Finite Group Databasebeta

← #10840287

Subgroups — #10840287

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#79256C211yesmaximal, characteristic
2#78494C211yesmaximal, characteristic, complemented
3C3×#1946C211yesmaximal, characteristic, complemented
4#70698C211yesmaximal, characteristic, complemented
5#70664C211yesmaximal, characteristic, complemented
6#70658C211yesmaximal, characteristic, complemented
7#70548C212yesmaximal, complemented
8#70547C212yesmaximal, complemented
9#69839C211yesmaximal, characteristic, complemented
10#68778C212yesmaximal, complemented
11#68737C212yesmaximal, complemented
12#24986—31nomaximal, Sylow
13#5748C2212yes—
14C2×#904C2211yescharacteristic
15C2×#901C2211yescharacteristic
16C4×#235C2212yes—
17C2×C3×#108C2211yescharacteristic
18C3×#427C2211yescharacteristic
19C3×#424C2212yes—
20C2×C3×#97C2211yescharacteristic
21#5132C2212yes—
22#5113C2211yescharacteristic
23#5113C2212yes—
24#5106C2211yescharacteristic
25#5101C2211yescharacteristic
26#5006C2212yes—
27#5006C2212yes—
28#5005C2212yes—
29C2×#805C2211yescharacteristic
30#4898—22no—
31#4897—21no—
32#4896—21no—
33#4716—21no—
34#4711—21no—
35#4710—22no—
36C3×#339C2212yes—
37#4559C2212yes—
38#4557C2212yes—
39#4552C2211yescharacteristic
40#4531C2212yes—
41#4523C2211yescharacteristic
42#3067—31no—
43#2943—31no—
44#2080—32no—
45#2053—32no—
46#1976—31no—
47#1946D311yescharacteristic, complemented
48#1906—32no—
49#1833—31no—
50#1827—32no—
51#1812—31no—
52#1765—31no—
53C2×#235C2312yes—
54C2×#235C2311yescharacteristic
55C2×C3×M16C2312yes—
56C22×C3×C8C2311yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C3×#117C2312yes—
59C2×C3×C42C2311yescharacteristic
60C4×#67—42no—
61#926C2312yes—
62#926C2311yescharacteristic
63C2×#208C2311yescharacteristic
64C2×#208C2311yescharacteristic
65#909—22no—
66#909—22no—
67#908—22no—
68#908—22no—
69C2×Dic12D411yescharacteristic
70#904—21no—
71#904—21no—
72#904—21no—
73#904—21no—
74#904—42no—
75C2×D24D411yescharacteristic
76C2×#204D412yes—
77#901—21no—
78#901—21no—
79#901—21no—
80#901—21no—
81C2×ΓL(1,𝔽25)—21no—
82C2×D3×C8—21no—
83C4×D12—22no—
84#872—22no—
85C42×D3—22no—
86C4×Dic6—22no—
87C3×#108—21no—
88C3×#108—21no—
89C3×#107—21no—
90C3×#106—21no—
91C3×C4⋊C8—42no—
92C3×#97—21no—
93C3×#97—21no—
94#824—22no—
95#823—22no—
96#821—22no—
97#821—21no—
98#818—21no—
99#817—21no—
100#816—22no—
101#816—21no—
102#815—21no—
103C8×Dic3—21no—
104#806—42no—
105#806—42no—
106#805—22no—
107#805—42no—
108#575—32no—
109C2×#135—31no—
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×#108C2×D311yescharacteristic
119#427—31no—
120#427C2×D311yescharacteristic
121#424C2×D312yes—
122#421—32no—
123C2×#104—31no—
124#416—31no—
125#416—32no—
126#413—32no—
127#410—32no—
128#410—32no—
129#405—31no—
130C2×#97C2×D311yescharacteristic
131#344—32no—
132#341—31no—
133#339—31no—
134#339C2×D312yes—
135H(3,ℤ/4)—32no—
136H(3,ℤ/4)—32no—
137C22×C3×C4—21no—
138C22×C3×C4C2411yescharacteristic
139C2×#67—21no—
140C2×#67—22no—
141#235—21no—
142#235—21no—
143#235—21no—
144#235—22no—
145#235—22no—
146#235—22no—
147#235—22no—
148#235—21no—
149C2×D12C2×D412yes—
150C2×D12#4111yescharacteristic
151C2×C4×D3—22no—
152C2×C4×D3—21no—
153C2×C4×D3—42no—
154C2×Dic6C2×D412yes—
155C2×Dic6#4111yescharacteristic
156C3×M16—22no—
157C3×M16—22no—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8—22no—
162C2×C3×C8C2×D411yescharacteristic
163C2×C3×C8C2×D411yescharacteristic
164C2×C3×C8#4111yescharacteristic
165C2×C3×C8#4111yescharacteristic
166C2×C3×C8#4111yescharacteristic
167C2×C3×C8#4111yescharacteristic
168C3×C4⋊C4—22no—
169C3×C22⋊C4—22no—
170C3×C42—21no—
171C3×C42—22no—
172C3×C42#4112yes—
173#217—21no—
174#217—22no—
175(C2×D3)⋊C4—42no—
176C4⋊Dic3—22no—
177C4⋊Dic3—21no—
178Dic3⋊C4—42no—
179C4×Dic3—21no—
180C4×Dic3—22no—
181C4×Dic3—42no—
182#208—21no—
183#208—21no—
184#208—21no—
185#208—21no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—21no—
188Dic12—21no—
189Dic12—21no—
190Dic12—42no—
191D24—21no—
192D24—21no—
193D24—42no—
194#204—22no—
195#204—22no—
196#204—42no—
197#204—42no—
198ΓL(1,𝔽25)—41no—
199ΓL(1,𝔽25)—41no—
200D3×C8—41no—
201D3×C8—41no—
202C2×#41—32no—
203C2×#41—31no—
204#137—62no—
205#137—62no—
206AGL(1,ℤ/8)—62no—
207AGL(1,ℤ/8)—62no—
208#135—61no—
209#135—61no—
210#135—61no—
211#135—61no—
212#135—122no—
213C2×Q16—31no—
214C2×SD16—32no—
215C2×D8—31no—
216#131—61no—
217#131—61no—
218#131—61no—
219#131—61no—
220C2×M16—31no—
221C2×M16—31no—
222C2×M16—61no—
223C2×M16C22×D312yes—
224C22×C8—61no—
225C22×C8C22×D311yescharacteristic
226C22×C8C22×D311yescharacteristic
227C4×Q8—62no—
228C4×D4—62no—
229C4×D4—122no—
230#117—32no—
231#117—31no—
232#117—62no—
233#117C22×D312yes—
234C2×C42—62no—
235C2×C42C22×D311yescharacteristic
236#108—21no—
237#108—21no—
238#107—21no—
239#107—61no—
240#106—21no—
241#106—61no—
242C4⋊C8—42no—
243#104—62no—
244#104—122no—
245#103—61no—
246#103—62no—
247#102—61no—
248#102—62no—
249#101—62no—
250#100—62no—
251#99—122no—
252#99—122no—
253#97—21no—
254#97—21no—
255#97—61no—
256C4×C8—61no—
257C23×C3H(5,𝔽2)11yescharacteristic
258C22×D3—22no—
259C22×D3—21no—
260C2×C3×C4—22no—
261C2×C3×C4—42no—
262C2×C3×C4—81no—
263C2×C3×C4C22×D411yescharacteristic
264C2×C3×C4C2×#4111yescharacteristic
265C2×C3×C4C2×#4111yescharacteristic
266C2×C3×C4C2×#4111yescharacteristic
267C2×C3×C4C2×#4111yescharacteristic
268C2×C3×C4H(5,𝔽2)11yescharacteristic
269#67—41no—
270#67—42no—
271#67—41no—
272#67—42no—
273C2×Dic3—22no—
274C2×Dic3—21no—
275C2×Dic3—42no—
276C2×Dic3—41no—
277D12—22no—
278D12—21no—
279D12—21no—
280D12—22no—
281C4×D3—41no—
282C4×D3—42no—
283C4×D3—42no—
284C4×D3—41no—
285C4×D3—84no—
286Dic6—22no—
287Dic6—21no—
288Dic6—21no—
289Dic6—22no—
290C3×C8—42no—
291C3×C8—41no—
292C3×C8#12311yescharacteristic
293C3×C8#12311yescharacteristic
294C3×C8#12311yescharacteristic
295C3×C8#12311yescharacteristic
296C3⋊C8—41no—
297C3⋊C8—41no—
298#41—61no—
299#41—62no—
300#41—61no—
301#41—62no—
302#41—61no—
303#41—61no—
304#41—62no—
305#41—62no—
306C2×Q8—32no—
307C2×Q8—31no—
308C2×D4—32no—
309C2×D4—31no—
310C2×D4—61no—
311C2×D4—62no—
312C22×C4—21no—
313C22×C4—62no—
314C22×C4—61no—
315C22×C4—122no—
316C22×C4C23×D311yescharacteristic
317Q16—61no—
318Q16—61no—
319Q16—122no—
320SD16—62no—
321SD16—62no—
322SD16—122no—
323SD16—122no—
324D8—61no—
325D8—61no—
326D8—122no—
327M16—22no—
328M16—22no—
329M16—61no—
330M16—61no—
331M16—61no—
332M16—61no—
333M16—121no—
334M16—121no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—22no—
338C2×C8—21no—
339C2×C8—61no—
340C2×C8—61no—
341C2×C8—121no—
342C2×C8—121no—
343C2×C8#23511yescharacteristic
344C2×C8#23511yescharacteristic
345C2×C8D3×D411yescharacteristic
346C2×C8D3×D411yescharacteristic
347C2×C8#23911yescharacteristic
348C2×C8#23911yescharacteristic
349C4⋊C4—22no—
350C4⋊C4—62no—
351C4⋊C4—61no—
352C4⋊C4—122no—
353C22⋊C4—22no—
354C22⋊C4—61no—
355C22⋊C4—62no—
356C22⋊C4—122no—
357C42—21no—
358C42—22no—
359C42—61no—
360C42—62no—
361C42—122no—
362C42#23912yes—
363C22×C3—41no—
364C22×C3#54011yescharacteristic
365C22×C3#54611yescharacteristic
366C22×C3#55311yescharacteristic
367C2×D3—41no—
368C2×D3—42no—
369C2×D3—82no—
370C2×D3—81no—
371C3×C4—42no—
372C3×C4—82no—
373C3×C4C2×#12311yescharacteristic
374C3×C4#52911yescharacteristic
375C3×C4#54611yescharacteristic
376C3×C4#55511yescharacteristic
377Dic3—42no—
378Dic3—41no—
379Dic3—82no—
380Dic3—81no—
381C23—61no—
382C23—62no—
383C23#100911yescharacteristic
384Q8—62no—
385Q8—61no—
386Q8—61no—
387Q8—62no—
388D4—62no—
389D4—61no—
390D4—61no—
391D4—62no—
392D4—122no—
393D4—121no—
394D4—122no—
395D4—121no—
396C2×C4—22no—
397C2×C4—42no—
398C2×C4—61no—
399C2×C4—62no—
400C2×C4—81no—
401C2×C4—122no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4—122no—
405C2×C4—121no—
406C2×C4—122no—
407C2×C4—244no—
408C2×C4C2×#23511yescharacteristic
409C2×C4C2×D3×D411yescharacteristic
410C2×C4#100411yescharacteristic
411C2×C4C2×#23911yescharacteristic
412C2×C4C2×#23911yescharacteristic
413C2×C4D3×#4111yescharacteristic
414C8—41no—
415C8—42no—
416C8—121no—
417C8—121no—
418C8#89411yescharacteristic
419C8#89411yescharacteristic
420C8#89411yescharacteristic
421C8#89411yescharacteristic
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3#242011yescharacteristic
426D3—82no—
427D3—81no—
428C22—41no—
429C22—121no—
430C22—122no—
431C22—242no—
432C22—241no—
433C22#556211yescharacteristic
434C22#563011yescharacteristic
435C22#569511yescharacteristic
436C4—42no—
437C4—82no—
438C4—121no—
439C4—122no—
440C4—242no—
441C4—241no—
442C4C2×#89411yescharacteristic
443C4#558011yescharacteristic
444C4#561211yescharacteristic
445C4#573411yescharacteristic
446C3#2498611yescharacteristic, complemented, Sylow
447C2—21no—
448C2—21no—
449C2—21no—
450C2—242no—
451C2—241no—
452C2#7621311yescharacteristic
subgroupG/H
centerC4#5734
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1946C2
socleC2×C3#2420
prime radical#79256C2
hypercenter#1946D3
hypocenterC3#24986
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1946

Complements

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

NH ≅ G/Ntype
2. #78494C2semidirect
3. C3×#1946C2semidirect
4. #70698C2semidirect
5. #70664C2semidirect
6. #70658C2semidirect
7. #70548C2semidirect
8. #70547C2semidirect
9. #69839C2semidirect
10. #68778C2semidirect
11. #68737C2semidirect
47. #1946D3semidirect
446. C3#24986semidirect

Maximal subgroups

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

subgroupindex
#792562
#784942
C3×#19462
#706982
#706642
#706582
#705482
#705472
#698392
#687782
#687372
#249863