

subgroupindex
C3×#30382
#825212
#717612
#717592
#716462
#677152
#677112
#93223