Finite Group Databasebeta

← #10801990

Subgroups — #10801990

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#83742C211yesmaximal, characteristic
2C2×#5101C211yesmaximal, characteristic, complemented
3C2×#4894C211yesmaximal, characteristic, complemented
4#78627C212yesmaximal, complemented
5C3×#1810C211yesmaximal, characteristic, complemented
6#70705C211yesmaximal, characteristic, complemented
7#70703C211yesmaximal, characteristic, complemented
8#70697C211yesmaximal, characteristic, complemented
9#69966C212yesmaximal, complemented
10#68926C212yesmaximal, complemented
11#68918C212yesmaximal, complemented
12#21547—31nomaximal, Sylow
13C23×C3×C8C2211yescharacteristic
14#5791C2212yes—
15C2×#926C2211yescharacteristic
16C22×#208C2211yescharacteristic
17C2×#907C2212yes—
18C2×#904C2211yescharacteristic
19C2×C3×#108C2211yescharacteristic
20C2×C3×#108C2211yescharacteristic
21C3×#407C2212yes—
22C2×#825C2212yes—
23C2×#825C2212yes—
24#5129C2212yes—
25#5122C2212yes—
26#5119C2212yes—
27#5116C412yescomplemented
28#5115C412yescomplemented
29#5114C414yescomplemented
30#5113C2212yes—
31#5106C2211yescharacteristic
32#5101—24no—
33C2×#815C2211yescharacteristic
34C2×#815C2211yescharacteristic
35C2×C8×Dic3C2211yescharacteristic
36C2×C8×Dic3C2211yescharacteristic
37#4894—24no—
38#4729—24no—
39#4728—24no—
40C3×#341C2212yes—
41#4555C2212yes—
42#4555C2212yes—
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#1810D311yescharacteristic, complemented
52#1773—31no—
53#1764—32no—
54C23×C3×C4C2311yescharacteristic
55C2×#235C2312yes—
56C2×C3×M16C2312yes—
57C2×C3×M16C2312yes—
58C22×C3×C8—22no—
59C22×C3×C8—22no—
60C22×C3×C8C2×C412yes—
61C22×C3×C8C2×C412yes—
62C22×C3×C8C2311yescharacteristic
63C22×C3×C8C2311yescharacteristic
64#953—22no—
65C2×#217—21no—
66#930C2×C414yes—
67#929—22no—
68C4×#67—24no—
69#926—22no—
70#926—22no—
71#926C2312yes—
72C2×C4⋊Dic3—21no—
73#924C2×C414yes—
74C2×C4×Dic3C2311yescharacteristic
75C2×C4×Dic3C2311yescharacteristic
76C2×#208—22no—
77C2×#208—22no—
78C2×#208D414yes—
79C22×C3⋊C8C2311yescharacteristic
80C22×C3⋊C8C2311yescharacteristic
81#907—24no—
82#907—24no—
83C2×Dic12C2×C411yescharacteristic
84#904—22no—
85#904—22no—
86C2×D24C2×C411yescharacteristic
87C2×#204C2×C412yes—
88C2×ΓL(1,𝔽25)—22no—
89C2×D3×C8—22no—
90C3×#108—22no—
91C3×#108—22no—
92C3×C22⋊C8—24no—
93#825—24no—
94#825—24no—
95#821C2×C414yes—
96(C2×D3)⋊C8—24no—
97#818C2×C412yes—
98#817C2×C412yes—
99#816C2×C414yes—
100#815—22no—
101#815—22no—
102#815D414yescomplemented
103Dic3⋊C8—24no—
104C8×Dic3—22no—
105C8×Dic3—22no—
106C8×Dic3—22no—
107C2×#135—31no—
108C2×#131—32no—
109C22×M16—31no—
110C23×C8C2×D311yescharacteristic
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×#108C2×D311yescharacteristic
120C2×#108C2×D311yescharacteristic
121#427—31no—
122#424—32no—
123C2×#104—32no—
124C2×#104—32no—
125#416—32no—
126#408—32no—
127#407C2×D312yes—
128#405—32no—
129#405—64no—
130C2×#97—31no—
131C2×#97—31no—
132C2×C4×C8—31no—
133C2×C4×C8—31no—
134#341C2×D312yes—
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×C4C22×C412yes—
143C22×C3×C4C2×D412yes—
144C22×C3×C4#4111yescharacteristic
145C22×C3×C4C2411yescharacteristic
146C2×#67—22no—
147C2×#67—24no—
148C22×Dic3—21no—
149C22×Dic3—21no—
150#235—22no—
151#235—22no—
152#235—22no—
153#235—22no—
154C2×D12C22×C412yes—
155C2×C4×D3—22no—
156C2×Dic6C22×C412yes—
157C3×M16—22no—
158C3×M16—22no—
159C3×M16—22no—
160C3×M16—22no—
161C2×C3×C8—21no—
162C2×C3×C8—21no—
163C2×C3×C8—24no—
164C2×C3×C8—24no—
165C2×C3×C8—22no—
166C2×C3×C8—22no—
167C2×C3×C8C22×C412yes—
168C2×C3×C8C22×C411yescharacteristic
169C2×C3×C8C22×C411yescharacteristic
170C2×C3×C8C2×D412yes—
171C2×C3×C8#4112yes—
172#217—24no—
173#217—24no—
174#217—22no—
175(C2×D3)⋊C4—24no—
176C4⋊Dic3—24no—
177C4⋊Dic3C22×C414yes—
178Dic3⋊C4—24no—
179C4×Dic3—21no—
180C4×Dic3—22no—
181C4×Dic3—21no—
182C4×Dic3C2×D412yes—
183C4×Dic3#4112yes—
184#208—28no—
185C2×C3⋊C8—21no—
186C2×C3⋊C8—21no—
187C2×C3⋊C8C2×D412yes—
188C2×C3⋊C8C2×D412yes—
189C2×C3⋊C8C2×D412yes—
190C2×C3⋊C8#4112yes—
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—34no—
206C2×M16—62no—
207C2×M16—62no—
208C2×M16—62no—
209C2×M16C22×D312yes—
210C2×M16C22×D312yes—
211C22×C8—22no—
212C22×C8—22no—
213C22×C8—31no—
214C22×C8—31no—
215C22×C8—62no—
216C22×C8C4×D312yes—
217C22×C8C4×D312yes—
218C22×C8C22×D311yescharacteristic
219C22×C8C22×D311yescharacteristic
220#123—62no—
221C22⋊Q8—34no—
222#121—34no—
223#120—62no—
224C4×D4—64no—
225#117—32no—
226#117—62no—
227#117—62no—
228C2×C4⋊C4—61no—
229C2×C22⋊C4—61no—
230C2×C42—31no—
231C2×C42—31no—
232#108—22no—
233#108—22no—
234#107—32no—
235#106—32no—
236C4⋊C8—64no—
237#104—64no—
238#104—64no—
239#103—34no—
240#102—34no—
241C22⋊C8—24no—
242C22⋊C8—64no—
243#97—34no—
244#97—62no—
245#97—62no—
246C4×C8—62no—
247C4×C8—62no—
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×C4C4×D412yes—
262C2×C3×C4C23×C411yescharacteristic
263C2×C3×C4C22×D411yescharacteristic
264C2×C3×C4C22×D411yescharacteristic
265C2×C3×C4C2×#4111yescharacteristic
266C2×C3×C4C2×#4111yescharacteristic
267C2×C3×C4C2×#4111yescharacteristic
268#67—42no—
269#67—42no—
270#67—84no—
271C2×Dic3—24no—
272C2×Dic3—22no—
273C2×Dic3—22no—
274C2×Dic3—22no—
275C2×Dic3—42no—
276C2×Dic3—42no—
277D12—22no—
278D12—22no—
279C4×D3—42no—
280C4×D3—42no—
281Dic6—22no—
282Dic6—22no—
283C3×C8—22no—
284C3×C8—42no—
285C3×C8—42no—
286C3×C8C4×D412yes—
287C3×C8C4×D412yes—
288C3⋊C8—22no—
289C3⋊C8#12114yes—
290C24#23911yescharacteristic
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—61no—
304C22×C4—61no—
305C22×C4—62no—
306C22×C4C2×C4×D312yes—
307C22×C4D3×D412yes—
308C22×C4#23911yescharacteristic
309C22×C4C23×D311yescharacteristic
310Q16—62no—
311SD16—64no—
312D8—62no—
313M16—22no—
314M16—22no—
315M16—22no—
316M16—22no—
317M16—68no—
318M16—124no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—24no—
322C2×C8—24no—
323C2×C8—22no—
324C2×C8—22no—
325C2×C8—32no—
326C2×C8—32no—
327C2×C8—32no—
328C2×C8—32no—
329C2×C8—61no—
330C2×C8—61no—
331C2×C8—124no—
332C2×C8C2×C4×D312yes—
333C2×C8C2×C4×D311yescharacteristic
334C2×C8C2×C4×D311yescharacteristic
335C2×C8D3×D412yes—
336C2×C8#23712yes—
337C4⋊C4—34no—
338C4⋊C4—64no—
339C4⋊C4—64no—
340C22⋊C4—64no—
341C22⋊C4—64no—
342C22⋊C4—62no—
343C22⋊C4—64no—
344C42—32no—
345C42—32no—
346C42—61no—
347C42—62no—
348C42—61no—
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×C4C2×C4×D411yescharacteristic
362C3×C4C4×#4111yescharacteristic
363C3×C4C2×#12111yescharacteristic
364C3×C4#53211yescharacteristic
365Dic3—42no—
366Dic3—42no—
367Dic3—44no—
368Dic3—42no—
369C23—21no—
370C23—21no—
371C23—42no—
372C23—62no—
373C23#89312yes—
374C23C2×#23911yescharacteristic
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—64no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—122no—
393C2×C4—122no—
394C2×C4—122no—
395C2×C4—122no—
396C2×C4#89312yes—
397C2×C4C22×C4×D311yescharacteristic
398C2×C4C2×D3×D411yescharacteristic
399C2×C4C2×D3×D411yescharacteristic
400C2×C4C2×#23711yescharacteristic
401C2×C4D3×#4111yescharacteristic
402C2×C4D3×#4111yescharacteristic
403C8—22no—
404C8—34no—
405C8—42no—
406C8—42no—
407C8—62no—
408C8#67⋊C412yes—
409C8#67⋊C412yes—
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×#89311yescharacteristic
425C22#562411yescharacteristic
426C22#562711yescharacteristic
427C4—22no—
428C4—122no—
429C4—122no—
430C4—122no—
431C4—124no—
432C4C2×#67⋊C411yescharacteristic
433C4#552211yescharacteristic
434C4#559611yescharacteristic
435C4D3×#12111yescharacteristic
436C3#2154711yescharacteristic, complemented, Sylow
437C2—21no—
438C2—21no—
439C2—42no—
440C2—242no—
441C2#7386511yescharacteristic
442C2#7924912yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1810C2
socleC22×C3C2×C4×D4
prime radical#83742C2
cosocleC22×C3×C4C24
hypercenter#1810D3
hypocenterC3#21547
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1810

Complements

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

NH ≅ G/Ntype
2. C2×#5101C2semidirect
3. C2×#4894C2semidirect
4. #78627C2semidirect
5. C3×#1810C2semidirect
6. #70705C2semidirect
7. #70703C2semidirect
8. #70697C2semidirect
9. #69966C2semidirect
10. #68926C2semidirect
11. #68918C2semidirect
27. #5116C4semidirect
28. #5115C4semidirect
29. #5114C4semidirect
51. #1810D3semidirect
102. #815D4semidirect
436. C3#21547semidirect

Maximal subgroups

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

subgroupindex
#837422
C2×#51012
C2×#48942
#786272
C3×#18102
#707052
#707032
#706972
#699662
#689262
#689182
#215473