Finite Group Databasebeta

← #10713682

Subgroups — #10713682

Order 768.

472 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83903C211yesmaximal, characteristic
2C3×#2143C211yesmaximal, characteristic, complemented
3#72076C211yesmaximal, characteristic, complemented
4#71450C211yesmaximal, characteristic, complemented
5C2×#4511C211yesmaximal, characteristic, complemented
6#70702C211yesmaximal, characteristic, complemented
7#70690C211yesmaximal, characteristic, complemented
8#13259—31nomaximal, Sylow
9C2×C3×#135C2211yescharacteristic
10#5832—21no—
11C2×#950—21no—
12C2×#904C2211yescharacteristic
13C2×C3×M32C2211yescharacteristic
14C2×C3×#108C2211yescharacteristic
15#5191C411yescharacteristic, complemented
16#5183C411yescharacteristic, complemented
17#5175—21no—
18#5168—21no—
19#5155C411yescharacteristic, complemented
20#5151C411yescharacteristic, complemented
21#5122—21no—
22#5106C2211yescharacteristic
23C2×C8×Dic3C2211yescharacteristic
24C2×#812C2211yescharacteristic
25C3×#359—22no—
26#4566—22no—
27#4555—21no—
28#4518—22no—
29#4511—21no—
30#4511—21no—
31#4497—22no—
32#3147—31no—
33C2×#364—31no—
34#2145—31no—
35#2145—31no—
36#2143—31no—
37#2143D311yescharacteristic, complemented
38#1772—31no—
39C2×C3×#41—21no—
40C2×#235—21no—
41C3×#135—21no—
42C3×#135—21no—
43C3×#135—21no—
44C3×#135—21no—
45C2×C3×Q16C2×C411yescharacteristic
46C2×C3×SD16—21no—
47C2×C3×D8C2×C411yescharacteristic
48C2×C3×M16—21no—
49C22×C3×C8C2311yescharacteristic
50#950—41no—
51#950—41no—
52#950—41no—
53#950—41no—
54#947—21no—
55#944—21no—
56C2×#216—21no—
57C2×#215—21no—
58#940—21no—
59#939—42no—
60#936—21no—
61C2×#214—21no—
62C2×#213—21no—
63C4×#67—42no—
64#926—21no—
65C2×C4×Dic3D411yescharacteristic
66C22×C3⋊C8D411yescharacteristic
67C2×Dic12C2×C411yescharacteristic
68#904—21no—
69#904—21no—
70#904—21no—
71#904—21no—
72C2×D24C2×C411yescharacteristic
73C2×#204—21no—
74C3×M32—21no—
75C3×M32—21no—
76C2×C3×C16—21no—
77C3×#108—21no—
78C3×#108D412yescomplemented
79#818—21no—
80#817D412yescomplemented
81#815—41no—
82C8×Dic3—41no—
83C8×Dic3C2×C411yescharacteristic
84C8×Dic3C2×C411yescharacteristic
85C8×Dic3D411yescharacteristic
86C8×Dic3D411yescharacteristic
87#812—21no—
88#812—21no—
89C2×C3⋊C16—21no—
90C2×#135—31no—
91C2×#135—61no—
92C2×#135C2×D311yescharacteristic
93#532—61no—
94C2×M32—31no—
95C2×M32C2×D311yescharacteristic
96#495—31no—
97#495—31no—
98#495—61no—
99#494—31no—
100#493—31no—
101#492—61no—
102C2×#108C2×D311yescharacteristic
103#427—31no—
104#405—61no—
105C2×C4×C8—31no—
106#364—61no—
107#364—61no—
108#360—62no—
109#360—62no—
110#359—22no—
111#359—62no—
112#339—61no—
113C3×#41—41no—
114C3×#41—41no—
115C3×#41—41no—
116C3×#41—41no—
117C2×C3×Q8—21no—
118C2×C3×D4—21no—
119C2×C3×D4—41no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C2×#67—42no—
123C2×#67—41no—
124C22×Dic3—21no—
125#235—41no—
126#235—41no—
127#235—41no—
128#235—41no—
129C2×D12—21no—
130C2×C4×D3—41no—
131C2×Dic6—21no—
132C3×Q16—21no—
133C3×Q16—21no—
134C3×SD16—41no—
135C3×SD16—41no—
136C3×D8—21no—
137C3×D8—21no—
138C3×M16—22no—
139C3×M16—41no—
140C2×C3×C8—41no—
141C2×C3×C8C22×C411yescharacteristic
142C2×C3×C8C2×D411yescharacteristic
143C2×C3×C8C2×D411yescharacteristic
144C2×C3×C8C2×D411yescharacteristic
145C2×C3×C8#4111yescharacteristic
146C2×C3×C8#4111yescharacteristic
147#217—42no—
148#217—41no—
149#216—41no—
150#216—41no—
151#215—41no—
152#215—41no—
153#214—41no—
154#214—41no—
155#213—41no—
156#213—41no—
157(C2×D3)⋊C4—42no—
158C4⋊Dic3—22no—
159Dic3⋊C4—42no—
160C4×Dic3—21no—
161C4×Dic3—41no—
162C4×Dic3C22⋊C411yescharacteristic
163C4×Dic3C22⋊C411yescharacteristic
164C2×C3⋊C8C22⋊C411yescharacteristic
165C2×C3⋊C8C22⋊C411yescharacteristic
166C2×C3⋊C8D811yescharacteristic
167C2×C3⋊C8D811yescharacteristic
168C2×C3⋊C8SD1611yescharacteristic
169C2×C3⋊C8SD1611yescharacteristic
170Dic12—21no—
171Dic12—21no—
172D24—21no—
173D24—21no—
174#204—41no—
175#204—41no—
176C3×C16—41no—
177C3⋊C16—41no—
178C2×#41—21no—
179C2×#41—61no—
180#135—21no—
181#135—21no—
182#135—21no—
183#135—21no—
184#135—61no—
185#135—61no—
186#135—61no—
187#135—61no—
188#135—121no—
189#135—121no—
190#135—121no—
191#135—121no—
192C2×Q16—31no—
193C2×Q16—61no—
194C2×Q16C4×D311yescharacteristic
195C2×SD16—21no—
196C2×SD16—61no—
197C2×SD16—61no—
198C2×SD16—61no—
199C2×D8—31no—
200C2×D8—61no—
201C2×D8C4×D311yescharacteristic
202C2×M16—21no—
203C22×C8—31no—
204C22×C8C22×D311yescharacteristic
205C4⋊Q8—61no—
206#127—61no—
207#124—61no—
208#124—61no—
209#121—122no—
210C4×D4—122no—
211#117—61no—
212C2×C42—31no—
213M32—21no—
214M32—21no—
215M32—61no—
216M32—61no—
217C2×C16—21no—
218C2×C16—61no—
219#108—21no—
220#108#6712yescomplemented
221#107—61no—
222#106—32no—
223#97—121no—
224C4×C8—31no—
225C4×C8—31no—
226C4×C8—31no—
227C4×C8—31no—
228C4×C8—121no—
229C23×C3—41no—
230C23×C3AGL(1,ℤ/8)11yescharacteristic
231C22×D3—41no—
232C3×Q8—41no—
233C3×Q8—41no—
234C3×D4—41no—
235C3×D4—41no—
236C3×D4—81no—
237C3×D4—81no—
238C2×C3×C4—41no—
239C2×C3×C4—81no—
240C2×C3×C4—81no—
241C2×C3×C4C2×C22⋊C411yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4C2×D811yescharacteristic
245C2×C3×C4C2×SD1611yescharacteristic
246C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
247#67—84no—
248#67—81no—
249#67—81no—
250C2×Dic3—21no—
251C2×Dic3—21no—
252C2×Dic3—42no—
253C2×Dic3—41no—
254C2×Dic3—42no—
255D12—41no—
256D12—41no—
257C4×D3—81no—
258C4×D3—81no—
259Dic6—41no—
260Dic6—41no—
261C3×C8—42no—
262C3×C8C4×D411yescharacteristic
263C3×C8C4×D411yescharacteristic
264C3×C8#12411yescharacteristic
265C3×C8#12711yescharacteristic
266C3⋊C8#10211yescharacteristic
267C3⋊C8#10211yescharacteristic
268C3⋊C8#10211yescharacteristic
269C3⋊C8#10211yescharacteristic
270#41—41no—
271#41—41no—
272#41—41no—
273#41—41no—
274#41—121no—
275#41—121no—
276#41—121no—
277#41—121no—
278C2×Q8—21no—
279C2×Q8—61no—
280C2×D4—21no—
281C2×D4—41no—
282C2×D4—61no—
283C2×D4—122no—
284C2×D4—121no—
285C22×C4—41no—
286C22×C4—61no—
287C22×C4—121no—
288C22×C4D3×D411yescharacteristic
289Q16—21no—
290Q16—21no—
291Q16—61no—
292Q16—61no—
293Q16—121no—
294Q16—121no—
295SD16—41no—
296SD16—41no—
297SD16—121no—
298SD16—121no—
299SD16—121no—
300SD16—121no—
301SD16—121no—
302SD16—121no—
303D8—21no—
304D8—21no—
305D8—61no—
306D8—61no—
307D8—121no—
308D8—121no—
309M16—22no—
310M16—41no—
311C2×C8—31no—
312C2×C8—31no—
313C2×C8—31no—
314C2×C8—31no—
315C2×C8—31no—
316C2×C8—31no—
317C2×C8—41no—
318C2×C8C2×C4×D311yescharacteristic
319C2×C8#23511yescharacteristic
320C2×C8D3×D411yescharacteristic
321C2×C8D3×D411yescharacteristic
322C2×C8#23711yescharacteristic
323C2×C8C2×#6711yescharacteristic
324C4⋊C4—62no—
325C4⋊C4—122no—
326C22⋊C4—122no—
327C22⋊C4—122no—
328C22⋊C4—121no—
329C42—31no—
330C42—31no—
331C42—61no—
332C42—121no—
333C16—41no—
334C16—121no—
335C22×C3—41no—
336C22×C3—82no—
337C22×C3—81no—
338C22×C3#41811yescharacteristic
339C22×C3#45911yescharacteristic
340C22×C3#46011yescharacteristic
341C2×D3—82no—
342C2×D3—81no—
343C3×C4—81no—
344C3×C4#39011yescharacteristic
345C3×C4C2×#10211yescharacteristic
346C3×C4#45911yescharacteristic
347C3×C4#46011yescharacteristic
348Dic3—22no—
349Dic3—41no—
350Dic3—82no—
351Dic3—81no—
352C23—41no—
353C23—121no—
354C23#91411yescharacteristic
355Q8—41no—
356Q8—41no—
357Q8—121no—
358Q8—121no—
359D4—41no—
360D4—41no—
361D4—81no—
362D4—81no—
363D4—121no—
364D4—121no—
365D4—244no—
366D4—241no—
367D4—241no—
368C2×C4—41no—
369C2×C4—61no—
370C2×C4—61no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—122no—
374C2×C4—121no—
375C2×C4—122no—
376C2×C4—241no—
377C2×C4—241no—
378C2×C4D3×C22⋊C411yescharacteristic
379C2×C4#88411yescharacteristic
380C2×C4D3×D811yescharacteristic
381C2×C4#91111yescharacteristic
382C2×C4D3×SD1611yescharacteristic
383C2×C4#93911yescharacteristic
384C8—31no—
385C8—31no—
386C8—31no—
387C8—31no—
388C8—42no—
389C8#67⋊C411yescharacteristic
390C8#88511yescharacteristic
391C8C4×#6711yescharacteristic
392C8#94011yescharacteristic
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—21no—
396C2×C3—161no—
397C2×C3#198211yescharacteristic
398D3—161no—
399C22—41no—
400C22—82no—
401C22—81no—
402C22—242no—
403C22—241no—
404C22#477011yescharacteristic
405C22#478611yescharacteristic
406C22#516311yescharacteristic
407C4—62no—
408C4—81no—
409C4—121no—
410C4—242no—
411C4—241no—
412C4#476211yescharacteristic
413C4D3×#10211yescharacteristic
414C4#495711yescharacteristic
415C4#515011yescharacteristic
416C3#1325911yescharacteristic, complemented, Sylow
417C2—21no—
418C2—21no—
419C2—21no—
420C2—161no—
421C2—481no—
422C2#7016111yescharacteristic
subgroupG/H
centerC4#5150
commutatorC2×C3×C8C22×C4
FrattiniC22×C8C22×D3
FittingC3×#2143C2
socleC2×C3#1982
prime radical#83903C2
cosocleC22×C3×C8C23
hypercenter#2143D3
hypocenterC3#13259
LCSG ▷ C2×C3×C8 ▷ C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ C22×C8 ◁ #2143

Complements

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

NH ≅ G/Ntype
2. C3×#2143C2semidirect
3. #72076C2semidirect
4. #71450C2semidirect
5. C2×#4511C2semidirect
6. #70702C2semidirect
7. #70690C2semidirect
15. #5191C4semidirect
16. #5183C4semidirect
19. #5155C4semidirect
20. #5151C4semidirect
37. #2143D3semidirect
78. C3×#108D4semidirect
80. #817D4semidirect
220. #108#67semidirect
416. C3#13259semidirect

Maximal subgroups

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

subgroupindex
#839032
C3×#21432
#720762
#714502
C2×#45112
#707022
#706902
#132593