Finite Group Databasebeta

← #10840273

Subgroups — #10840273

Order 768.

646 conjugacy classes of subgroups.

The subgroup lattice has 462 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83928C211yesmaximal, characteristic
2Dic3×#135C211yesmaximal, characteristic, complemented
3C3×#1976C211yesmaximal, characteristic, complemented
4#70664C211yesmaximal, characteristic, complemented
5#70662C211yesmaximal, characteristic, complemented
6#70404C212yesmaximal, complemented
7#70395C212yesmaximal, complemented
8#70043C212yesmaximal, complemented
9#70039C212yesmaximal, complemented
10#69896C211yesmaximal, characteristic, complemented
11#69887C211yesmaximal, characteristic, complemented
12#24985—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—
24Dic3×Q16—21no—
25#5177—22no—
26Dic3×SD16—22no—
27#5160—21no—
28Dic3×D8—21no—
29C2×#825C2211yescharacteristic
30C2×#819C2211yescharacteristic
31#5106C2211yescharacteristic
32C2×C8×Dic3C2211yescharacteristic
33#5006C2212yes—
34#5002C2211yescharacteristic
35#4999C2212yes—
36#4558C2211yescharacteristic
37#4556C2212yes—
38#4532C2212yes—
39#4532C2211yescharacteristic
40#4531C2212yes—
41#4531C2212yes—
42#3067—31no—
43C4×#135—31no—
44#2080—32no—
45#2053—32no—
46#1976D311yescharacteristic, complemented
47#1945—31no—
48#1906—32no—
49#1833—31no—
50#1827—32no—
51#1812—31no—
52#1764—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—21no—
98#825—21no—
99#819—21no—
100#819—21no—
101#818—21no—
102#817—21no—
103C8×Dic3—21no—
104C8×Dic3—21no—
105#810—21no—
106#809—21no—
107#806—42no—
108C4⋊(C3⋊C8)—42no—
109#575—32no—
110C2×#135C2×D311yescharacteristic
111C2×#131—31no—
112C4×#41—32no—
113#473—61no—
114#472—61no—
115#471—62no—
116C4×Q16—61no—
117C4×SD16—62no—
118C4×D8—61no—
119C2×#108—31no—
120#427—31no—
121#427C2×D311yescharacteristic
122#424—32no—
123#421C2×D312yes—
124C2×#104—31no—
125#416—32no—
126#416—31no—
127#413—32no—
128#410—32no—
129#410C2×D312yes—
130#405C2×D311yescharacteristic
131C2×C4×C8—31no—
132#344—32no—
133#341—31no—
134#339—31no—
135#339—32no—
136H(3,ℤ/4)—32no—
137H(3,ℤ/4)—32no—
138C3×#41—22no—
139C3×#41—22no—
140C3×#41—22no—
141C3×#41—22no—
142C2×C3×Q8C2×D412yes—
143C2×C3×D4—22no—
144C2×C3×D4C2×D412yes—
145C22×C3×C4—22no—
146C22×C3×C4C2411yescharacteristic
147C2×#67—21no—
148C22×Dic3—21no—
149C22×Dic3—42no—
150#235—21no—
151#235—21no—
152#235—21no—
153#235—21no—
154C2×D12#4111yescharacteristic
155C2×C4×D3—21no—
156C2×Dic6#4111yescharacteristic
157C3×Q16—21no—
158C3×Q16—21no—
159C3×SD16—22no—
160C3×SD16—22no—
161C3×D8—21no—
162C3×D8—21no—
163C3×M16—21no—
164C3×M16—21no—
165C2×C3×C8—21no—
166C2×C3×C8C2×D411yescharacteristic
167C2×C3×C8C2×D411yescharacteristic
168C2×C3×C8#4111yescharacteristic
169C2×C3×C8#4111yescharacteristic
170C2×C3×C8#4111yescharacteristic
171C2×C3×C8#4111yescharacteristic
172C3×C4⋊C4—21no—
173C3×C4⋊C4—22no—
174C3×C22⋊C4—22no—
175C3×C22⋊C4—21no—
176C3×C42—21no—
177C3×C42—22no—
178#217—22no—
179#217—42no—
180#216—42no—
181#215—42no—
182#214—42no—
183#213—42no—
184C4⋊Dic3—22no—
185C4⋊Dic3—42no—
186C4×Dic3—21no—
187C4×Dic3—22no—
188C4×Dic3—42no—
189C4×Dic3#4111yescharacteristic
190C4×Dic3#4111yescharacteristic
191#208—22no—
192#208—21no—
193#208—21no—
194#208—22no—
195C2×C3⋊C8—21no—
196C2×C3⋊C8—21no—
197C2×C3⋊C8—21no—
198C2×C3⋊C8—21no—
199C2×C3⋊C8—22no—
200ΓL(1,𝔽25)—41no—
201ΓL(1,𝔽25)—41no—
202D3×C8—41no—
203D3×C8—41no—
204C2×#41—31no—
205C2×#41C22×D312yes—
206#137—62no—
207#137—62no—
208AGL(1,ℤ/8)—62no—
209AGL(1,ℤ/8)—62no—
210#135—21no—
211#135—21no—
212#135—21no—
213#135—21no—
214#135—122no—
215C2×Q16D1211yescharacteristic
216C2×SD16D1212yes—
217C2×D8D1211yescharacteristic
218#131—61no—
219#131—61no—
220#131—61no—
221#131—61no—
222C2×M16—31no—
223C2×M16—32no—
224C2×M16—61no—
225C2×M16C22×D311yescharacteristic
226C22×C8—31no—
227C22×C8—61no—
228C22×C8C22×D311yescharacteristic
229C4×Q8—62no—
230C4×D4—62no—
231C4×D4—122no—
232#117—32no—
233#117—62no—
234#117C22×D311yescharacteristic
235#117C22×D312yes—
236C2×C42—31no—
237C2×C42—62no—
238#108—61no—
239#108—61no—
240#107—21no—
241#107—61no—
242#106—21no—
243#106—61no—
244C4⋊C8—122no—
245#104—42no—
246#104—61no—
247#104—61no—
248#103—61no—
249#103—62no—
250#102—61no—
251#102—62no—
252#101—62no—
253#100—62no—
254#99—42no—
255#99—122no—
256#97—21no—
257C4×C8—21no—
258C4×C8—61no—
259C4×C8—61no—
260C23×C3—22no—
261C23×C3H(5,𝔽2)11yescharacteristic
262C22×D3—21no—
263C3×Q8—22no—
264C3×Q8—22no—
265C3×D4—22no—
266C3×D4—22no—
267C3×D4—42no—
268C3×D4—42no—
269C2×C3×C4—22no—
270C2×C3×C4—42no—
271C2×C3×C4—42no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4C22×D411yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4C2×#4111yescharacteristic
279C2×C3×C4H(5,𝔽2)11yescharacteristic
280#67—41no—
281#67—41no—
282C2×Dic3—21no—
283C2×Dic3—21no—
284C2×Dic3—21no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287C2×Dic3—84no—
288D12—21no—
289D12—21no—
290C4×D3—41no—
291C4×D3—41no—
292Dic6—21no—
293Dic6—21no—
294C3×C8—41no—
295C3×C8#12311yescharacteristic
296C3×C8#12311yescharacteristic
297C3×C8#12311yescharacteristic
298C3×C8#12311yescharacteristic
299C3⋊C8—41no—
300C3⋊C8—41no—
301C3⋊C8—42no—
302#41—22no—
303#41—22no—
304#41—22no—
305#41—22no—
306#41—61no—
307#41—61no—
308#41—61no—
309#41—61no—
310C2×Q8—31no—
311C2×Q8C2×D1212yes—
312C2×D4—22no—
313C2×D4—31no—
314C2×D4—61no—
315C2×D4C2×D1212yes—
316C22×C4—22no—
317C22×C4—61no—
318C22×C4—61no—
319C22×C4—122no—
320C22×C4C23×D311yescharacteristic
321Q16—21no—
322Q16—21no—
323Q16—122no—
324SD16—22no—
325SD16—22no—
326SD16—122no—
327SD16—122no—
328D8—21no—
329D8—21no—
330D8—122no—
331M16—21no—
332M16—21no—
333M16—61no—
334M16—62no—
335M16—61no—
336M16—62no—
337M16—121no—
338M16—121no—
339C2×C8—21no—
340C2×C8—61no—
341C2×C8—61no—
342C2×C8—61no—
343C2×C8—62no—
344C2×C8—61no—
345C2×C8—121no—
346C2×C8—121no—
347C2×C8C2×D1211yescharacteristic
348C2×C8C2×D1211yescharacteristic
349C2×C8#23711yescharacteristic
350C2×C8#23711yescharacteristic
351C2×C8#23711yescharacteristic
352C2×C8#23711yescharacteristic
353C4⋊C4—22no—
354C4⋊C4—21no—
355C4⋊C4—62no—
356C4⋊C4—122no—
357C22⋊C4—21no—
358C22⋊C4—22no—
359C22⋊C4—62no—
360C22⋊C4—122no—
361C42—21no—
362C42—22no—
363C42—31no—
364C42—31no—
365C42—61no—
366C42—62no—
367C42—122no—
368C22×C3—41no—
369C22×C3—42no—
370C22×C3—82no—
371C22×C3#54011yescharacteristic
372C22×C3#54611yescharacteristic
373C22×C3#55311yescharacteristic
374C2×D3—41no—
375C2×D3—81no—
376C3×C4—42no—
377C3×C4—82no—
378C3×C4—81no—
379C3×C4C2×#12311yescharacteristic
380C3×C4#52911yescharacteristic
381C3×C4#54611yescharacteristic
382C3×C4#55511yescharacteristic
383Dic3—22no—
384Dic3—41no—
385Dic3—41no—
386Dic3—82no—
387C23—22no—
388C23—61no—
389C23#100411yescharacteristic
390Q8—22no—
391Q8—22no—
392Q8—61no—
393Q8—61no—
394D4—22no—
395D4—22no—
396D4—42no—
397D4—42no—
398D4—61no—
399D4—61no—
400D4—121no—
401D4—121no—
402C2×C4—22no—
403C2×C4—42no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—61no—
408C2×C4—61no—
409C2×C4—61no—
410C2×C4—122no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—122no—
414C2×C4—244no—
415C2×C4C22×D1211yescharacteristic
416C2×C4C2×#23711yescharacteristic
417C2×C4C2×#23711yescharacteristic
418C2×C4#100411yescharacteristic
419C2×C4D3×#4111yescharacteristic
420C2×C4D3×#4111yescharacteristic
421C8—41no—
422C8—121no—
423C8—121no—
424C8—122no—
425C8#88611yescharacteristic
426C8#88611yescharacteristic
427C8#88611yescharacteristic
428C8#88611yescharacteristic
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—82no—
433C2×C3#242011yescharacteristic
434D3—81no—
435C22—41no—
436C22—42no—
437C22—82no—
438C22—121no—
439C22—241no—
440C22#555411yescharacteristic
441C22#561411yescharacteristic
442C22#571911yescharacteristic
443C4—42no—
444C4—62no—
445C4—82no—
446C4—81no—
447C4—121no—
448C4—121no—
449C4—242no—
450C4C2×#88611yescharacteristic
451C4#555411yescharacteristic
452C4#561611yescharacteristic
453C4#568011yescharacteristic
454C3#2498511yescharacteristic, complemented, Sylow
455C2—21no—
456C2—21no—
457C2—21no—
458C2—82no—
459C2—241no—
460C2#7644011yescharacteristic
subgroupG/H
centerC4#5616
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1976C2
socleC2×C3#2420
prime radical#83928C2
hypercenter#1976D3
hypocenterC3#24985
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. Dic3×#135C2semidirect
3. C3×#1976C2semidirect
4. #70664C2semidirect
5. #70662C2semidirect
6. #70404C2semidirect
7. #70395C2semidirect
8. #70043C2semidirect
9. #70039C2semidirect
10. #69896C2semidirect
11. #69887C2semidirect
46. #1976D3semidirect
454. C3#24985semidirect

Maximal subgroups

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

subgroupindex
#839282
Dic3×#1352
C3×#19762
#706642
#706622
#704042
#703952
#700432
#700392
#698962
#698872
#249853