Finite Group Databasebeta

← #10840284

Subgroups — #10840284

Order 768.

642 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83928C211yesmaximal, characteristic
2#83897C211yesmaximal, characteristic, complemented
3C3×#1976C211yesmaximal, characteristic, complemented
4#70664C211yesmaximal, characteristic, complemented
5#70663C211yesmaximal, characteristic, complemented
6#70404C212yesmaximal, complemented
7#70395C212yesmaximal, complemented
8#70043C212yesmaximal, complemented
9#70039C212yesmaximal, complemented
10#69896C211yesmaximal, characteristic, complemented
11#69888C211yesmaximal, characteristic, complemented
12#24986—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14Dic3×#41C2212yes—
15#5822C2212yes—
16C2×#901C2211yescharacteristic
17C3×#427C2211yescharacteristic
18C3×#421C2212yes—
19C3×#410C2212yes—
20C3×#405C2211yescharacteristic
21#5237C2212yes—
22#5233C2212yes—
23#5192—21no—
24#5184—21no—
25#5177—22no—
26#5164—22no—
27#5160—21no—
28#5152—21no—
29C2×#825C2211yescharacteristic
30C2×#819C2211yescharacteristic
31#5106C2211yescharacteristic
32C2×#815C2211yescharacteristic
33#5006C2212yes—
34#5002C2211yescharacteristic
35#4999C2212yes—
36#4558C2211yescharacteristic
37#4556C2212yes—
38#4532C2211yescharacteristic
39#4532C2212yes—
40#4531C2212yes—
41#4531C2212yes—
42#3067—31no—
43#2943—31no—
44#2080—32no—
45#2053—32no—
46#1976D311yescharacteristic, complemented
47#1946—31no—
48#1906—32no—
49#1833—31no—
50#1827—32no—
51#1812—31no—
52#1765—31no—
53C2×C3×#41C2312yes—
54C2×#235C2311yescharacteristic
55C3×#135—21no—
56C3×#135—21no—
57C3×#135—21no—
58C3×#135—21no—
59C2×C3×Q16D411yescharacteristic
60C2×C3×SD16D412yes—
61C2×C3×D8D411yescharacteristic
62C2×C3×M16C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×#117C2312yes—
65C3×#117C2311yescharacteristic
66#951—22no—
67#950—42no—
68#949—22no—
69Q8×Dic3—22no—
70#942—22no—
71D4×Dic3—22no—
72D4×Dic3—42no—
73#932—22no—
74#926—22no—
75#926C2312yes—
76C2×C4×Dic3—22no—
77C2×C4×Dic3C2311yescharacteristic
78C2×#208C2311yescharacteristic
79C2×#208C2312yes—
80C22×C3⋊C8C2311yescharacteristic
81#901—21no—
82#901—21no—
83#901—21no—
84#901—21no—
85C2×ΓL(1,𝔽25)—21no—
86C2×D3×C8—21no—
87C3×#107—21no—
88C3×#106—21no—
89C3×#104—42no—
90C3×#99—42no—
91C3×#97—21no—
92C3×C4×C8—21no—
93#836—22no—
94#835—22no—
95#833—22no—
96#832—22no—
97#825—22no—
98#819—21no—
99#819—21no—
100#818—21no—
101#817—21no—
102#815—21no—
103#815—21no—
104#810—21no—
105#809—21no—
106#806—42no—
107C4⋊(C3⋊C8)—42no—
108#575—32no—
109C2×#135C2×D311yescharacteristic
110C2×#131—31no—
111C4×#41—32no—
112#473—61no—
113#472—61no—
114#471—62no—
115#442—61no—
116#441—61no—
117#440—62no—
118C2×#108—31no—
119#427—31no—
120#427C2×D311yescharacteristic
121#424—32no—
122#421C2×D312yes—
123C2×#104—31no—
124#416—32no—
125#416—31no—
126#413—32no—
127#410—32no—
128#410C2×D312yes—
129#405C2×D311yescharacteristic
130C2×#97—31no—
131#344—32no—
132#341—31no—
133#339—31no—
134#339—32no—
135H(3,ℤ/4)—32no—
136H(3,ℤ/4)—32no—
137C3×#41—22no—
138C3×#41—22no—
139C3×#41—22no—
140C3×#41—22no—
141C2×C3×Q8C2×D412yes—
142C2×C3×D4—22no—
143C2×C3×D4C2×D412yes—
144C22×C3×C4—22no—
145C22×C3×C4C2411yescharacteristic
146C2×#67—21no—
147C22×Dic3—21no—
148C22×Dic3—42no—
149#235—21no—
150#235—21no—
151#235—21no—
152#235—21no—
153C2×D12#4111yescharacteristic
154C2×C4×D3—21no—
155C2×Dic6#4111yescharacteristic
156C3×Q16—21no—
157C3×Q16—21no—
158C3×SD16—22no—
159C3×SD16—22no—
160C3×D8—21no—
161C3×D8—21no—
162C3×M16—21no—
163C3×M16—21no—
164C2×C3×C8—21no—
165C2×C3×C8C2×D411yescharacteristic
166C2×C3×C8C2×D411yescharacteristic
167C2×C3×C8#4111yescharacteristic
168C2×C3×C8#4111yescharacteristic
169C2×C3×C8#4111yescharacteristic
170C2×C3×C8#4111yescharacteristic
171C3×C4⋊C4—21no—
172C3×C4⋊C4—22no—
173C3×C22⋊C4—22no—
174C3×C22⋊C4—21no—
175C3×C42—21no—
176C3×C42—22no—
177#217—22no—
178#217—42no—
179#216—42no—
180#215—42no—
181#214—42no—
182#213—42no—
183C4⋊Dic3—22no—
184C4⋊Dic3—42no—
185C4×Dic3—21no—
186C4×Dic3—22no—
187C4×Dic3—42no—
188C4×Dic3#4112yes—
189#208—22no—
190#208—21no—
191#208—21no—
192#208—22no—
193C2×C3⋊C8—21no—
194C2×C3⋊C8—21no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—22no—
198ΓL(1,𝔽25)—41no—
199ΓL(1,𝔽25)—41no—
200D3×C8—41no—
201D3×C8—41no—
202C2×#41—31no—
203C2×#41C22×D312yes—
204#137—62no—
205#137—62no—
206AGL(1,ℤ/8)—62no—
207AGL(1,ℤ/8)—62no—
208#135—21no—
209#135—21no—
210#135—21no—
211#135—21no—
212#135—122no—
213C2×Q16D1211yescharacteristic
214C2×SD16D1212yes—
215C2×D8D1211yescharacteristic
216#131—61no—
217#131—61no—
218#131—61no—
219#131—61no—
220C2×M16—31no—
221C2×M16—32no—
222C2×M16—61no—
223C2×M16C22×D311yescharacteristic
224C22×C8—31no—
225C22×C8—61no—
226C22×C8C22×D311yescharacteristic
227C4×Q8—62no—
228C4×D4—62no—
229C4×D4—122no—
230#117—32no—
231#117—62no—
232#117C22×D311yescharacteristic
233#117C22×D312yes—
234C2×C42—31no—
235C2×C42—62no—
236#108—61no—
237#108—61no—
238#107—21no—
239#107—61no—
240#106—21no—
241#106—61no—
242C4⋊C8—122no—
243#104—42no—
244#104—62no—
245#103—61no—
246#103—62no—
247#102—61no—
248#102—62no—
249#101—62no—
250#100—62no—
251#99—42no—
252#99—122no—
253#97—21no—
254#97—61no—
255#97—61no—
256C4×C8—21no—
257C23×C3—22no—
258C23×C3H(5,𝔽2)11yescharacteristic
259C22×D3—21no—
260C3×Q8—22no—
261C3×Q8—22no—
262C3×D4—22no—
263C3×D4—22no—
264C3×D4—42no—
265C3×D4—42no—
266C2×C3×C4—22no—
267C2×C3×C4—42no—
268C2×C3×C4—42no—
269C2×C3×C4—42no—
270C2×C3×C4—41no—
271C2×C3×C4C22×D411yescharacteristic
272C2×C3×C4C2×#4111yescharacteristic
273C2×C3×C4C2×#4111yescharacteristic
274C2×C3×C4C2×#4111yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×C3×C4H(5,𝔽2)11yescharacteristic
277#67—41no—
278#67—41no—
279C2×Dic3—22no—
280C2×Dic3—21no—
281C2×Dic3—42no—
282C2×Dic3—81no—
283C2×Dic3—84no—
284D12—21no—
285D12—21no—
286C4×D3—41no—
287C4×D3—41no—
288Dic6—21no—
289Dic6—21no—
290C3×C8—41no—
291C3×C8#12311yescharacteristic
292C3×C8#12311yescharacteristic
293C3×C8#12311yescharacteristic
294C3×C8#12311yescharacteristic
295C3⋊C8—41no—
296C3⋊C8—41no—
297C3⋊C8—42no—
298#41—22no—
299#41—22no—
300#41—22no—
301#41—22no—
302#41—61no—
303#41—61no—
304#41—61no—
305#41—61no—
306C2×Q8—31no—
307C2×Q8C2×D1212yes—
308C2×D4—22no—
309C2×D4—31no—
310C2×D4—61no—
311C2×D4C2×D1212yes—
312C22×C4—22no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—122no—
316C22×C4C23×D311yescharacteristic
317Q16—21no—
318Q16—21no—
319Q16—122no—
320SD16—22no—
321SD16—22no—
322SD16—122no—
323SD16—122no—
324D8—21no—
325D8—21no—
326D8—122no—
327M16—21no—
328M16—21no—
329M16—61no—
330M16—62no—
331M16—61no—
332M16—62no—
333M16—121no—
334M16—121no—
335C2×C8—21no—
336C2×C8—61no—
337C2×C8—61no—
338C2×C8—61no—
339C2×C8—62no—
340C2×C8—61no—
341C2×C8—121no—
342C2×C8—121no—
343C2×C8C2×D1211yescharacteristic
344C2×C8C2×D1211yescharacteristic
345C2×C8#23711yescharacteristic
346C2×C8#23711yescharacteristic
347C2×C8#23711yescharacteristic
348C2×C8#23711yescharacteristic
349C4⋊C4—22no—
350C4⋊C4—21no—
351C4⋊C4—62no—
352C4⋊C4—122no—
353C22⋊C4—21no—
354C22⋊C4—22no—
355C22⋊C4—62no—
356C22⋊C4—122no—
357C42—21no—
358C42—22no—
359C42—32no—
360C42—61no—
361C42—62no—
362C42—122no—
363C22×C3—41no—
364C22×C3—42no—
365C22×C3—82no—
366C22×C3#54011yescharacteristic
367C22×C3#54611yescharacteristic
368C22×C3#55311yescharacteristic
369C2×D3—41no—
370C2×D3—81no—
371C3×C4—42no—
372C3×C4—82no—
373C3×C4—81no—
374C3×C4C2×#12311yescharacteristic
375C3×C4#52911yescharacteristic
376C3×C4#54611yescharacteristic
377C3×C4#55511yescharacteristic
378Dic3—42no—
379Dic3—41no—
380Dic3—82no—
381C23—22no—
382C23—61no—
383C23#100411yescharacteristic
384Q8—22no—
385Q8—22no—
386Q8—61no—
387Q8—61no—
388D4—22no—
389D4—22no—
390D4—42no—
391D4—42no—
392D4—61no—
393D4—61no—
394D4—121no—
395D4—121no—
396C2×C4—22no—
397C2×C4—42no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—62no—
402C2×C4—61no—
403C2×C4—121no—
404C2×C4—121no—
405C2×C4—122no—
406C2×C4—241no—
407C2×C4—244no—
408C2×C4C22×D1211yescharacteristic
409C2×C4C2×#23711yescharacteristic
410C2×C4C2×#23711yescharacteristic
411C2×C4#100411yescharacteristic
412C2×C4D3×#4111yescharacteristic
413C2×C4D3×#4111yescharacteristic
414C8—41no—
415C8—121no—
416C8—121no—
417C8—122no—
418C8#88611yescharacteristic
419C8#88611yescharacteristic
420C8#88611yescharacteristic
421C8#88611yescharacteristic
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—82no—
426C2×C3#242011yescharacteristic
427D3—81no—
428C22—41no—
429C22—42no—
430C22—82no—
431C22—121no—
432C22—241no—
433C22#555411yescharacteristic
434C22#561411yescharacteristic
435C22#571911yescharacteristic
436C4—42no—
437C4—82no—
438C4—81no—
439C4—122no—
440C4—121no—
441C4—242no—
442C4C2×#88611yescharacteristic
443C4#555411yescharacteristic
444C4#561611yescharacteristic
445C4#568011yescharacteristic
446C3#2498611yescharacteristic, complemented, Sylow
447C2—21no—
448C2—21no—
449C2—21no—
450C2—82no—
451C2—241no—
452C2#7644011yescharacteristic
subgroupG/H
centerC4#5616
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1976C2
socleC2×C3#2420
prime radical#83928C2
hypercenter#1976D3
hypocenterC3#24986
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1976

Complements

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

NH ≅ G/Ntype
2. #83897C2semidirect
3. C3×#1976C2semidirect
4. #70664C2semidirect
5. #70663C2semidirect
6. #70404C2semidirect
7. #70395C2semidirect
8. #70043C2semidirect
9. #70039C2semidirect
10. #69896C2semidirect
11. #69888C2semidirect
46. #1976D3semidirect
446. C3#24986semidirect

Maximal subgroups

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

subgroupindex
#839282
#838972
C3×#19762
#706642
#706632
#704042
#703952
#700432
#700392
#698962
#698882
#249863