Finite Group Databasebeta

← #10840286

Subgroups — #10840286

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#83805C211yesmaximal, characteristic
2#78494C211yesmaximal, characteristic, complemented
3C3×#1765C211yesmaximal, characteristic, complemented
4#72012C211yesmaximal, characteristic, complemented
5#70391C212yesmaximal, complemented
6#70296C212yesmaximal, complemented
7#69897C211yesmaximal, characteristic, complemented
8#69896C211yesmaximal, characteristic, complemented
9#69844C211yesmaximal, characteristic, complemented
10#68935C212yesmaximal, complemented
11#68737C212yesmaximal, complemented
12#24986—31nomaximal, Sylow
13C2×C3×#131C2211yescharacteristic
14#5822C2212yes—
15C2×#904C2211yescharacteristic
16C4×#235C2212yes—
17C2×C3×#104C2211yescharacteristic
18C3×#416C2211yescharacteristic
19C3×#405C2211yescharacteristic
20C2×C3×#97C2211yescharacteristic
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—
33#4716—21no—
34#4711—21no—
35#4710—22no—
36C3×#341C2211yescharacteristic
37C3×#339C2211yescharacteristic
38#4557C2212yes—
39#4556C2212yes—
40#4531C2212yes—
41#4523C2212yes—
42#3067—31no—
43#2943—31no—
44#2080—32no—
45#2053—32no—
46#1976—31no—
47#1946—31no—
48#1906—32no—
49#1833—31no—
50#1827—32no—
51#1812—31no—
52#1765D311yescharacteristic, 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—22no—
88C3×#103—21no—
89C3×#102—21no—
90C3×#97—21no—
91C3×#97—21no—
92C3×#97—21no—
93C3×C4×C8—21no—
94#834—42no—
95#825—42no—
96#824—22no—
97#823—22no—
98#821—22no—
99#819—21no—
100#819—21no—
101#818—21no—
102#818—21no—
103#817—21no—
104#817—21no—
105#816—22no—
106Dic3⋊C8—42no—
107#806—42no—
108#575—32no—
109C2×#135—31no—
110C2×#131C2×D311yescharacteristic
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#427—31no—
121#424—32no—
122#421—32no—
123C2×#104C2×D311yescharacteristic
124#416—32no—
125#416C2×D311yescharacteristic
126#413—32no—
127#410—32no—
128#410—32no—
129#405C2×D311yescharacteristic
130C2×#97C2×D311yescharacteristic
131#344—32no—
132#341C2×D311yescharacteristic
133#339—32no—
134#339C2×D311yescharacteristic
135H(3,ℤ/4)—32no—
136H(3,ℤ/4)—32no—
137C3×#41—21no—
138C3×#41—21no—
139C3×#41—21no—
140C3×#41—21no—
141C2×C3×Q8#4111yescharacteristic
142C2×C3×D4—21no—
143C2×C3×D4#4111yescharacteristic
144C22×C3×C4—21no—
145C22×C3×C4—21no—
146C22×C3×C4C2411yescharacteristic
147C2×#67—22no—
148#235—22no—
149#235—22no—
150#235—22no—
151#235—22no—
152C2×D12C2×D412yes—
153C2×C4×D3—22no—
154C2×C4×D3—42no—
155C2×Dic6C2×D412yes—
156C3×M16—21no—
157C3×M16—21no—
158C3×M16—21no—
159C3×M16—21no—
160C3×M16—41no—
161C3×M16—41no—
162C2×C3×C8—21no—
163C2×C3×C8—21no—
164C2×C3×C8—21no—
165C2×C3×C8—21no—
166C2×C3×C8—21no—
167C2×C3×C8—41no—
168C2×C3×C8—41no—
169C2×C3×C8C2×D411yescharacteristic
170C2×C3×C8C2×D411yescharacteristic
171C2×C3×C8#4111yescharacteristic
172C2×C3×C8#4111yescharacteristic
173C2×C3×C8#4111yescharacteristic
174C2×C3×C8#4111yescharacteristic
175C3×C4⋊C4—21no—
176C3×C22⋊C4—21no—
177C3×C42—21no—
178C3×C42—21no—
179C3×C42#4112yes—
180#217—22no—
181#217—22no—
182#216—42no—
183#215—42no—
184#214—42no—
185#213—42no—
186(C2×D3)⋊C4—42no—
187C4⋊Dic3—22no—
188C4⋊Dic3—22no—
189Dic3⋊C4—42no—
190C4×Dic3—22no—
191C4×Dic3—22no—
192C4×Dic3—42no—
193#208—22no—
194#208—22no—
195C2×C3⋊C8—22no—
196Dic12—21no—
197Dic12—21no—
198D24—21no—
199D24—21no—
200#204—22no—
201#204—22no—
202C2×#41—32no—
203C2×#41C22×D311yescharacteristic
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—21no—
217#131—21no—
218#131—21no—
219#131—21no—
220C2×M16—21no—
221C2×M16—32no—
222C2×M16C22×D311yescharacteristic
223C2×M16C22×D311yescharacteristic
224C22×C8—21no—
225C22×C8C22×D311yescharacteristic
226C22×C8C22×D311yescharacteristic
227C4×Q8—62no—
228C4×D4—62no—
229C4×D4—122no—
230#117—32no—
231#117—32no—
232#117—62no—
233#117C22×D311yescharacteristic
234C2×C42—62no—
235C2×C42C22×D311yescharacteristic
236#108—61no—
237#108—61no—
238#107—61no—
239#107—61no—
240#106—61no—
241#106—61no—
242C4⋊C8—122no—
243#104—22no—
244#104—122no—
245#103—21no—
246#103—62no—
247#102—21no—
248#102—62no—
249#101—62no—
250#100—62no—
251#99—122no—
252#99—122no—
253#97—21no—
254#97—21no—
255#97—21no—
256C4×C8—21no—
257C23×C3—21no—
258C23×C3H(5,𝔽2)11yescharacteristic
259C22×D3—22no—
260C3×Q8—21no—
261C3×Q8—21no—
262C3×D4—21no—
263C3×D4—21no—
264C3×D4—41no—
265C3×D4—41no—
266C2×C3×C4—22no—
267C2×C3×C4—21no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—81no—
272C2×C3×C4C22×D411yescharacteristic
273C2×C3×C4C2×#4111yescharacteristic
274C2×C3×C4C2×#4111yescharacteristic
275C2×C3×C4C2×#4111yescharacteristic
276C2×C3×C4C2×#4111yescharacteristic
277C2×C3×C4H(5,𝔽2)11yescharacteristic
278#67—42no—
279#67—42no—
280C2×Dic3—22no—
281C2×Dic3—42no—
282C2×Dic3—42no—
283D12—22no—
284D12—22no—
285C4×D3—42no—
286C4×D3—42no—
287C4×D3—84no—
288Dic6—22no—
289Dic6—22no—
290C3×C8—41no—
291C3×C8—41no—
292C3×C8—41no—
293C3×C8#12311yescharacteristic
294C3×C8#12311yescharacteristic
295C3×C8#12311yescharacteristic
296C3×C8#12311yescharacteristic
297C3⋊C8—42no—
298#41—21no—
299#41—21no—
300#41—21no—
301#41—21no—
302#41—62no—
303#41—62no—
304#41—62no—
305#41—62no—
306C2×Q8—32no—
307C2×Q8#23511yescharacteristic
308C2×D4—21no—
309C2×D4—32no—
310C2×D4—62no—
311C2×D4#23511yescharacteristic
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—62no—
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—21no—
328M16—21no—
329M16—21no—
330M16—21no—
331M16—41no—
332M16—41no—
333M16—62no—
334M16—62no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—21no—
338C2×C8—21no—
339C2×C8—21no—
340C2×C8—41no—
341C2×C8—41no—
342C2×C8—62no—
343C2×C8#23511yescharacteristic
344C2×C8#23511yescharacteristic
345C2×C8D3×D411yescharacteristic
346C2×C8D3×D411yescharacteristic
347C2×C8#23711yescharacteristic
348C2×C8#23711yescharacteristic
349C4⋊C4—21no—
350C4⋊C4—62no—
351C4⋊C4—62no—
352C4⋊C4—122no—
353C22⋊C4—21no—
354C22⋊C4—62no—
355C22⋊C4—62no—
356C22⋊C4—122no—
357C42—21no—
358C42—21no—
359C42—62no—
360C42—62no—
361C42—122no—
362C42#23512yes—
363C22×C3—41no—
364C22×C3—41no—
365C22×C3—81no—
366C22×C3#54011yescharacteristic
367C22×C3#54611yescharacteristic
368C22×C3#55311yescharacteristic
369C2×D3—42no—
370C2×D3—82no—
371C3×C4—42no—
372C3×C4—41no—
373C3×C4—81no—
374C3×C4C2×#12311yescharacteristic
375C3×C4#52911yescharacteristic
376C3×C4#54611yescharacteristic
377C3×C4#55511yescharacteristic
378Dic3—42no—
379Dic3—82no—
380Dic3—82no—
381C23—21no—
382C23—62no—
383C23#100911yescharacteristic
384Q8—21no—
385Q8—21no—
386Q8—62no—
387Q8—62no—
388D4—21no—
389D4—21no—
390D4—41no—
391D4—41no—
392D4—62no—
393D4—62no—
394D4—122no—
395D4—122no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—62no—
402C2×C4—81no—
403C2×C4—122no—
404C2×C4—122no—
405C2×C4—122no—
406C2×C4—122no—
407C2×C4—244no—
408C2×C4C2×#23511yescharacteristic
409C2×C4C2×#23511yescharacteristic
410C2×C4C2×#23511yescharacteristic
411C2×C4C2×D3×D411yescharacteristic
412C2×C4C2×#23711yescharacteristic
413C2×C4#100911yescharacteristic
414C8—41no—
415C8—41no—
416C8—41no—
417C8—122no—
418C8#88311yescharacteristic
419C8#88311yescharacteristic
420C8#88311yescharacteristic
421C8#88311yescharacteristic
422C2×C3—21no—
423C2×C3—21no—
424C2×C3—21no—
425C2×C3—81no—
426C2×C3#242011yescharacteristic
427D3—82no—
428C22—41no—
429C22—41no—
430C22—81no—
431C22—122no—
432C22—242no—
433C22#553511yescharacteristic
434C22#555011yescharacteristic
435C22#556511yescharacteristic
436C4—42no—
437C4—41no—
438C4—81no—
439C4—122no—
440C4—242no—
441C4—242no—
442C4#547911yescharacteristic
443C4C2×#88311yescharacteristic
444C4#555011yescharacteristic
445C4#556311yescharacteristic
446C3#2498611yescharacteristic, complemented, Sylow
447C2—21no—
448C2—21no—
449C2—21no—
450C2—81no—
451C2—242no—
452C2#7358411yescharacteristic
subgroupG/H
centerC4#5563
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×#1765C2
socleC2×C3#2420
prime radical#83805C2
hypercenter#1765D3
hypocenterC3#24986
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ #1765

Complements

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

NH ≅ G/Ntype
2. #78494C2semidirect
3. C3×#1765C2semidirect
4. #72012C2semidirect
5. #70391C2semidirect
6. #70296C2semidirect
7. #69897C2semidirect
8. #69896C2semidirect
9. #69844C2semidirect
10. #68935C2semidirect
11. #68737C2semidirect
52. #1765D3semidirect
446. C3#24986semidirect

Maximal subgroups

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

subgroupindex
#838052
#784942
C3×#17652
#720122
#703912
#702962
#698972
#698962
#698442
#689352
#687372
#249863