Finite Group Databasebeta

← #10840278

Subgroups — #10840278

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#79256C211yesmaximal, characteristic
2#79165C211yesmaximal, characteristic, complemented
3C3×#1833C211yesmaximal, characteristic, complemented
4#70704C211yesmaximal, characteristic, complemented
5#70664C211yesmaximal, characteristic, complemented
6#69897C211yesmaximal, characteristic, complemented
7#69887C211yesmaximal, characteristic, complemented
8#69043C212yesmaximal, complemented
9#69039C212yesmaximal, complemented
10#68711C212yesmaximal, complemented
11#68698C212yesmaximal, complemented
12#24985—31nomaximal, Sylow
13#5748C2212yes—
14C2×#904C2211yescharacteristic
15C2×#901C2211yescharacteristic
16#5522C2212yes—
17C2×C3×#108C2211yescharacteristic
18C3×#427C2211yescharacteristic
19C3×#405C2211yescharacteristic
20#5139C2212yes—
21C2×#825C2211yescharacteristic
22#5132C2212yes—
23#5119C2212yes—
24#5106C2211yescharacteristic
25C2×C8×Dic3C2211yescharacteristic
26#5006C2212yes—
27#5006C2212yes—
28#5002C2212yes—
29#5002C2211yescharacteristic
30#4898—22no—
31#4897—21no—
32#4896—21no—
33#4873—21no—
34#4872—21no—
35#4852—22no—
36C3×#344C2212yes—
37C3×H(3,ℤ/4)C2212yes—
38#4558C2211yescharacteristic
39#4555C2212yes—
40#4532C2211yescharacteristic
41#4531C2212yes—
42#3067—31no—
43C4×#135—31no—
44#2080—32no—
45#2053—32no—
46#1976—31no—
47#1945—31no—
48#1906—32no—
49#1833D311yescharacteristic, complemented
50#1827—32no—
51#1812—31no—
52#1764—31no—
53C2×#235C2312yes—
54C2×#235C2311yescharacteristic
55C2×C3×M16C2312yes—
56C2×C3×M16C2311yescharacteristic
57C22×C3×C8C2311yescharacteristic
58C3×#117C2311yescharacteristic
59C3×#117C2312yes—
60#926C2312yes—
61C2×C4×Dic3C2311yescharacteristic
62C2×#208C2311yescharacteristic
63C22×C3⋊C8C2311yescharacteristic
64#909—22no—
65#909—22no—
66#908—22no—
67#908—22no—
68C2×Dic12D411yescharacteristic
69#904—21no—
70#904—21no—
71#904—21no—
72#904—21no—
73#904—42no—
74C2×D24D411yescharacteristic
75C2×#204D412yes—
76#901—21no—
77#901—21no—
78#901—21no—
79#901—21no—
80C2×ΓL(1,𝔽25)—21no—
81C2×D3×C8—21no—
82#893—22no—
83#887—22no—
84#67⋊C4—42no—
85#872—22no—
86C42×D3—22no—
87C3×#108—21no—
88C3×#108—21no—
89C3×#107—21no—
90C3×#106—21no—
91C3×#97—21no—
92C3×C4×C8—21no—
93#825—21no—
94#825—21no—
95#825—42no—
96#824—22no—
97#823—22no—
98#818—21no—
99#817—21no—
100Dic3⋊C8—42no—
101C8×Dic3—21no—
102C8×Dic3—21no—
103#810—22no—
104#810—21no—
105#809—22no—
106#809—21no—
107#806—42no—
108#806—42no—
109#575—32no—
110C2×#135—31no—
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×#108C2×D311yescharacteristic
120#427—31no—
121#427C2×D311yescharacteristic
122#424—32no—
123#421—32no—
124C2×#104—31no—
125#416—31no—
126#416—32no—
127#413—32no—
128#410—32no—
129#410—32no—
130#405C2×D311yescharacteristic
131C2×C4×C8—31no—
132#344C2×D312yes—
133#341—31no—
134#339—32no—
135#339—31no—
136H(3,ℤ/4)—32no—
137H(3,ℤ/4)C2×D312yes—
138C22×C3×C4C2411yescharacteristic
139C2×#67—21no—
140C2×#67—22no—
141C22×Dic3—21no—
142#235—21no—
143#235—21no—
144#235—21no—
145#235—22no—
146#235—22no—
147#235—22no—
148#235—22no—
149#235—21no—
150C2×D12C2×D412yes—
151C2×D12#4111yescharacteristic
152C2×C4×D3—22no—
153C2×C4×D3—21no—
154C2×C4×D3—42no—
155C2×Dic6C2×D412yes—
156C2×Dic6#4111yescharacteristic
157C3×M16—22no—
158C3×M16—21no—
159C3×M16—22no—
160C3×M16—21no—
161C2×C3×C8—21no—
162C2×C3×C8—22no—
163C2×C3×C8C2×D411yescharacteristic
164C2×C3×C8C2×D411yescharacteristic
165C2×C3×C8#4111yescharacteristic
166C2×C3×C8#4111yescharacteristic
167C2×C3×C8#4111yescharacteristic
168C2×C3×C8#4111yescharacteristic
169C3×C4⋊C4—22no—
170C3×C4⋊C4—21no—
171C3×C22⋊C4—21no—
172C3×C22⋊C4—22no—
173C3×C42—21no—
174C3×C42—22no—
175#217—22no—
176(C2×D3)⋊C4—42no—
177C4⋊Dic3—22no—
178Dic3⋊C4—42no—
179C4×Dic3—21no—
180C4×Dic3—22no—
181C4×Dic3—42no—
182C4×Dic3#4111yescharacteristic
183C4×Dic3#4111yescharacteristic
184#208—21no—
185#208—21no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8—21no—
188C2×C3⋊C8—21no—
189C2×C3⋊C8—21no—
190Dic12—21no—
191Dic12—21no—
192Dic12—42no—
193D24—21no—
194D24—21no—
195D24—42no—
196#204—22no—
197#204—22no—
198#204—42no—
199#204—42no—
200ΓL(1,𝔽25)—41no—
201ΓL(1,𝔽25)—41no—
202D3×C8—41no—
203D3×C8—41no—
204C2×#41—32no—
205C2×#41—31no—
206#137—62no—
207#137—62no—
208AGL(1,ℤ/8)—62no—
209AGL(1,ℤ/8)—62no—
210#135—61no—
211#135—61no—
212#135—61no—
213#135—61no—
214#135—122no—
215C2×Q16—31no—
216C2×SD16—32no—
217C2×D8—31no—
218#131—61no—
219#131—61no—
220#131—61no—
221#131—61no—
222C2×M16—31no—
223C2×M16—61no—
224C2×M16C22×D312yes—
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×D312yes—
235#117C22×D311yescharacteristic
236C2×C42—31no—
237C2×C42—62no—
238#108—21no—
239#108—21no—
240#107—21no—
241#107—61no—
242#106—21no—
243#106—61no—
244C4⋊C8—122no—
245#104—61no—
246#104—61no—
247#104—122no—
248#103—61no—
249#103—62no—
250#102—61no—
251#102—62no—
252#101—62no—
253#100—62no—
254#99—122no—
255#99—122no—
256#97—21no—
257C4×C8—21no—
258C4×C8—61no—
259C4×C8—61no—
260C23×C3H(5,𝔽2)11yescharacteristic
261C22×D3—22no—
262C22×D3—21no—
263C2×C3×C4—42no—
264C2×C3×C4—41no—
265C2×C3×C4C22×D411yescharacteristic
266C2×C3×C4C2×#4111yescharacteristic
267C2×C3×C4C2×#4111yescharacteristic
268C2×C3×C4C2×#4111yescharacteristic
269C2×C3×C4C2×#4111yescharacteristic
270C2×C3×C4H(5,𝔽2)11yescharacteristic
271#67—41no—
272#67—42no—
273#67—41no—
274#67—42no—
275C2×Dic3—21no—
276C2×Dic3—21no—
277C2×Dic3—22no—
278C2×Dic3—21no—
279C2×Dic3—42no—
280C2×Dic3—42no—
281D12—22no—
282D12—21no—
283D12—21no—
284D12—22no—
285C4×D3—41no—
286C4×D3—42no—
287C4×D3—42no—
288C4×D3—41no—
289C4×D3—84no—
290Dic6—22no—
291Dic6—21no—
292Dic6—21no—
293Dic6—22no—
294C3×C8—42no—
295C3×C8—41no—
296C3×C8#12311yescharacteristic
297C3×C8#12311yescharacteristic
298C3×C8#12311yescharacteristic
299C3×C8#12311yescharacteristic
300C3⋊C8—41no—
301C3⋊C8—41no—
302#41—62no—
303#41—61no—
304#41—61no—
305#41—62no—
306#41—61no—
307#41—62no—
308#41—62no—
309#41—61no—
310C2×Q8—32no—
311C2×Q8—31no—
312C2×D4—32no—
313C2×D4—31no—
314C2×D4—61no—
315C2×D4—62no—
316C22×C4—61no—
317C22×C4—61no—
318C22×C4—62no—
319C22×C4—122no—
320C22×C4C23×D311yescharacteristic
321Q16—61no—
322Q16—61no—
323Q16—122no—
324SD16—62no—
325SD16—62no—
326SD16—122no—
327SD16—122no—
328D8—61no—
329D8—61no—
330D8—122no—
331M16—22no—
332M16—21no—
333M16—22no—
334M16—21no—
335M16—61no—
336M16—61no—
337M16—121no—
338M16—121no—
339C2×C8—21no—
340C2×C8—22no—
341C2×C8—61no—
342C2×C8—61no—
343C2×C8—61no—
344C2×C8—61no—
345C2×C8—121no—
346C2×C8—121no—
347C2×C8#23511yescharacteristic
348C2×C8#23511yescharacteristic
349C2×C8D3×D411yescharacteristic
350C2×C8D3×D411yescharacteristic
351C2×C8#23911yescharacteristic
352C2×C8#23911yescharacteristic
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#54011yescharacteristic
370C22×C3#54611yescharacteristic
371C22×C3#55311yescharacteristic
372C2×D3—41no—
373C2×D3—42no—
374C2×D3—82no—
375C2×D3—81no—
376C3×C4—82no—
377C3×C4—81no—
378C3×C4C2×#12311yescharacteristic
379C3×C4#52911yescharacteristic
380C3×C4#54611yescharacteristic
381C3×C4#55511yescharacteristic
382Dic3—22no—
383Dic3—41no—
384Dic3—42no—
385Dic3—41no—
386Dic3—82no—
387C23—62no—
388C23—61no—
389C23#100911yescharacteristic
390Q8—61no—
391Q8—62no—
392Q8—61no—
393Q8—62no—
394D4—61no—
395D4—62no—
396D4—61no—
397D4—62no—
398D4—121no—
399D4—121no—
400D4—122no—
401D4—122no—
402C2×C4—42no—
403C2×C4—41no—
404C2×C4—61no—
405C2×C4—61no—
406C2×C4—62no—
407C2×C4—61no—
408C2×C4—122no—
409C2×C4—122no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—244no—
415C2×C4C2×#23511yescharacteristic
416C2×C4C2×D3×D411yescharacteristic
417C2×C4C2×#23911yescharacteristic
418C2×C4D3×#4111yescharacteristic
419C2×C4D3×#4111yescharacteristic
420C2×C4#100911yescharacteristic
421C8—42no—
422C8—41no—
423C8—121no—
424C8—121no—
425C8#89411yescharacteristic
426C8#89411yescharacteristic
427C8#89411yescharacteristic
428C8#89411yescharacteristic
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3#242011yescharacteristic
433D3—82no—
434D3—81no—
435C22—41no—
436C22—122no—
437C22—121no—
438C22—242no—
439C22—241no—
440C22#553011yescharacteristic
441C22#567611yescharacteristic
442C22#569711yescharacteristic
443C4—62no—
444C4—82no—
445C4—81no—
446C4—121no—
447C4—122no—
448C4—121no—
449C4—242no—
450C4C2×#89411yescharacteristic
451C4#553211yescharacteristic
452C4#567611yescharacteristic
453C4#569111yescharacteristic
454C3#2498511yescharacteristic, complemented, Sylow
455C2—21no—
456C2—21no—
457C2—21no—
458C2—242no—
459C2—241no—
460C2#7423111yescharacteristic
subgroupG/H
centerC4#5532
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1833C2
socleC2×C3#2420
prime radical#79256C2
hypercenter#1833D3
hypocenterC3#24985
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1833

Complements

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

NH ≅ G/Ntype
2. #79165C2semidirect
3. C3×#1833C2semidirect
4. #70704C2semidirect
5. #70664C2semidirect
6. #69897C2semidirect
7. #69887C2semidirect
8. #69043C2semidirect
9. #69039C2semidirect
10. #68711C2semidirect
11. #68698C2semidirect
49. #1833D3semidirect
454. C3#24985semidirect

Maximal subgroups

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

subgroupindex
#792562
#791652
C3×#18332
#707042
#706642
#698972
#698872
#690432
#690392
#687112
#686982
#249853