Finite Group Databasebeta

← #10802000

Subgroups — #10802000

Order 768.

906 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83914C211yesmaximal, characteristic
2C2×#5197C211yesmaximal, characteristic, complemented
3#78656C212yesmaximal, complemented
4C2×#4708C211yesmaximal, characteristic, complemented
5C3×#1975C211yesmaximal, characteristic, complemented
6#70711C211yesmaximal, characteristic, complemented
7#70703C211yesmaximal, characteristic, complemented
8#70383C212yesmaximal, complemented
9#70369C212yesmaximal, complemented
10#69997C212yesmaximal, complemented
11#69844C211yesmaximal, characteristic, complemented
12#21547—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14#5828C2212yes—
15C2×#948C2212yes—
16C22×ΓL(1,𝔽25)C2211yescharacteristic
17C22×D3×C8C2211yescharacteristic
18C2×#872C2211yescharacteristic
19C2×C3×#108C2211yescharacteristic
20C2×C3×#104C2212yes—
21C3×#408C2212yes—
22C2×C3×#97C2211yescharacteristic
23C2×#837C2212yes—
24#5233C2212yes—
25#5197—24no—
26#5188C412yescomplemented
27#5171C414yescomplemented
28#5157C412yescomplemented
29#5126C2212yes—
30#5122C2212yes—
31C2×#819C2211yescharacteristic
32#5106C2211yescharacteristic
33#5102C2212yes—
34C2×#815C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36C2×C4×C3⋊C8C2211yescharacteristic
37#4730—24no—
38#4729—24no—
39#4708—24no—
40#4558C2212yes—
41#4555C2212yes—
42#4523C2212yes—
43#3050—31no—
44C2×#444—31no—
45#2930—32no—
46C2×#405—31no—
47#1975D311yescharacteristic, complemented
48#1946—31no—
49#1937—32no—
50#1897—32no—
51#1810—31no—
52#1773—31no—
53#1764—32no—
54C2×C3×#41C2312yes—
55C22×C4×D3C2311yescharacteristic
56C3×#135—22no—
57C3×#135—22no—
58C2×C3×Q16C2×C411yescharacteristic
59C2×C3×SD16C2×C412yes—
60C2×C3×D8C2×C411yescharacteristic
61C2×C3×M16C2312yes—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C2×C3×C42C2311yescharacteristic
65#948—24no—
66#948—24no—
67#946C2×C414yes—
68#938C2×C414yes—
69#937—22no—
70#935—22no—
71C4×#67—24no—
72C2×(C2×D3)⋊C4—21no—
73#926C2312yes—
74C2×Dic3⋊C4—21no—
75C2×C4×Dic3C2311yescharacteristic
76C2×#208—22no—
77C2×#208C2312yes—
78C22×C3⋊C8—22no—
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2311yescharacteristic
81C2×ΓL(1,𝔽25)—22no—
82C2×ΓL(1,𝔽25)—22no—
83C2×ΓL(1,𝔽25)D414yes—
84C2×D3×C8—22no—
85C2×D3×C8—22no—
86C2×D3×C8C2×C412yes—
87C2×D3×C8C2×C412yes—
88#872—22no—
89#872—22no—
90C3×#108—22no—
91C3×#104—24no—
92C3×C22⋊C8—24no—
93C3×#97—22no—
94#837—24no—
95#835C2×C414yes—
96#832C2×C414yes—
97(C2×D3)⋊C8—24no—
98#819—22no—
99#818C2×C412yes—
100#817C2×C412yes—
101#815—22no—
102#815D414yescomplemented
103Dic3⋊C8—24no—
104C8×Dic3—22no—
105C8×Dic3—22no—
106C4×C3⋊C8—22no—
107C2×#135C2×D311yescharacteristic
108C2×#131—32no—
109C22×M16—31no—
110C23×C8—31no—
111#525—32no—
112C2×#117—31no—
113#467—32no—
114#466—32no—
115#465—34no—
116#444—64no—
117#436—64no—
118#435—64no—
119C2×#108—31no—
120C2×#108C2×D311yescharacteristic
121#427—31no—
122#424—32no—
123C2×#104—32no—
124C2×#104C2×D312yes—
125#416—32no—
126#408C2×D312yes—
127#407—32no—
128#405—32no—
129#405—64no—
130C2×#97—31no—
131C2×#97C2×D311yescharacteristic
132C2×C4×C8—31no—
133C2×C4×C8—31no—
134#341—32no—
135#339—32no—
136#339—32no—
137C23×D3#4111yescharacteristic
138C3×#41—22no—
139C3×#41—22no—
140C3×#41—22no—
141C3×#41—22no—
142C2×C3×Q8C22×C412yes—
143C2×C3×D4—22no—
144C2×C3×D4C22×C412yes—
145C22×C3×C4—21no—
146C22×C3×C4—22no—
147C22×C3×C4C2411yescharacteristic
148C2×#67—24no—
149C22×Dic3—21no—
150C22×Dic3#4111yescharacteristic
151C2×C4×D3—21no—
152C2×C4×D3—21no—
153C2×C4×D3—21no—
154C2×C4×D3—21no—
155C2×C4×D3C22×C412yes—
156C2×C4×D3C2×D412yes—
157C3×Q16—22no—
158C3×SD16—24no—
159C3×D8—22no—
160C3×M16—22no—
161C3×M16—22no—
162C2×C3×C8—21no—
163C2×C3×C8—21no—
164C2×C3×C8—22no—
165C2×C3×C8C22×C411yescharacteristic
166C2×C3×C8C22×C411yescharacteristic
167C2×C3×C8C2×D412yes—
168C2×C3×C8C2×D412yes—
169C2×C3×C8#4112yes—
170C3×C42—21no—
171C3×C42#4112yes—
172#217—22no—
173#217—24no—
174(C2×D3)⋊C4—24no—
175(C2×D3)⋊C4—24no—
176C4⋊Dic3C22×C414yes—
177Dic3⋊C4—24no—
178Dic3⋊C4—24no—
179C4×Dic3—22no—
180C4×Dic3—21no—
181C4×Dic3C2×D412yes—
182#208—22no—
183#208—22no—
184#208—44no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—22no—
188C2×C3⋊C8—44no—
189C2×C3⋊C8C22×C412yes—
190C2×C3⋊C8C2×D412yes—
191C2×C3⋊C8C2×D412yes—
192C2×C3⋊C8#4112yes—
193ΓL(1,𝔽25)—28no—
194D3×C8—24no—
195D3×C8—24no—
196C2×#41C22×D312yes—
197C23×C4—31no—
198#135—22no—
199#135—22no—
200C2×Q16C4×D311yescharacteristic
201C2×SD16C4×D312yes—
202C2×D8C4×D311yescharacteristic
203#131—64no—
204#131—64no—
205C2×M16—34no—
206C2×M16—32no—
207C2×M16—62no—
208C2×M16—62no—
209C2×M16—62no—
210C2×M16C22×D312yes—
211C22×C8—31no—
212C22×C8—31no—
213C22×C8—32no—
214C22×C8—32no—
215C22×C8—62no—
216C22×C8—62no—
217C22×C8—62no—
218C22×C8C22×D311yescharacteristic
219C22×C8C22×D311yescharacteristic
220#123—62no—
221C22⋊Q8—34no—
222#121—34no—
223#120—62no—
224C4×D4—64no—
225#117—32no—
226#117—62no—
227#117—62no—
228C2×C4⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C42—31no—
231C2×C42C22×D311yescharacteristic
232#108—22no—
233#108—62no—
234#107—32no—
235#106—32no—
236C4⋊C8—64no—
237#104—24no—
238#104—64no—
239#103—34no—
240#102—34no—
241C22⋊C8—24no—
242C22⋊C8—64no—
243#97—22no—
244#97—34no—
245#97—62no—
246C4×C8—62no—
247C4×C8—62no—
248C4×C8—62no—
249C23×C3—22no—
250C23×C3C2×#4111yescharacteristic
251C22×D3—21no—
252C22×D3—21no—
253C22×D3—42no—
254C22×D3C4×D412yes—
255C3×Q8—22no—
256C3×Q8—22no—
257C3×D4—22no—
258C3×D4—22no—
259C3×D4—42no—
260C3×D4—42no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4—42no—
266C2×C3×C4C23×C411yescharacteristic
267C2×C3×C4C22×D411yescharacteristic
268C2×C3×C4C22×D411yescharacteristic
269C2×C3×C4C2×#4111yescharacteristic
270C2×C3×C4C2×#4111yescharacteristic
271C2×C3×C4C2×#4111yescharacteristic
272#67—84no—
273C2×Dic3—21no—
274C2×Dic3—21no—
275C2×Dic3—24no—
276C2×Dic3—22no—
277C2×Dic3—42no—
278C2×Dic3C4×D412yes—
279C4×D3—22no—
280C4×D3—22no—
281C4×D3—22no—
282C4×D3—22no—
283C3×C8—22no—
284C3×C8—42no—
285C3×C8C4×D412yes—
286C3×C8C4×D412yes—
287C3⋊C8—22no—
288C3⋊C8—42no—
289C3⋊C8#12114yes—
290C24—31no—
291#41—22no—
292#41—22no—
293#41—22no—
294#41—22no—
295C2×Q8C2×C4×D312yes—
296C2×D4—22no—
297C2×D4—64no—
298C2×D4C2×C4×D312yes—
299C22×C4—21no—
300C22×C4—22no—
301C22×C4—31no—
302C22×C4—32no—
303C22×C4—32no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—61no—
309C22×C4C23×D311yescharacteristic
310Q16—22no—
311SD16—24no—
312D8—22no—
313M16—22no—
314M16—22no—
315M16—68no—
316M16—62no—
317M16—62no—
318M16—124no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—22no—
322C2×C8—32no—
323C2×C8—32no—
324C2×C8—32no—
325C2×C8—32no—
326C2×C8—61no—
327C2×C8—61no—
328C2×C8—64no—
329C2×C8—64no—
330C2×C8—62no—
331C2×C8—124no—
332C2×C8C2×C4×D311yescharacteristic
333C2×C8C2×C4×D311yescharacteristic
334C2×C8D3×D412yes—
335C2×C8D3×D412yes—
336C2×C8#23712yes—
337C4⋊C4—34no—
338C4⋊C4—64no—
339C4⋊C4—64no—
340C22⋊C4—64no—
341C22⋊C4—64no—
342C22⋊C4—64no—
343C22⋊C4—62no—
344C42—21no—
345C42—32no—
346C42—62no—
347C42—61no—
348C42#23712yes—
349C22×C3—22no—
350C22×C3—44no—
351C22×C3—42no—
352C22×C3C2×C4×D411yescharacteristic
353C22×C3#52511yescharacteristic
354C22×C3#52911yescharacteristic
355C2×D3—22no—
356C2×D3—22no—
357C2×D3—22no—
358C2×D3—42no—
359C2×D3—42no—
360C3×C4—42no—
361C3×C4—42no—
362C3×C4C2×C4×D411yescharacteristic
363C3×C4C4×#4111yescharacteristic
364C3×C4C2×#12111yescharacteristic
365C3×C4#53211yescharacteristic
366Dic3—22no—
367Dic3—42no—
368Dic3—44no—
369C23—22no—
370C23—32no—
371C23—61no—
372C23—61no—
373C23—122no—
374C23D3×#4111yescharacteristic
375Q8—22no—
376Q8—22no—
377D4—22no—
378D4—22no—
379D4—42no—
380D4—42no—
381D4—244no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—32no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—62no—
393C2×C4—62no—
394C2×C4—64no—
395C2×C4—62no—
396C2×C4—122no—
397C2×C4C22×C4×D311yescharacteristic
398C2×C4C2×D3×D411yescharacteristic
399C2×C4C2×D3×D411yescharacteristic
400C2×C4C2×#23711yescharacteristic
401C2×C4C2×#23711yescharacteristic
402C2×C4D3×#4111yescharacteristic
403C8—22no—
404C8—34no—
405C8—42no—
406C8—62no—
407C8—122no—
408C8#67⋊C412yes—
409C8#67⋊C412yes—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—82no—
413C2×C3C4×#12111yescharacteristic
414C2×C3#295712yes—
415D3—42no—
416C22—22no—
417C22—42no—
418C22—44no—
419C22—62no—
420C22—62no—
421C22—62no—
422C22—122no—
423C22—122no—
424C22C4×D3×D411yescharacteristic
425C22#560611yescharacteristic
426C22#568011yescharacteristic
427C4—42no—
428C4—42no—
429C4—62no—
430C4—122no—
431C4—124no—
432C4C2×#67⋊C411yescharacteristic
433C4C4×#23711yescharacteristic
434C4D3×#12111yescharacteristic
435C4#571611yescharacteristic
436C3#2154711yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—82no—
440C2—122no—
441C2#7629111yescharacteristic
442C2#7989912yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1975C2
socleC22×C3C2×C4×D4
prime radical#83914C2
cosocleC22×C3×C4C24
hypercenter#1975D3
hypocenterC3#21547
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1975

Complements

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

NH ≅ G/Ntype
2. C2×#5197C2semidirect
3. #78656C2semidirect
4. C2×#4708C2semidirect
5. C3×#1975C2semidirect
6. #70711C2semidirect
7. #70703C2semidirect
8. #70383C2semidirect
9. #70369C2semidirect
10. #69997C2semidirect
11. #69844C2semidirect
26. #5188C4semidirect
27. #5171C4semidirect
28. #5157C4semidirect
47. #1975D3semidirect
102. #815D4semidirect
436. C3#21547semidirect

Maximal subgroups

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

subgroupindex
#839142
C2×#51972
#786562
C2×#47082
C3×#19752
#707112
#707032
#703832
#703692
#699972
#698442
#215473