Finite Group Databasebeta

← #10801997

Subgroups — #10801997

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
1C2×C3×#405C211yesmaximal, characteristic, complemented
2#83756C212yesmaximal, complemented
3#83742C211yesmaximal, characteristic
4C2×#4717C211yesmaximal, characteristic, complemented
5#71977C211yesmaximal, characteristic, complemented
6#71591C212yesmaximal, complemented
7#69845C211yesmaximal, characteristic, complemented
8#69844C211yesmaximal, characteristic, complemented
9#69829C211yesmaximal, characteristic, complemented
10#69793C212yesmaximal, complemented
11#69789C212yesmaximal, complemented
12#21547—31nomaximal, Sylow
13C22×C3×M16C2211yescharacteristic
14C23×C3×C8C2211yescharacteristic
15C2×C3×#117C2211yescharacteristic
16#5791C2212yes—
17C2×#904C2211yescharacteristic
18C2×#901C2212yes—
19C3×#405—24no—
20C2×C3×#97C2211yescharacteristic
21C2×C3×#97C2211yescharacteristic
22C2×C3×C4×C8C2211yescharacteristic
23C2×C3×C4×C8C2211yescharacteristic
24#5207C2212yes—
25#5128—24no—
26#5127—24no—
27#5116C412yescomplemented
28#5115C412yescomplemented
29#5114C414yescomplemented
30#5113C2212yes—
31#5110C2212yes—
32C2×#819C2211yescharacteristic
33C2×#819C2211yescharacteristic
34#5106C2211yescharacteristic
35#5102C2212yes—
36#5101C2212yes—
37C2×#805C2212yes—
38C2×#805C2212yes—
39#4717—24no—
40#4552C2212yes—
41#4523C2212yes—
42#4523C2212yes—
43#3050—31no—
44C2×#444—31no—
45#2930—32no—
46C2×#405D311yescharacteristic, complemented
47#1975—31no—
48#1946—31no—
49#1937—32no—
50#1897—32no—
51#1810—31no—
52#1773—31no—
53#1764—32no—
54C23×C3×C4C2311yescharacteristic
55C2×#235C2312yes—
56C2×C3×M16—22no—
57C2×C3×M16—22no—
58C2×C3×M16D414yes—
59C22×C3×C8—22no—
60C22×C3×C8—22no—
61C22×C3×C8C2×C412yes—
62C22×C3×C8C2×C412yes—
63C22×C3×C8C2311yescharacteristic
64C22×C3×C8C2311yescharacteristic
65C22×C3×C8C2311yescharacteristic
66C22×C3×C8C2311yescharacteristic
67C3×#117—22no—
68C3×#117—22no—
69C2×C3×C4⋊C4—21no—
70C2×C3×C22⋊C4—21no—
71C2×C3×C42C2311yescharacteristic
72C2×C3×C42C2311yescharacteristic
73#953—22no—
74#930C2×C414yes—
75#929—22no—
76C4×#67—24no—
77#926C2312yes—
78#924C2×C414yes—
79C2×#208C2312yes—
80C2×#208C2312yes—
81C2×Dic12C2×C411yescharacteristic
82#904—22no—
83#904—22no—
84C2×D24C2×C411yescharacteristic
85C2×#204C2×C412yes—
86#901—24no—
87#901—24no—
88C2×ΓL(1,𝔽25)—22no—
89C2×D3×C8—22no—
90C3×#97—22no—
91C3×#97D414yescomplemented
92C3×C4×C8—22no—
93C3×C4×C8—22no—
94#830—24no—
95#821C2×C414yes—
96(C2×D3)⋊C8—24no—
97#819—22no—
98#819—22no—
99#818C2×C412yes—
100#817C2×C412yes—
101#816C2×C414yes—
102#815—22no—
103Dic3⋊C8—24no—
104C8×Dic3—22no—
105#805—24no—
106#805—24no—
107C2×#135—31no—
108C2×#131—32no—
109C22×M16C2×D311yescharacteristic
110C23×C8C2×D311yescharacteristic
111#525—32no—
112C2×#117C2×D311yescharacteristic
113#467—32no—
114#466—32no—
115#465—34no—
116#444—64no—
117#436—64no—
118#435—64no—
119C2×#108—31no—
120C2×#108—31no—
121#427—31no—
122#424—32no—
123C2×#104—32no—
124C2×#104—32no—
125#416—32no—
126#408—32no—
127#407—32no—
128#405—24no—
129#405—32no—
130C2×#97C2×D311yescharacteristic
131C2×#97C2×D311yescharacteristic
132C2×C4×C8C2×D311yescharacteristic
133C2×C4×C8C2×D311yescharacteristic
134#341—32no—
135#339—32no—
136#339—32no—
137C24×C3#4111yescharacteristic
138C22×C3×C4—21no—
139C22×C3×C4—21no—
140C22×C3×C4—21no—
141C22×C3×C4—21no—
142C22×C3×C4—21no—
143C22×C3×C4—21no—
144C22×C3×C4C22×C412yes—
145C22×C3×C4C2×D412yes—
146C22×C3×C4#4111yescharacteristic
147C22×C3×C4C2411yescharacteristic
148C2×#67—22no—
149C2×#67—24no—
150#235—22no—
151#235—22no—
152#235—22no—
153#235—22no—
154C2×D12C22×C412yes—
155C2×C4×D3—22no—
156C2×Dic6C22×C412yes—
157C3×M16—28no—
158C2×C3×C8—21no—
159C2×C3×C8—21no—
160C2×C3×C8—21no—
161C2×C3×C8—21no—
162C2×C3×C8—24no—
163C2×C3×C8—24no—
164C2×C3×C8C22×C412yes—
165C2×C3×C8C22×C411yescharacteristic
166C2×C3×C8C22×C411yescharacteristic
167C2×C3×C8C2×D412yes—
168C2×C3×C8C2×D412yes—
169C2×C3×C8C2×D412yes—
170C2×C3×C8C2×D412yes—
171C2×C3×C8#4112yes—
172C2×C3×C8#4112yes—
173C3×C4⋊C4—24no—
174C3×C22⋊C4—24no—
175C3×C42—21no—
176C3×C42—21no—
177C3×C42C2×D412yes—
178C3×C42#4112yes—
179#217—24no—
180#217—22no—
181(C2×D3)⋊C4—24no—
182C4⋊Dic3C22×C414yes—
183Dic3⋊C4—24no—
184C4×Dic3—22no—
185#208—22no—
186#208—22no—
187#208—22no—
188#208—22no—
189C2×C3⋊C8—22no—
190C2×C3⋊C8—22no—
191Dic12—22no—
192D24—22no—
193#204—24no—
194ΓL(1,𝔽25)—44no—
195D3×C8—44no—
196C2×#41—32no—
197C23×C4C22×D311yescharacteristic
198#135—62no—
199#135—62no—
200C2×Q16—31no—
201C2×SD16—32no—
202C2×D8—31no—
203#131—64no—
204#131—64no—
205C2×M16—22no—
206C2×M16—22no—
207C2×M16—32no—
208C2×M16—32no—
209C2×M16—62no—
210C2×M16D1214yes—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—62no—
214C22×C8C4×D312yes—
215C22×C8C4×D312yes—
216C22×C8C22×D311yescharacteristic
217C22×C8C22×D311yescharacteristic
218C22×C8C22×D311yescharacteristic
219C22×C8C22×D311yescharacteristic
220#123—62no—
221C22⋊Q8—34no—
222#121—34no—
223#120—62no—
224C4×D4—64no—
225#117—22no—
226#117—22no—
227#117—32no—
228C2×C4⋊C4—21no—
229C2×C22⋊C4—21no—
230C2×C42C22×D311yescharacteristic
231C2×C42C22×D311yescharacteristic
232#108—62no—
233#108—62no—
234#107—32no—
235#106—32no—
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#97#6714yescomplemented
246C4×C8—22no—
247C4×C8—22no—
248C4×C8—62no—
249C23×C3—21no—
250C23×C3—21no—
251C23×C3—42no—
252C23×C3C4×D412yes—
253C23×C3C2×#4111yescharacteristic
254C22×D3—22no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—22no—
258C2×C3×C4—22no—
259C2×C3×C4—22no—
260C2×C3×C4—22no—
261C2×C3×C4—22no—
262C2×C3×C4—22no—
263C2×C3×C4—42no—
264C2×C3×C4—42no—
265C2×C3×C4C4×D412yes—
266C2×C3×C4C23×C411yescharacteristic
267C2×C3×C4C22×D411yescharacteristic
268C2×C3×C4C22×D411yescharacteristic
269C2×C3×C4C2×#4111yescharacteristic
270C2×C3×C4C2×#4111yescharacteristic
271C2×C3×C4C2×#4111yescharacteristic
272#67—42no—
273#67—42no—
274#67—84no—
275C2×Dic3—24no—
276C2×Dic3—22no—
277D12—22no—
278D12—22no—
279C4×D3—42no—
280C4×D3—42no—
281Dic6—22no—
282Dic6—22no—
283C3×C8—22no—
284C3×C8—22no—
285C3×C8C4×D412yes—
286C3×C8C4×D412yes—
287C3×C8#12114yes—
288C3⋊C8—42no—
289C3⋊C8—42no—
290C24#23511yescharacteristic
291#41—62no—
292#41—62no—
293#41—62no—
294#41—62no—
295C2×Q8—32no—
296C2×D4—32no—
297C2×D4—62no—
298C2×D4—64no—
299C22×C4—21no—
300C22×C4—21no—
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4—21no—
305C22×C4—62no—
306C22×C4C2×C4×D312yes—
307C22×C4C2×D1212yes—
308C22×C4#23511yescharacteristic
309C22×C4C23×D311yescharacteristic
310Q16—62no—
311SD16—64no—
312D8—62no—
313M16—28no—
314M16—62no—
315M16—62no—
316M16—62no—
317M16—62no—
318M16—124no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—21no—
322C2×C8—21no—
323C2×C8—24no—
324C2×C8—24no—
325C2×C8—62no—
326C2×C8—62no—
327C2×C8—124no—
328C2×C8C2×C4×D312yes—
329C2×C8C2×C4×D311yescharacteristic
330C2×C8C2×C4×D311yescharacteristic
331C2×C8C2×D1212yes—
332C2×C8C2×D1212yes—
333C2×C8#23512yes—
334C2×C8#23512yes—
335C2×C8C2×#6712yes—
336C2×C8C2×#6712yes—
337C4⋊C4—24no—
338C4⋊C4—34no—
339C4⋊C4—64no—
340C22⋊C4—24no—
341C22⋊C4—64no—
342C22⋊C4—62no—
343C22⋊C4—64no—
344C42—21no—
345C42—21no—
346C42—62no—
347C42#23512yes—
348C42C2×#6712yes—
349C22×C3—22no—
350C22×C3—22no—
351C22×C3—22no—
352C22×C3—22no—
353C22×C3—42no—
354C22×C3—42no—
355C22×C3C2×C4×D411yescharacteristic
356C22×C3#52511yescharacteristic
357C22×C3#52911yescharacteristic
358C2×D3—44no—
359C2×D3—42no—
360C3×C4—22no—
361C3×C4—42no—
362C3×C4—42no—
363C3×C4C2×C4×D411yescharacteristic
364C3×C4C4×#4111yescharacteristic
365C3×C4C2×#12111yescharacteristic
366C3×C4#53211yescharacteristic
367Dic3—44no—
368Dic3—42no—
369C23—21no—
370C23—21no—
371C23—42no—
372C23—62no—
373C23C4×D1212yes—
374C23C2×#23511yescharacteristic
375Q8—62no—
376Q8—62no—
377D4—62no—
378D4—62no—
379D4—122no—
380D4—122no—
381D4—244no—
382C2×C4—21no—
383C2×C4—21no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—42no—
391C2×C4—42no—
392C2×C4—64no—
393C2×C4—62no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4C4×D1212yes—
397C2×C4C22×C4×D311yescharacteristic
398C2×C4C22×D1211yescharacteristic
399C2×C4C2×#23511yescharacteristic
400C2×C4C2×#23511yescharacteristic
401C2×C4C2×#23511yescharacteristic
402C2×C4C22×#6711yescharacteristic
403C8—22no—
404C8—22no—
405C8—122no—
406C8—122no—
407C8C4×#6712yes—
408C8C4×#6712yes—
409C8#93014yes—
410C2×C3—21no—
411C2×C3—21no—
412C2×C3—42no—
413C2×C3C4×#12111yescharacteristic
414C2×C3#295712yes—
415D3—82no—
416C22—22no—
417C22—22no—
418C22—22no—
419C22—22no—
420C22—42no—
421C22—42no—
422C22—122no—
423C22—124no—
424C22C2×C4×D1211yescharacteristic
425C22#547911yescharacteristic
426C22#579111yescharacteristic
427C4—22no—
428C4—42no—
429C4—42no—
430C4—122no—
431C4—124no—
432C4C4×#23511yescharacteristic
433C4#547711yescharacteristic
434C4C2×C4×#6711yescharacteristic
435C4C2×#93011yescharacteristic
436C3#2154711yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—42no—
440C2—242no—
441C2#7858012yes—
442C2C4×#93011yescharacteristic
subgroupG/H
centerC2×C4C22×D12
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC2×C3×#405C2
socleC22×C3C2×C4×D4
prime radical#83742C2
cosocleC22×C3×C4C24
hypercenterC2×#405D3
hypocenterC3#21547
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×M16 ◁ C2×#405

Complements

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

NH ≅ G/Ntype
1. C2×C3×#405C2semidirect
2. #83756C2semidirect
4. C2×#4717C2semidirect
5. #71977C2semidirect
6. #71591C2semidirect
7. #69845C2semidirect
8. #69844C2semidirect
9. #69829C2semidirect
10. #69793C2semidirect
11. #69789C2semidirect
27. #5116C4semidirect
28. #5115C4semidirect
29. #5114C4semidirect
46. C2×#405D3semidirect
91. C3×#97D4semidirect
245. #97#67semidirect
436. C3#21547semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#4052
#837562
#837422
C2×#47172
#719772
#715912
#698452
#698442
#698292
#697932
#697892
#215473