Finite Group Databasebeta

← #10840269

Subgroups — #10840269

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
1C3×C4×#135C211yesmaximal, characteristic, complemented
2#83928C211yesmaximal, characteristic
3#72086C211yesmaximal, characteristic, complemented
4#71645C212yesmaximal, complemented
5#71644C212yesmaximal, complemented
6#70664C211yesmaximal, characteristic, complemented
7#70657C211yesmaximal, characteristic, complemented
8#70296C212yesmaximal, complemented
9#70294C212yesmaximal, complemented
10#69794C211yesmaximal, characteristic, complemented
11#69789C211yesmaximal, characteristic, complemented
12#24985—31nomaximal, Sylow
13C2×C3×#135C2211yescharacteristic
14C3×C4×#41C2212yes—
15#5822C2212yes—
16C2×#901C2211yescharacteristic
17C3×C4×Q16—21no—
18C3×C4×SD16—22no—
19C3×C4×D8—21no—
20C3×#427C2211yescharacteristic
21C3×#416C2212yes—
22C2×C3×C4×C8C2211yescharacteristic
23#5240C2212yes—
24#5237C2212yes—
25#5236C2212yes—
26#5192—21no—
27#5177—22no—
28#5160—21no—
29#5113C2211yescharacteristic
30C2×#819C2211yescharacteristic
31#5106C2211yescharacteristic
32#5102C2212yes—
33#5101C2211yescharacteristic
34#5006C2212yes—
35C2×#805C2211yescharacteristic
36#4552C2211yescharacteristic
37#4533C2212yes—
38#4531C2212yes—
39#4531C2212yes—
40#4523C2211yescharacteristic
41#4523C2212yes—
42#3067—31no—
43C4×#135D311yescharacteristic, complemented
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#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
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64C3×C4×Q8—22no—
65C3×C4×D4—22no—
66C3×C4×D4—42no—
67C3×#117—22no—
68C3×#117C2312yes—
69C2×C3×C42—22no—
70C2×C3×C42C2311yescharacteristic
71#951—22no—
72#950—42no—
73#949—22no—
74#942—22no—
75#932—22no—
76#926C2312yes—
77#926C2311yescharacteristic
78C2×#208C2311yescharacteristic
79C2×#208C2311yescharacteristic
80C2×#208C2312yes—
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×#103—22no—
90C3×#102—22no—
91C3×C4×C8—21no—
92C3×C4×C8—21no—
93#837—42no—
94#836—22no—
95#834—42no—
96#833—22no—
97#821—21no—
98#819—21no—
99#819—21no—
100#818—21no—
101#817—21no—
102#816—21no—
103#815—21no—
104Dic3⋊C8—42no—
105C8×Dic3—21no—
106#806—42no—
107#805—21no—
108#805—21no—
109#575—32no—
110C2×#135C2×D311yescharacteristic
111C2×#131—31no—
112C4×#41C2×D312yes—
113#473—61no—
114#472—61no—
115#471—62no—
116C4×Q16—21no—
117C4×SD16—22no—
118C4×D8—21no—
119C2×#108—31no—
120#427—31no—
121#427C2×D311yescharacteristic
122#424—32no—
123#421—32no—
124C2×#104—31no—
125#416—31no—
126#416C2×D312yes—
127#413—32no—
128#410—32no—
129#410—32no—
130#405—31no—
131C2×C4×C8C2×D311yescharacteristic
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—21no—
146C22×C3×C4—22no—
147C22×C3×C4—42no—
148C22×C3×C4C2411yescharacteristic
149C2×#67—21no—
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—
163C2×C3×C8—21no—
164C2×C3×C8—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—22no—
173C3×C4⋊C4—42no—
174C3×C22⋊C4—22no—
175C3×C22⋊C4—42no—
176C3×C42—21no—
177C3×C42—22no—
178C3×C42—42no—
179C3×C42#4111yescharacteristic
180C3×C42#4111yescharacteristic
181#217—22no—
182#217—21no—
183#216—42no—
184#215—42no—
185#214—42no—
186#213—42no—
187C4⋊Dic3—21no—
188C4⋊Dic3—22no—
189C4×Dic3—21no—
190C4×Dic3—22no—
191#208—22no—
192#208—21no—
193#208—21no—
194#208—21no—
195#208—22no—
196#208—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×Q16#6711yescharacteristic
216C2×SD16#6712yes—
217C2×D8#6711yescharacteristic
218#131—61no—
219#131—61no—
220#131—61no—
221#131—61no—
222C2×M16—31no—
223C2×M16—32no—
224C2×M16—31no—
225C2×M16—61no—
226C22×C8—61no—
227C22×C8C22×D311yescharacteristic
228C22×C8C22×D311yescharacteristic
229C4×Q8—22no—
230C4×D4—22no—
231C4×D4—42no—
232#117—22no—
233#117—31no—
234#117—32no—
235#117C22×D312yes—
236C2×C42—22no—
237C2×C42C22×D311yescharacteristic
238#108—61no—
239#108—61no—
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—22no—
249#103—61no—
250#102—22no—
251#102—61no—
252#101—62no—
253#100—62no—
254#99—122no—
255#99—122no—
256#97—61no—
257C4×C8—21no—
258C4×C8—21no—
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—21no—
270C2×C3×C4—21no—
271C2×C3×C4—22no—
272C2×C3×C4—42no—
273C2×C3×C4—42no—
274C2×C3×C4—42no—
275C2×C3×C4—42no—
276C2×C3×C4—84no—
277C2×C3×C4C22×D411yescharacteristic
278C2×C3×C4C2×#4111yescharacteristic
279C2×C3×C4C2×#4111yescharacteristic
280C2×C3×C4C2×#4111yescharacteristic
281C2×C3×C4C2×#4111yescharacteristic
282C2×C3×C4H(5,𝔽2)11yescharacteristic
283#67—41no—
284#67—41no—
285C2×Dic3—21no—
286C2×Dic3—42no—
287C2×Dic3—41no—
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—42no—
301C3⋊C8—41no—
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×#6712yes—
312C2×D4—22no—
313C2×D4—31no—
314C2×D4—61no—
315C2×D4C2×#6712yes—
316C22×C4—21no—
317C22×C4—22no—
318C22×C4—42no—
319C22×C4—61no—
320C22×C4C23×D311yescharacteristic
321Q16—21no—
322Q16—21no—
323Q16—122no—
324SD16—22no—
325SD16—22no—
326SD16—122no—
327SD16—122no—
328D8—21no—
329D8—21no—
330D8—122no—
331M16—61no—
332M16—61no—
333M16—62no—
334M16—61no—
335M16—61no—
336M16—62no—
337M16—121no—
338M16—121no—
339C2×C8—21no—
340C2×C8—21no—
341C2×C8—21no—
342C2×C8—61no—
343C2×C8—61no—
344C2×C8—62no—
345C2×C8—121no—
346C2×C8—121no—
347C2×C8#23511yescharacteristic
348C2×C8#23511yescharacteristic
349C2×C8#23511yescharacteristic
350C2×C8#23511yescharacteristic
351C2×C8C2×#6711yescharacteristic
352C2×C8C2×#6711yescharacteristic
353C4⋊C4—22no—
354C4⋊C4—42no—
355C4⋊C4—62no—
356C4⋊C4—61no—
357C22⋊C4—22no—
358C22⋊C4—42no—
359C22⋊C4—61no—
360C22⋊C4—62no—
361C42—21no—
362C42—22no—
363C42—42no—
364C42—61no—
365C42—62no—
366C42#23711yescharacteristic
367C42#23711yescharacteristic
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—22no—
377C3×C4—41no—
378C3×C4—42no—
379C3×C4—82no—
380C3×C4C2×#12311yescharacteristic
381C3×C4#52911yescharacteristic
382C3×C4#54611yescharacteristic
383C3×C4#55511yescharacteristic
384Dic3—41no—
385Dic3—82no—
386Dic3—81no—
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—21no—
403C2×C4—21no—
404C2×C4—22no—
405C2×C4—42no—
406C2×C4—42no—
407C2×C4—42no—
408C2×C4—42no—
409C2×C4—61no—
410C2×C4—84no—
411C2×C4—121no—
412C2×C4—121no—
413C2×C4—121no—
414C2×C4—122no—
415C2×C4C2×#23511yescharacteristic
416C2×C4C2×#23511yescharacteristic
417C2×C4C2×#23711yescharacteristic
418C2×C4D3×#4111yescharacteristic
419C2×C4#100911yescharacteristic
420C2×C4C22×#6711yescharacteristic
421C8—41no—
422C8—121no—
423C8—121no—
424C8—122no—
425C8#92911yescharacteristic
426C8#92911yescharacteristic
427C8#92911yescharacteristic
428C8#92911yescharacteristic
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#555811yescharacteristic
441C22#556211yescharacteristic
442C22#580611yescharacteristic
443C4—22no—
444C4—41no—
445C4—42no—
446C4—82no—
447C4—121no—
448C4—242no—
449C4—241no—
450C4#555711yescharacteristic
451C4#556311yescharacteristic
452C4C2×#92911yescharacteristic
453C4#582911yescharacteristic
454C3#2498511yescharacteristic, complemented, Sylow
455C2—21no—
456C2—21no—
457C2—21no—
458C2—82no—
459C2—241no—
460C2#8240511yescharacteristic
subgroupG/H
centerC4#5563
commutator, cosocleC22×C3×C4C24
FrattiniC22×C4C23×D3
FittingC3×C4×#135C2
socleC2×C3#2420
prime radical#83928C2
hypercenterC4×#135D3
hypocenterC3#24985
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4 ◁ C22×C4 ◁ C4×#135

Complements

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

NH ≅ G/Ntype
1. C3×C4×#135C2semidirect
3. #72086C2semidirect
4. #71645C2semidirect
5. #71644C2semidirect
6. #70664C2semidirect
7. #70657C2semidirect
8. #70296C2semidirect
9. #70294C2semidirect
10. #69794C2semidirect
11. #69789C2semidirect
43. C4×#135D3semidirect
454. C3#24985semidirect

Maximal subgroups

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

subgroupindex
C3×C4×#1352
#839282
#720862
#716452
#716442
#706642
#706572
#702962
#702942
#697942
#697892
#249853