Finite Group Databasebeta

← #10801999

Subgroups — #10801999

Order 768.

906 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79180C211yesmaximal, characteristic
2#78554C212yesmaximal, complemented
3C2×#4717C211yesmaximal, characteristic, complemented
4C2×#4708C211yesmaximal, characteristic, complemented
5C3×#1773C211yesmaximal, characteristic, complemented
6#70711C211yesmaximal, characteristic, complemented
7#70697C211yesmaximal, characteristic, complemented
8#70663C211yesmaximal, characteristic, complemented
9#69887C212yesmaximal, complemented
10#68757C212yesmaximal, complemented
11#68745C212yesmaximal, complemented
12#21547—31nomaximal, Sylow
13C2×#904C2211yescharacteristic
14C2×#901C2212yes—
15C22×ΓL(1,𝔽25)C2211yescharacteristic
16C22×D3×C8C2211yescharacteristic
17#5524C2212yes—
18C2×#872C2211yescharacteristic
19C2×C3×#108C2211yescharacteristic
20C3×#427C2211yescharacteristic
21C3×#405C2212yes—
22C2×C3×#97C2211yescharacteristic
23C2×#825C2212yes—
24#5129C2212yes—
25#5126C2212yes—
26C2×#819C2211yescharacteristic
27C2×#815C2211yescharacteristic
28C2×C8×Dic3C2211yescharacteristic
29#5002C2212yes—
30#4999C2212yes—
31C2×#805C2212yes—
32C2×C4×C3⋊C8C2211yescharacteristic
33#4887C412yescomplemented
34#4883C412yescomplemented
35#4864C414yescomplemented
36#4717—24no—
37#4708—24no—
38#4706—24no—
39#4688—24no—
40C3×#339C2212yes—
41#4558C2212yes—
42#4532C2212yes—
43#3050—31no—
44C2×#444—31no—
45#2930—32no—
46C2×#405—31no—
47#1975—31no—
48#1946—31no—
49#1937—32no—
50#1897—32no—
51#1810—31no—
52#1773D311yescharacteristic, complemented
53#1764—32no—
54C2×#235C2312yes—
55C22×C4×D3C2311yescharacteristic
56C2×C3×M16C2312yes—
57C22×C3×C8C2311yescharacteristic
58C22×C3×C8C2311yescharacteristic
59C3×#117C2312yes—
60C2×C3×C42C2311yescharacteristic
61C2×(C2×D3)⋊C4—21no—
62C2×Dic3⋊C4—21no—
63C2×C4×Dic3C2311yescharacteristic
64C2×#208C2312yes—
65C22×C3⋊C8C2311yescharacteristic
66C22×C3⋊C8C2311yescharacteristic
67C2×Dic12C2×C411yescharacteristic
68#904—22no—
69#904—22no—
70C2×D24C2×C411yescharacteristic
71C2×#204C2×C412yes—
72#901—24no—
73#901—24no—
74C2×ΓL(1,𝔽25)—22no—
75C2×ΓL(1,𝔽25)—22no—
76C2×ΓL(1,𝔽25)—22no—
77C2×ΓL(1,𝔽25)D414yes—
78C2×D3×C8—22no—
79C2×D3×C8—22no—
80C2×D3×C8—22no—
81C2×D3×C8C2×C412yes—
82C2×D3×C8C2×C412yes—
83#897C2×C414yes—
84#895C2×C414yes—
85#886—22no—
86#882—22no—
87C4×D12—24no—
88#872—22no—
89#872—22no—
90C3×#108—22no—
91C3×#107C2×C412yes—
92C3×#106C2×C412yes—
93C3×#97—22no—
94C3×#97D414yescomplemented
95C3×C4×C8—22no—
96#825—24no—
97(C2×D3)⋊C8—24no—
98(C2×D3)⋊C8—24no—
99#819—22no—
100#815—22no—
101C8×Dic3—22no—
102#810C2×C414yes—
103#809C2×C414yes—
104#805—24no—
105C4⋊(C3⋊C8)—24no—
106C4×C3⋊C8—22no—
107C2×#135—31no—
108C2×#131—32no—
109C22×M16—31no—
110C23×C8—31no—
111#525—32no—
112C2×#117—31no—
113#467—32no—
114#466—32no—
115#465—34no—
116#444—64no—
117#436—64no—
118#435—64no—
119C2×#108—31no—
120C2×#108C2×D311yescharacteristic
121#427C2×D311yescharacteristic
122#424—32no—
123C2×#104—32no—
124C2×#104—32no—
125#416—32no—
126#408—32no—
127#407—32no—
128#405—64no—
129#405C2×D312yes—
130C2×#97—31no—
131C2×#97C2×D311yescharacteristic
132C2×C4×C8—31no—
133C2×C4×C8—31no—
134#341—32no—
135#339—32no—
136#339C2×D312yes—
137C23×D3#4111yescharacteristic
138C22×C3×C4—21no—
139C22×C3×C4C2411yescharacteristic
140C2×#67—22no—
141C22×Dic3—21no—
142C22×Dic3#4111yescharacteristic
143#235—22no—
144#235—22no—
145#235—22no—
146#235—22no—
147C2×D12—24no—
148C2×D12C22×C412yes—
149C2×C4×D3—21no—
150C2×C4×D3—21no—
151C2×C4×D3—21no—
152C2×C4×D3—21no—
153C2×C4×D3—22no—
154C2×C4×D3C22×C412yes—
155C2×C4×D3C2×D412yes—
156C2×Dic6C22×C412yes—
157C3×M16—22no—
158C3×M16—22no—
159C2×C3×C8—22no—
160C2×C3×C8C22×C411yescharacteristic
161C2×C3×C8C22×C411yescharacteristic
162C2×C3×C8C2×D412yes—
163C2×C3×C8C2×D412yes—
164C2×C3×C8C2×D412yes—
165C2×C3×C8#4112yes—
166C2×C3×C8#4112yes—
167C3×C4⋊C4C22×C414yes—
168C3×C22⋊C4—22no—
169C3×C42—22no—
170C3×C42—21no—
171C3×C42C2×D412yes—
172(C2×D3)⋊C4—24no—
173(C2×D3)⋊C4—24no—
174(C2×D3)⋊C4—24no—
175C4⋊Dic3—24no—
176Dic3⋊C4—24no—
177C4×Dic3—21no—
178C4×Dic3#4112yes—
179#208—22no—
180#208—22no—
181C2×C3⋊C8—21no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—21no—
184C2×C3⋊C8—21no—
185C2×C3⋊C8—22no—
186C2×C3⋊C8C22×C412yes—
187C2×C3⋊C8C2×D412yes—
188Dic12—22no—
189D24—22no—
190#204—24no—
191ΓL(1,𝔽25)—28no—
192ΓL(1,𝔽25)—44no—
193D3×C8—24no—
194D3×C8—24no—
195D3×C8—44no—
196C2×#41—32no—
197C23×C4—31no—
198#135—62no—
199#135—62no—
200C2×Q16—31no—
201C2×SD16—32no—
202C2×D8—31no—
203#131—64no—
204#131—64no—
205C2×M16—34no—
206C2×M16—32no—
207C2×M16—62no—
208C2×M16—62no—
209C2×M16—62no—
210C2×M16C22×D312yes—
211C22×C8—31no—
212C22×C8—31no—
213C22×C8—32no—
214C22×C8—32no—
215C22×C8—62no—
216C22×C8—62no—
217C22×C8—62no—
218C22×C8C22×D311yescharacteristic
219C22×C8C22×D311yescharacteristic
220#123—62no—
221C22⋊Q8—34no—
222#121—34no—
223#120—62no—
224C4×D4—64no—
225#117—62no—
226#117—62no—
227#117C22×D312yes—
228C2×C4⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C42—31no—
231C2×C42C22×D311yescharacteristic
232#108—22no—
233#108—62no—
234#107C4×D312yes—
235#106C4×D312yes—
236C4⋊C8—64no—
237#104—64no—
238#104—64no—
239#103—34no—
240#102—34no—
241C22⋊C8—64no—
242C22⋊C8—64no—
243#97—22no—
244#97—62no—
245#97D1214yescomplemented
246C4×C8—22no—
247C4×C8—62no—
248C4×C8—62no—
249C23×C3C2×#4111yescharacteristic
250C22×D3—21no—
251C22×D3—21no—
252C22×D3—22no—
253C22×D3—42no—
254C22×D3C4×D412yes—
255C2×C3×C4—24no—
256C2×C3×C4—22no—
257C2×C3×C4—42no—
258C2×C3×C4C23×C411yescharacteristic
259C2×C3×C4C22×D411yescharacteristic
260C2×C3×C4C22×D411yescharacteristic
261C2×C3×C4C2×#4111yescharacteristic
262C2×C3×C4C2×#4111yescharacteristic
263C2×C3×C4C2×#4111yescharacteristic
264#67—42no—
265#67—42no—
266C2×Dic3—21no—
267C2×Dic3—21no—
268C2×Dic3—22no—
269C2×Dic3—22no—
270C2×Dic3—42no—
271C2×Dic3C4×D412yes—
272D12—22no—
273D12—22no—
274D12—84no—
275C4×D3—22no—
276C4×D3—22no—
277C4×D3—22no—
278C4×D3—22no—
279C4×D3—42no—
280C4×D3—42no—
281Dic6—22no—
282Dic6—22no—
283C3×C8—42no—
284C3×C8C4×D412yes—
285C3×C8C4×D412yes—
286C3×C8#12114yes—
287C3⋊C8—22no—
288C3⋊C8—22no—
289C3⋊C8—42no—
290C24—31no—
291#41—62no—
292#41—62no—
293#41—62no—
294#41—62no—
295C2×Q8—32no—
296C2×D4—32no—
297C2×D4—64no—
298C2×D4—62no—
299C22×C4—21no—
300C22×C4—31no—
301C22×C4—32no—
302C22×C4—32no—
303C22×C4—61no—
304C22×C4—61no—
305C22×C4—61no—
306C22×C4—61no—
307C22×C4—61no—
308C22×C4—62no—
309C22×C4C23×D311yescharacteristic
310Q16—62no—
311SD16—64no—
312D8—62no—
313M16—22no—
314M16—22no—
315M16—68no—
316M16—62no—
317M16—62no—
318M16—124no—
319C2×C8—22no—
320C2×C8—32no—
321C2×C8—32no—
322C2×C8—61no—
323C2×C8—61no—
324C2×C8—61no—
325C2×C8—61no—
326C2×C8—64no—
327C2×C8—64no—
328C2×C8—62no—
329C2×C8—124no—
330C2×C8C2×C4×D311yescharacteristic
331C2×C8C2×C4×D311yescharacteristic
332C2×C8C2×D1212yes—
333C2×C8C2×D1212yes—
334C2×C8#23512yes—
335C2×C8D3×D412yes—
336C2×C8#23912yes—
337C4⋊C4—64no—
338C4⋊C4—64no—
339C4⋊C4C2×C4×D314yes—
340C22⋊C4—22no—
341C22⋊C4—64no—
342C22⋊C4—64no—
343C22⋊C4—64no—
344C42—22no—
345C42—21no—
346C42—32no—
347C42—61no—
348C42C2×D1212yes—
349C22×C3—22no—
350C22×C3C2×C4×D411yescharacteristic
351C22×C3#52511yescharacteristic
352C22×C3#52911yescharacteristic
353C2×D3—22no—
354C2×D3—22no—
355C2×D3—22no—
356C2×D3—42no—
357C2×D3—42no—
358C2×D3—44no—
359C2×D3—42no—
360C3×C4—42no—
361C3×C4—44no—
362C3×C4C2×C4×D411yescharacteristic
363C3×C4C4×#4111yescharacteristic
364C3×C4C2×#12111yescharacteristic
365C3×C4#53211yescharacteristic
366Dic3—22no—
367Dic3—42no—
368Dic3—42no—
369C23—32no—
370C23—61no—
371C23—61no—
372C23—62no—
373C23—122no—
374C23D3×#4111yescharacteristic
375Q8—62no—
376Q8—62no—
377D4—62no—
378D4—62no—
379D4—122no—
380D4—122no—
381D4—244no—
382C2×C4—22no—
383C2×C4—24no—
384C2×C4—32no—
385C2×C4—42no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—62no—
393C2×C4—62no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4—122no—
397C2×C4C22×C4×D311yescharacteristic
398C2×C4C22×D1211yescharacteristic
399C2×C4C2×#23511yescharacteristic
400C2×C4C2×D3×D411yescharacteristic
401C2×C4C2×#23911yescharacteristic
402C2×C4D3×#4111yescharacteristic
403C8—42no—
404C8—62no—
405C8—62no—
406C8—122no—
407C8C4×D1212yes—
408C8C4×D1212yes—
409C8#89514yes—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3C4×#12111yescharacteristic
413C2×C3#295712yes—
414D3—42no—
415D3—82no—
416C22—22no—
417C22—62no—
418C22—62no—
419C22—62no—
420C22—122no—
421C22—122no—
422C22—124no—
423C22—122no—
424C22#552411yescharacteristic
425C22C4×D3×D411yescharacteristic
426C22#558011yescharacteristic
427C4—42no—
428C4—44no—
429C4—62no—
430C4—122no—
431C4—122no—
432C4C2×C4×D1211yescharacteristic
433C4C2×#89511yescharacteristic
434C4#554811yescharacteristic
435C4C4×#23911yescharacteristic
436C3#2154711yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—122no—
440C2—242no—
441C2C4×#89511yescharacteristic
442C2#7989912yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1773C2
socleC22×C3C2×C4×D4
prime radical#79180C2
cosocleC22×C3×C4C24
hypercenter#1773D3
hypocenterC3#21547
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1773

Complements

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

NH ≅ G/Ntype
2. #78554C2semidirect
3. C2×#4717C2semidirect
4. C2×#4708C2semidirect
5. C3×#1773C2semidirect
6. #70711C2semidirect
7. #70697C2semidirect
8. #70663C2semidirect
9. #69887C2semidirect
10. #68757C2semidirect
11. #68745C2semidirect
33. #4887C4semidirect
34. #4883C4semidirect
35. #4864C4semidirect
52. #1773D3semidirect
94. C3×#97D4semidirect
245. #97D12semidirect
436. C3#21547semidirect

Maximal subgroups

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

subgroupindex
#791802
#785542
C2×#47172
C2×#47082
C3×#17732
#707112
#706972
#706632
#698872
#687572
#687452
#215473