Finite Group Databasebeta

← #10840275

Subgroups — #10840275

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#83805C211yesmaximal, characteristic
2C4×#904C211yesmaximal, characteristic, complemented
3C3×#1764C211yesmaximal, characteristic, complemented
4#72012C211yesmaximal, characteristic, complemented
5#70391C212yesmaximal, complemented
6#70296C212yesmaximal, complemented
7#69897C211yesmaximal, characteristic, complemented
8#69896C211yesmaximal, characteristic, complemented
9#69794C211yesmaximal, characteristic, complemented
10#68935C212yesmaximal, complemented
11#68737C212yesmaximal, complemented
12#24985—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14#5822C2212yes—
15C2×#904C2211yescharacteristic
16C4×#235C2212yes—
17C2×C3×#104C2211yescharacteristic
18C3×#416C2211yescharacteristic
19C3×#405C2211yescharacteristic
20C2×C3×C4×C8C2211yescharacteristic
21#5240C2212yes—
22#5143—21no—
23#5142—21no—
24#5141—22no—
25#5139C2212yes—
26#5132C2212yes—
27#5113C2212yes—
28C2×#819C2211yescharacteristic
29#5106C2211yescharacteristic
30#5106C2211yescharacteristic
31#5102C2212yes—
32#5006C2212yes—
33C4×Dic12—21no—
34C4×D24—21no—
35C4×#204—22no—
36C3×#341C2211yescharacteristic
37C3×#339C2211yescharacteristic
38#4557C2212yes—
39#4556C2212yes—
40#4531C2212yes—
41#4523C2212yes—
42#3067—31no—
43C4×#135—31no—
44#2080—32no—
45#2053—32no—
46#1976—31no—
47#1945—31no—
48#1906—32no—
49#1833—31no—
50#1827—32no—
51#1812—31no—
52#1764D311yescharacteristic, complemented
53C2×C3×#41C2311yescharacteristic
54C2×#235C2312yes—
55C3×#131—21no—
56C3×#131—21no—
57C3×#131—21no—
58C3×#131—21no—
59C2×C3×M16—21no—
60C2×C3×M16C2311yescharacteristic
61C2×C3×M16C2311yescharacteristic
62C22×C3×C8—21no—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C3×#117C2311yescharacteristic
66C2×C3×C42C2311yescharacteristic
67#951—22no—
68#950—42no—
69#949—22no—
70#942—22no—
71#932—22no—
72C4×#67—42no—
73#926C2312yes—
74#926C2312yes—
75C2×#208C2312yes—
76C2×Dic12D411yescharacteristic
77#904—21no—
78#904—21no—
79#904—21no—
80#904—21no—
81C2×D24D411yescharacteristic
82C2×#204D412yes—
83C4×D12—22no—
84#872—22no—
85C42×D3—22no—
86C4×Dic6—22no—
87C3×#104—21no—
88C3×#104—21no—
89C3×#103—21no—
90C3×#102—21no—
91C3×#97—21no—
92C3×C4×C8—21no—
93C3×C4×C8—21no—
94C3×C4×C8—21no—
95#834—42no—
96#825—42no—
97#824—22no—
98#823—22no—
99#821—22no—
100#819—21no—
101#819—21no—
102#818—21no—
103#818—21no—
104#817—21no—
105#817—21no—
106#816—22no—
107Dic3⋊C8—42no—
108#806—42no—
109#575—32no—
110C2×#135—31no—
111C2×#131C2×D311yescharacteristic
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#427—31no—
122#424—32no—
123#421—32no—
124C2×#104C2×D311yescharacteristic
125#416—32no—
126#416C2×D311yescharacteristic
127#413—32no—
128#410—32no—
129#410—32no—
130#405C2×D311yescharacteristic
131C2×C4×C8C2×D311yescharacteristic
132#344—32no—
133#341C2×D311yescharacteristic
134#339—32no—
135#339C2×D311yescharacteristic
136H(3,ℤ/4)—32no—
137H(3,ℤ/4)—32no—
138C3×#41—21no—
139C3×#41—21no—
140C3×#41—21no—
141C3×#41—21no—
142C2×C3×Q8#4111yescharacteristic
143C2×C3×D4—21no—
144C2×C3×D4#4111yescharacteristic
145C22×C3×C4—21no—
146C22×C3×C4—21no—
147C22×C3×C4C2411yescharacteristic
148C2×#67—22no—
149#235—22no—
150#235—22no—
151#235—22no—
152#235—22no—
153C2×D12C2×D412yes—
154C2×C4×D3—22no—
155C2×C4×D3—42no—
156C2×Dic6C2×D412yes—
157C3×M16—21no—
158C3×M16—21no—
159C3×M16—21no—
160C3×M16—21no—
161C3×M16—41no—
162C3×M16—41no—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—21no—
168C2×C3×C8—41no—
169C2×C3×C8—41no—
170C2×C3×C8C2×D411yescharacteristic
171C2×C3×C8C2×D411yescharacteristic
172C2×C3×C8#4111yescharacteristic
173C2×C3×C8#4111yescharacteristic
174C2×C3×C8#4111yescharacteristic
175C2×C3×C8#4111yescharacteristic
176C3×C4⋊C4—21no—
177C3×C22⋊C4—21no—
178C3×C42—21no—
179C3×C42—21no—
180C3×C42#4111yescharacteristic
181C3×C42#4111yescharacteristic
182#217—22no—
183#217—22no—
184#216—42no—
185#215—42no—
186#214—42no—
187#213—42no—
188(C2×D3)⋊C4—42no—
189C4⋊Dic3—22no—
190C4⋊Dic3—22no—
191Dic3⋊C4—42no—
192C4×Dic3—22no—
193C4×Dic3—22no—
194C4×Dic3—42no—
195#208—22no—
196#208—22no—
197C2×C3⋊C8—22no—
198Dic12—21no—
199Dic12—21no—
200D24—21no—
201D24—21no—
202#204—22no—
203#204—22no—
204C2×#41—32no—
205C2×#41C22×D311yescharacteristic
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—21no—
219#131—21no—
220#131—21no—
221#131—21no—
222C2×M16—21no—
223C2×M16—32no—
224C2×M16C22×D311yescharacteristic
225C2×M16C22×D311yescharacteristic
226C22×C8—21no—
227C22×C8C22×D311yescharacteristic
228C22×C8C22×D311yescharacteristic
229C4×Q8—62no—
230C4×D4—62no—
231C4×D4—122no—
232#117—32no—
233#117—32no—
234#117—62no—
235#117C22×D311yescharacteristic
236C2×C42—62no—
237C2×C42C22×D311yescharacteristic
238#108—61no—
239#108—61no—
240#107—61no—
241#107—61no—
242#106—61no—
243#106—61no—
244C4⋊C8—122no—
245#104—21no—
246#104—21no—
247#104—122no—
248#103—21no—
249#103—62no—
250#102—21no—
251#102—62no—
252#101—62no—
253#100—62no—
254#99—122no—
255#99—122no—
256#97—21no—
257C4×C8—21no—
258C4×C8—21no—
259C4×C8—21no—
260C23×C3—21no—
261C23×C3H(5,𝔽2)11yescharacteristic
262C22×D3—22no—
263C3×Q8—21no—
264C3×Q8—21no—
265C3×D4—21no—
266C3×D4—21no—
267C3×D4—41no—
268C3×D4—41no—
269C2×C3×C4—21no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—42no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4C22×D411yescharacteristic
277C2×C3×C4C2×#4111yescharacteristic
278C2×C3×C4C2×#4111yescharacteristic
279C2×C3×C4C2×#4111yescharacteristic
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C4H(5,𝔽2)11yescharacteristic
282#67—42no—
283#67—42no—
284C2×Dic3—22no—
285C2×Dic3—42no—
286C2×Dic3—42no—
287D12—22no—
288D12—22no—
289C4×D3—42no—
290C4×D3—42no—
291C4×D3—84no—
292Dic6—22no—
293Dic6—22no—
294C3×C8—41no—
295C3×C8—41no—
296C3×C8—41no—
297C3×C8#12311yescharacteristic
298C3×C8#12311yescharacteristic
299C3×C8#12311yescharacteristic
300C3×C8#12311yescharacteristic
301C3⋊C8—42no—
302#41—21no—
303#41—21no—
304#41—21no—
305#41—21no—
306#41—62no—
307#41—62no—
308#41—62no—
309#41—62no—
310C2×Q8—32no—
311C2×Q8#23511yescharacteristic
312C2×D4—21no—
313C2×D4—32no—
314C2×D4—62no—
315C2×D4#23511yescharacteristic
316C22×C4—21no—
317C22×C4—21no—
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—21no—
332M16—21no—
333M16—21no—
334M16—21no—
335M16—41no—
336M16—41no—
337M16—62no—
338M16—62no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—21no—
343C2×C8—21no—
344C2×C8—41no—
345C2×C8—41no—
346C2×C8—62no—
347C2×C8#23511yescharacteristic
348C2×C8#23511yescharacteristic
349C2×C8D3×D411yescharacteristic
350C2×C8D3×D411yescharacteristic
351C2×C8#23711yescharacteristic
352C2×C8#23711yescharacteristic
353C4⋊C4—21no—
354C4⋊C4—62no—
355C4⋊C4—62no—
356C4⋊C4—122no—
357C22⋊C4—21no—
358C22⋊C4—62no—
359C22⋊C4—62no—
360C22⋊C4—122no—
361C42—21no—
362C42—21no—
363C42—62no—
364C42—62no—
365C42—122no—
366C42#23511yescharacteristic
367C42#23511yescharacteristic
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—81no—
371C22×C3#54011yescharacteristic
372C22×C3#54611yescharacteristic
373C22×C3#55311yescharacteristic
374C2×D3—42no—
375C2×D3—82no—
376C3×C4—22no—
377C3×C4—41no—
378C3×C4—41no—
379C3×C4—81no—
380C3×C4C2×#12311yescharacteristic
381C3×C4#52911yescharacteristic
382C3×C4#54611yescharacteristic
383C3×C4#55511yescharacteristic
384Dic3—42no—
385Dic3—82no—
386Dic3—82no—
387C23—21no—
388C23—62no—
389C23#100911yescharacteristic
390Q8—21no—
391Q8—21no—
392Q8—62no—
393Q8—62no—
394D4—21no—
395D4—21no—
396D4—41no—
397D4—41no—
398D4—62no—
399D4—62no—
400D4—122no—
401D4—122no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—62no—
410C2×C4—122no—
411C2×C4—122no—
412C2×C4—122no—
413C2×C4—122no—
414C2×C4—244no—
415C2×C4C2×#23511yescharacteristic
416C2×C4C2×#23511yescharacteristic
417C2×C4C2×#23511yescharacteristic
418C2×C4C2×D3×D411yescharacteristic
419C2×C4C2×#23711yescharacteristic
420C2×C4#100911yescharacteristic
421C8—41no—
422C8—41no—
423C8—41no—
424C8—122no—
425C8#88311yescharacteristic
426C8#88311yescharacteristic
427C8#88311yescharacteristic
428C8#88311yescharacteristic
429C2×C3—21no—
430C2×C3—21no—
431C2×C3—21no—
432C2×C3—81no—
433C2×C3#242011yescharacteristic
434D3—82no—
435C22—41no—
436C22—41no—
437C22—81no—
438C22—122no—
439C22—242no—
440C22#553511yescharacteristic
441C22#555011yescharacteristic
442C22#556511yescharacteristic
443C4—22no—
444C4—41no—
445C4—41no—
446C4—81no—
447C4—122no—
448C4—242no—
449C4—242no—
450C4#547911yescharacteristic
451C4C2×#88311yescharacteristic
452C4#555011yescharacteristic
453C4#556311yescharacteristic
454C3#2498511yescharacteristic, complemented, Sylow
455C2—21no—
456C2—21no—
457C2—21no—
458C2—81no—
459C2—242no—
460C2#7358411yescharacteristic
subgroupG/H
centerC4#5563
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1764C2
socleC2×C3#2420
prime radical#83805C2
hypercenter#1764D3
hypocenterC3#24985
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1764

Complements

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

NH ≅ G/Ntype
2. C4×#904C2semidirect
3. C3×#1764C2semidirect
4. #72012C2semidirect
5. #70391C2semidirect
6. #70296C2semidirect
7. #69897C2semidirect
8. #69896C2semidirect
9. #69794C2semidirect
10. #68935C2semidirect
11. #68737C2semidirect
52. #1764D3semidirect
454. C3#24985semidirect

Maximal subgroups

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

subgroupindex
#838052
C4×#9042
C3×#17642
#720122
#703912
#702962
#698972
#698962
#697942
#689352
#687372
#249853