Finite Group Databasebeta

← #10798423

Subgroups — #10798423

Order 768.

772 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#78494C211yesmaximal, characteristic
2#78455C211yesmaximal, characteristic, complemented
3C4×#901C211yesmaximal, characteristic, complemented
4C3×#1746C211yesmaximal, characteristic, complemented
5#69840C211yesmaximal, characteristic, complemented
6#69430C211yesmaximal, characteristic, complemented
7#69430C211yesmaximal, characteristic, complemented
8#68695C212yesmaximal
9#68694C212yesmaximal, complemented
10#68646C212yesmaximal, complemented
11#68645C212yesmaximal, complemented
12#21138—31nomaximal, Sylow
13C2×#904C2211yescharacteristic
14C2×#901C2212yes—
15C4×#235C2211yescharacteristic
16C4×#235C2211yescharacteristic
17C2×C3×#97C2211yescharacteristic
18C2×C3×C4×C8C2211yescharacteristic
19C3×#376C2211yescharacteristic
20#5113C2211yescharacteristic
21#5113C2211yescharacteristic
22#5110C2212yes—
23C2×#819C2212yes—
24#5106C2211yescharacteristic
25#5102C2212yes—
26#5101C2212yes—
27C2×#805C2212yes—
28C2×#805C2212yes—
29C4×#208C2211yescharacteristic
30C4×#208C2211yescharacteristic
31#4716C412yescomplemented
32#4711C412yescomplemented
33#4710C412yescomplemented
34#4710C412yescomplemented
35#4690—21no—
36#4690—21no—
37#4689—22no—
38C4×ΓL(1,𝔽25)—21no—
39#4685—22no—
40C4×D3×C8—21no—
41C3×#336C2212yes—
42C3×#336C2212yes—
43#4552C2212yes—
44#4523C2212yes—
45#4523C2212yes—
46#2943—31no—
47#2922—31no—
48C4×#131—31no—
49#1929—32no—
50#1922—32no—
51#1876—32no—
52#1866—32no—
53#1778—31no—
54#1759—31no—
55#1759—31no—
56#1746D311yescharacteristic, complemented
57C2×#235C2311yescharacteristic
58C2×#235C2311yescharacteristic
59C22×C3×C8C2311yescharacteristic
60C22×C3×C8C2311yescharacteristic
61C22×C3×C8C2312yes—
62C2×C3×C42C2312yes—
63C2×C3×C42C2311yescharacteristic
64C4×#67—22no—
65C4×#67—22no—
66#926C2311yescharacteristic
67#926C2311yescharacteristic
68C2×#208C2312yes—
69C2×#208C2312yes—
70C2×Dic12C2×C411yescharacteristic
71#904—24no—
72C2×D24C2×C411yescharacteristic
73C2×#204C2×C411yescharacteristic
74C2×#204C2×C411yescharacteristic
75#901—24no—
76#901—24no—
77C2×ΓL(1,𝔽25)—22no—
78C2×D3×C8—22no—
79C4×D12C2×C412yes—
80C4×D12C2×C412yes—
81#872—21no—
82#872—21no—
83C42×D3—21no—
84C42×D3—21no—
85C4×Dic6C2×C412yes—
86C4×Dic6C2×C412yes—
87C3×#97C2×C412yes—
88C3×#97C2×C412yes—
89C3×C4×C8D411yescharacteristic
90C3×C4×C8D412yes—
91C3×C4×C8D411yescharacteristic
92C3×#95—22no—
93#821C2×C412yes—
94#821C2×C412yes—
95(C2×D3)⋊C8—22no—
96(C2×D3)⋊C8—22no—
97#819—24no—
98#818C2×C412yes—
99#817C2×C412yes—
100#816C2×C412yes—
101#816C2×C412yes—
102#815—22no—
103Dic3⋊C8—22no—
104Dic3⋊C8—22no—
105C8×Dic3—22no—
106#805—24no—
107#805—24no—
108#803—21no—
109#803—21no—
110C4×C3⋊C8—21no—
111C4×C3⋊C8—21no—
112C2×#135—31no—
113C2×#131—32no—
114C4×#41—31no—
115C4×#41—31no—
116#442—32no—
117#441—32no—
118#440—32no—
119#440—32no—
120#433—61no—
121#433—61no—
122#431—62no—
123C2×#108—32no—
124#427—31no—
125#424—32no—
126C2×#104—32no—
127C2×#104—32no—
128#416—31no—
129#416—31no—
130#408—32no—
131#405—32no—
132#405—62no—
133C4×M16—31no—
134C4×M16—31no—
135C4×M16—61no—
136C2×#97C2×D311yescharacteristic
137C2×C4×C8—61no—
138C2×C4×C8C2×D311yescharacteristic
139#376C2×D311yescharacteristic
140#341—32no—
141#339—32no—
142#339—32no—
143#336C2×D312yes—
144#336C2×D312yes—
145C22×C3×C4—22no—
146C22×C3×C4#4112yes—
147C22×C3×C4C2411yescharacteristic
148C2×#67—21no—
149C2×#67—21no—
150#235—22no—
151#235—22no—
152#235—22no—
153#235#4114yes—
154C2×D12C22×C411yescharacteristic
155C2×D12C22×C411yescharacteristic
156C2×C4×D3—21no—
157C2×C4×D3—22no—
158C2×C4×D3—22no—
159C2×C4×D3—21no—
160C2×Dic6C22×C411yescharacteristic
161C2×Dic6C22×C411yescharacteristic
162C2×C3×C8—22no—
163C2×C3×C8—22no—
164C2×C3×C8—22no—
165C2×C3×C8—22no—
166C2×C3×C8C22×C412yes—
167C2×C3×C8C22×C411yescharacteristic
168C2×C3×C8C22×C411yescharacteristic
169C2×C3×C8C2×D412yes—
170C2×C3×C8C2×D412yes—
171C3×C42—22no—
172C3×C42—22no—
173C3×C42C22×C412yes—
174C3×C42C2×D411yescharacteristic
175C3×C42C2×D411yescharacteristic
176#217—21no—
177#217—21no—
178(C2×D3)⋊C4—22no—
179(C2×D3)⋊C4—22no—
180C4⋊Dic3C22×C412yes—
181C4⋊Dic3C22×C412yes—
182Dic3⋊C4—22no—
183Dic3⋊C4—22no—
184C4×Dic3—22no—
185C4×Dic3—22no—
186C4×Dic3—21no—
187C4×Dic3—21no—
188#208—22no—
189#208—22no—
190#208—22no—
191#208#4114yes—
192C2×C3⋊C8—22no—
193C2×C3⋊C8—22no—
194Dic12—22no—
195D24—22no—
196#204—22no—
197#204—22no—
198ΓL(1,𝔽25)—28no—
199D3×C8—28no—
200C2×#41—31no—
201C2×#41—31no—
202#135—64no—
203C2×Q16—31no—
204C2×SD16—31no—
205C2×SD16—31no—
206C2×D8—31no—
207#131—64no—
208#131—64no—
209C2×M16—32no—
210C2×M16—32no—
211C2×M16—62no—
212C22×C8—62no—
213C22×C8C22×D311yescharacteristic
214C22×C8C22×D311yescharacteristic
215C22×C8C22×D312yes—
216C4×Q8—32no—
217C4×Q8—32no—
218C4×D4—32no—
219C4×D4—32no—
220C4×D4—62no—
221C4×D4—62no—
222#117—31no—
223#117—31no—
224#117—61no—
225#117—61no—
226C2×C42—61no—
227C2×C42—61no—
228C2×C42C22×D312yes—
229C2×C42C22×D311yescharacteristic
230#108—64no—
231#107—32no—
232#106—32no—
233C4⋊C8—62no—
234C4⋊C8—62no—
235#104—64no—
236#104—64no—
237#103—32no—
238#103—32no—
239#102—32no—
240#102—32no—
241C22⋊C8—62no—
242C22⋊C8—62no—
243#97—62no—
244#97—61no—
245#97—61no—
246#97C4×D312yes—
247#97C4×D312yes—
248C4×C8—62no—
249C4×C8—61no—
250C4×C8—61no—
251C4×C8D1211yescharacteristic
252C4×C8D1211yescharacteristic
253C4×C8D1212yes—
254#95—22no—
255C23×C3C2×#4111yescharacteristic
256C22×D3—21no—
257C22×D3—21no—
258C2×C3×C4—42no—
259C2×C3×C4—44no—
260C2×C3×C4—42no—
261C2×C3×C4C4×D412yes—
262C2×C3×C4C4×D412yes—
263C2×C3×C4C23×C411yescharacteristic
264C2×C3×C4C22×D411yescharacteristic
265C2×C3×C4C2×#4112yes—
266C2×C3×C4C2×#4112yes—
267#67—24no—
268#67—42no—
269C2×Dic3—21no—
270C2×Dic3—21no—
271C2×Dic3—22no—
272C2×Dic3—22no—
273D12—21no—
274D12—21no—
275D12—21no—
276D12#11712yes—
277C4×D3—24no—
278C4×D3—44no—
279C4×D3—44no—
280C4×D3—42no—
281Dic6—21no—
282Dic6—21no—
283Dic6—21no—
284Dic6#11712yes—
285C3×C8—21no—
286C3×C8—21no—
287C3×C8—22no—
288C3×C8—22no—
289C3×C8—22no—
290C3⋊C8—24no—
291C3⋊C8—42no—
292#41—34no—
293#41—62no—
294#41—62no—
295#41—62no—
296C2×Q8—31no—
297C2×Q8—31no—
298C2×D4—31no—
299C2×D4—31no—
300C2×D4—61no—
301C2×D4—61no—
302C22×C4—22no—
303C22×C4—61no—
304C22×C4—62no—
305C22×C4—62no—
306C22×C4—61no—
307C22×C4#23512yes—
308C22×C4C23×D311yescharacteristic
309Q16—62no—
310SD16—62no—
311SD16—62no—
312D8—62no—
313M16—34no—
314M16—68no—
315M16—62no—
316M16—62no—
317M16—62no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—22no—
321C2×C8—22no—
322C2×C8—68no—
323C2×C8—62no—
324C2×C8—62no—
325C2×C8C2×C4×D312yes—
326C2×C8C2×C4×D311yescharacteristic
327C2×C8C2×C4×D311yescharacteristic
328C2×C8C2×D1212yes—
329C2×C8C2×D1212yes—
330C4⋊C4—32no—
331C4⋊C4—32no—
332C4⋊C4—62no—
333C4⋊C4—62no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—62no—
337C22⋊C4—62no—
338C42—22no—
339C42—22no—
340C42—62no—
341C42—62no—
342C42—61no—
343C42—61no—
344C42C2×C4×D312yes—
345C42C2×D1211yescharacteristic
346C42C2×D1211yescharacteristic
347C22×C3—22no—
348C22×C3C2×C4×D411yescharacteristic
349C22×C3#52912yes—
350C2×D3—22no—
351C2×D3—42no—
352C2×D3—42no—
353C2×D3—41no—
354C3×C4—22no—
355C3×C4—22no—
356C3×C4—82no—
357C3×C4C2×#11711yescharacteristic
358C3×C4C4×#4111yescharacteristic
359C3×C4#52512yes—
360Dic3—22no—
361Dic3—42no—
362Dic3—42no—
363Dic3—41no—
364C23—61no—
365C23—61no—
366C23C2×#23511yescharacteristic
367Q8—32no—
368Q8—61no—
369Q8—61no—
370Q8—61no—
371D4—32no—
372D4—64no—
373D4—61no—
374D4—61no—
375D4—61no—
376D4—122no—
377C2×C4—42no—
378C2×C4—44no—
379C2×C4—42no—
380C2×C4—61no—
381C2×C4—64no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—61no—
385C2×C4—124no—
386C2×C4—124no—
387C2×C4—122no—
388C2×C4C4×D1212yes—
389C2×C4C4×D1212yes—
390C2×C4C22×C4×D311yescharacteristic
391C2×C4C22×D1211yescharacteristic
392C2×C4D3×#4112yes—
393C2×C4D3×#4112yes—
394C8—21no—
395C8—21no—
396C8—22no—
397C8—22no—
398C8—22no—
399C8—64no—
400C8—122no—
401C2×C3—22no—
402C2×C3#229011yescharacteristic
403C2×C3#295512yes—
404D3—42no—
405D3—81no—
406C22—22no—
407C22—62no—
408C22—122no—
409C22—121no—
410C22—122no—
411C22C2×C4×D1211yescharacteristic
412C22#552812yes—
413C4—22no—
414C4—22no—
415C4—62no—
416C4—82no—
417C4—121no—
418C4—122no—
419C4—122no—
420C4D3×#11711yescharacteristic
421C4#552211yescharacteristic
422C4#552412yes—
423C3#2113811yescharacteristic, complemented, Sylow
424C2—22no—
425C2—122no—
426C2—241no—
427C2#7341411yescharacteristic
428C2#7857912yes—
subgroupG/H
centerC2×C4C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1746C2
socleC22×C3C2×C4×D4
prime radicalC2×#904C22
cosocleC22×C3×C4C24
hypercenter#1746D3
hypocenterC3#21138
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C4×C8 ◁ #1746

Complements

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

NH ≅ G/Ntype
2. #78455C2semidirect
3. C4×#901C2semidirect
4. C3×#1746C2semidirect
5. #69840C2semidirect
6. #69430C2semidirect
7. #69430C2semidirect
9. #68694C2semidirect
10. #68646C2semidirect
11. #68645C2semidirect
31. #4716C4semidirect
32. #4711C4semidirect
33. #4710C4semidirect
34. #4710C4semidirect
56. #1746D3semidirect
423. C3#21138semidirect

Maximal subgroups

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

subgroupindex
#784942
#784552
C4×#9012
C3×#17462
#698402
#694302
#686952
#686942
#686462
#686452
#211383