Finite Group Databasebeta

← #10802003

Subgroups — #10802003

Order 768.

914 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#4895C211yesmaximal, characteristic, complemented
2#79181C211yesmaximal, characteristic
3#79073C212yesmaximal, complemented
4C2×#4708C211yesmaximal, characteristic, complemented
5C3×#1946C211yesmaximal, characteristic, complemented
6#70712C211yesmaximal, characteristic, complemented
7#70659C211yesmaximal, characteristic, complemented
8#70536C212yesmaximal, complemented
9#69845C211yesmaximal, characteristic, complemented
10#68757C212yesmaximal, complemented
11#68739C212yesmaximal, complemented
12#21548—31nomaximal, Sylow
13C2×#907C2212yes—
14C2×#904C2211yescharacteristic
15C22×ΓL(1,𝔽25)C2211yescharacteristic
16C22×D3×C8C2211yescharacteristic
17#5524C2212yes—
18C2×#872C2211yescharacteristic
19C2×C3×#108C2211yescharacteristic
20C3×#427C2211yescharacteristic
21C3×#424C2212yes—
22C2×C3×#97C2211yescharacteristic
23C2×#825C2212yes—
24#5126C2212yes—
25#5110C2212yes—
26C2×#819C2211yescharacteristic
27C2×#815C2211yescharacteristic
28C2×C8×Dic3C2211yescharacteristic
29#5002C2212yes—
30#4997C2212yes—
31C2×#805C2212yes—
32C2×C4×C3⋊C8C2211yescharacteristic
33#4895—24no—
34#4886C414yescomplemented
35#4867C412yescomplemented
36#4865C412yescomplemented
37#4836—24no—
38#4834—24no—
39#4708—24no—
40C3×#339C2212yes—
41#4558C2212yes—
42#4532C2212yes—
43#3054—31no—
44C2×#443—31no—
45#2929—32no—
46C2×#405—31no—
47#1975—31no—
48#1946D311yescharacteristic, complemented
49#1938—32no—
50#1897—32no—
51#1811—31no—
52#1772—31no—
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
67#907—24no—
68#907—24no—
69C2×Dic12C2×C411yescharacteristic
70#904—22no—
71#904—22no—
72C2×D24C2×C411yescharacteristic
73C2×#204C2×C412yes—
74C2×ΓL(1,𝔽25)—22no—
75C2×ΓL(1,𝔽25)—22no—
76C2×ΓL(1,𝔽25)—22no—
77C2×ΓL(1,𝔽25)C2×C412yes—
78C2×ΓL(1,𝔽25)C2×C412yes—
79C2×D3×C8—22no—
80C2×D3×C8—22no—
81C2×D3×C8—22no—
82C2×D3×C8D412yes—
83C2×D3×C8D412yes—
84#897C2×C414yes—
85#895C2×C414yes—
86#886—22no—
87#882—22no—
88C4×D12—24no—
89#872—22no—
90#872—22no—
91C3×#108—22no—
92C3×#107C2×C412yes—
93C3×#106C2×C412yes—
94C3×C4⋊C8—24no—
95C3×#97—22no—
96#825—24no—
97(C2×D3)⋊C8—24no—
98(C2×D3)⋊C8—24no—
99#819—22no—
100#815—22no—
101C8×Dic3D414yescomplemented
102#810C2×C414yes—
103#809C2×C414yes—
104#805—24no—
105#803—22no—
106C4×C3⋊C8—22no—
107C4×C3⋊C8—22no—
108C2×#135—31no—
109C2×#131—32no—
110C22×M16—31no—
111C23×C8—31no—
112#525—32no—
113C2×#117—31no—
114#470—32no—
115#469—32no—
116#468—34no—
117#443—64no—
118#435—64no—
119C8×D4—64no—
120C2×#108—31no—
121C2×#108C2×D311yescharacteristic
122#427C2×D311yescharacteristic
123#424C2×D312yes—
124C2×#104—32no—
125C2×#104—32no—
126#416—32no—
127#408—32no—
128#407—32no—
129#405—32no—
130#405—64no—
131C2×#97—31no—
132C2×#97C2×D311yescharacteristic
133C2×C4×C8—31no—
134C2×C4×C8—31no—
135#341—32no—
136#339—32no—
137#339C2×D312yes—
138C23×D3#4111yescharacteristic
139C22×C3×C4—21no—
140C22×C3×C4C2411yescharacteristic
141C2×#67—22no—
142C22×Dic3—21no—
143C22×Dic3#4111yescharacteristic
144#235—22no—
145#235—22no—
146#235—22no—
147#235—22no—
148C2×D12—24no—
149C2×D12C22×C412yes—
150C2×C4×D3—21no—
151C2×C4×D3—21no—
152C2×C4×D3—21no—
153C2×C4×D3—21no—
154C2×C4×D3—22no—
155C2×C4×D3C22×C412yes—
156C2×C4×D3C2×D412yes—
157C2×Dic6C22×C412yes—
158C3×M16—22no—
159C3×M16—22no—
160C2×C3×C8—21no—
161C2×C3×C8—21no—
162C2×C3×C8—22no—
163C2×C3×C8C22×C411yescharacteristic
164C2×C3×C8C22×C411yescharacteristic
165C2×C3×C8C2×D412yes—
166C2×C3×C8C2×D412yes—
167C2×C3×C8#4112yes—
168C3×C4⋊C4C22×C414yes—
169C3×C22⋊C4—22no—
170C3×C42—21no—
171C3×C42—22no—
172C3×C42#4112yes—
173(C2×D3)⋊C4—24no—
174(C2×D3)⋊C4—24no—
175(C2×D3)⋊C4—24no—
176C4⋊Dic3—24no—
177Dic3⋊C4—24no—
178C4×Dic3—21no—
179C4×Dic3C2×D412yes—
180#208—22no—
181#208—22no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—21no—
184C2×C3⋊C8—22no—
185C2×C3⋊C8C22×C412yes—
186C2×C3⋊C8C2×D411yescharacteristic
187C2×C3⋊C8C2×D412yes—
188C2×C3⋊C8C2×D411yescharacteristic
189C2×C3⋊C8#4112yes—
190Dic12—22no—
191D24—22no—
192#204—24no—
193ΓL(1,𝔽25)—24no—
194ΓL(1,𝔽25)—24no—
195ΓL(1,𝔽25)—44no—
196D3×C8—24no—
197D3×C8—24no—
198D3×C8—44no—
199C2×#41—32no—
200C23×C4—31no—
201#135—62no—
202#135—62no—
203C2×Q16—31no—
204C2×SD16—32no—
205C2×D8—31no—
206#131—64no—
207#131—64no—
208C2×M16—32no—
209C2×M16—32no—
210C2×M16—32no—
211C2×M16—62no—
212C2×M16—62no—
213C2×M16—62no—
214C2×M16C22×D312yes—
215C22×C8—31no—
216C22×C8—31no—
217C22×C8—32no—
218C22×C8—32no—
219C22×C8—62no—
220C22×C8—62no—
221C22×C8—62no—
222C22×C8C22×D311yescharacteristic
223C22×C8C22×D311yescharacteristic
224#123—62no—
225C22⋊Q8—34no—
226#121—34no—
227#120—62no—
228C4×D4—64no—
229#117—62no—
230#117—62no—
231#117C22×D312yes—
232C2×C4⋊C4—61no—
233C2×C22⋊C4—61no—
234C2×C42—31no—
235C2×C42C22×D311yescharacteristic
236#108—22no—
237#108—62no—
238#107C4×D312yes—
239#106C4×D312yes—
240C4⋊C8—24no—
241#104—64no—
242#104—64no—
243#103—34no—
244#102—34no—
245C22⋊C8—64no—
246C22⋊C8—64no—
247#97—22no—
248#97—62no—
249#97—62no—
250C4×C8—34no—
251C4×C8—62no—
252C4×C8—62no—
253C23×C3C2×#4111yescharacteristic
254C22×D3—21no—
255C22×D3—21no—
256C22×D3—22no—
257C22×D3—42no—
258C22×D3C4×D412yes—
259C2×C3×C4—24no—
260C2×C3×C4—22no—
261C2×C3×C4—42no—
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—
270C2×Dic3—21no—
271C2×Dic3—21no—
272C2×Dic3—22no—
273C2×Dic3—22no—
274C2×Dic3—42no—
275C2×Dic3C4×D412yes—
276D12—22no—
277D12—22no—
278D12—48no—
279C4×D3—22no—
280C4×D3—22no—
281C4×D3—22no—
282C4×D3—22no—
283C4×D3—42no—
284C4×D3—42no—
285Dic6—22no—
286Dic6—22no—
287C3×C8—22no—
288C3×C8—42no—
289C3×C8C4×D412yes—
290C3×C8C4×D412yes—
291C3⋊C8—22no—
292C3⋊C8—42no—
293C3⋊C8#12112yes—
294C3⋊C8#12112yes—
295C24—31no—
296#41—62no—
297#41—62no—
298#41—62no—
299#41—62no—
300C2×Q8—32no—
301C2×D4—32no—
302C2×D4—64no—
303C2×D4—62no—
304C22×C4—21no—
305C22×C4—31no—
306C22×C4—32no—
307C22×C4—32no—
308C22×C4—61no—
309C22×C4—61no—
310C22×C4—61no—
311C22×C4—61no—
312C22×C4—61no—
313C22×C4—62no—
314C22×C4C23×D311yescharacteristic
315Q16—62no—
316SD16—64no—
317D8—62no—
318M16—22no—
319M16—22no—
320M16—64no—
321M16—64no—
322M16—62no—
323M16—62no—
324M16—124no—
325C2×C8—21no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—32no—
329C2×C8—31no—
330C2×C8—31no—
331C2×C8—32no—
332C2×C8—32no—
333C2×C8—64no—
334C2×C8—64no—
335C2×C8—61no—
336C2×C8—61no—
337C2×C8—62no—
338C2×C8—124no—
339C2×C8C2×C4×D311yescharacteristic
340C2×C8C2×C4×D311yescharacteristic
341C2×C8D3×D412yes—
342C2×C8D3×D412yes—
343C2×C8#23912yes—
344C4⋊C4—64no—
345C4⋊C4—64no—
346C4⋊C4C2×C4×D314yes—
347C22⋊C4—22no—
348C22⋊C4—64no—
349C22⋊C4—64no—
350C22⋊C4—64no—
351C42—21no—
352C42—22no—
353C42—32no—
354C42—61no—
355C42#23912yes—
356C22×C3—22no—
357C22×C3C2×C4×D411yescharacteristic
358C22×C3#52511yescharacteristic
359C22×C3#52911yescharacteristic
360C2×D3—24no—
361C2×D3—22no—
362C2×D3—42no—
363C2×D3—42no—
364C2×D3—44no—
365C2×D3—42no—
366C3×C4—42no—
367C3×C4—44no—
368C3×C4C2×C4×D411yescharacteristic
369C3×C4C4×#4111yescharacteristic
370C3×C4C2×#12111yescharacteristic
371C3×C4#53211yescharacteristic
372Dic3—22no—
373Dic3—42no—
374Dic3—42no—
375C23—32no—
376C23—61no—
377C23—61no—
378C23—62no—
379C23—122no—
380C23D3×#4111yescharacteristic
381Q8—62no—
382Q8—62no—
383D4—62no—
384D4—62no—
385D4—128no—
386D4—122no—
387D4—122no—
388C2×C4—24no—
389C2×C4—22no—
390C2×C4—32no—
391C2×C4—42no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—62no—
395C2×C4—62no—
396C2×C4—62no—
397C2×C4—62no—
398C2×C4—62no—
399C2×C4—62no—
400C2×C4—122no—
401C2×C4—122no—
402C2×C4—122no—
403C2×C4C22×C4×D311yescharacteristic
404C2×C4C2×D3×D411yescharacteristic
405C2×C4C2×D3×D411yescharacteristic
406C2×C4C2×#23911yescharacteristic
407C2×C4C2×#23911yescharacteristic
408C2×C4D3×#4111yescharacteristic
409C8—22no—
410C8—32no—
411C8—32no—
412C8—42no—
413C8—62no—
414C8—122no—
415C8#89312yes—
416C8#89312yes—
417C2×C3—21no—
418C2×C3—21no—
419C2×C3C4×#12111yescharacteristic
420C2×C3#295512yes—
421D3—42no—
422D3—82no—
423C22—22no—
424C22—64no—
425C22—62no—
426C22—122no—
427C22—122no—
428C22—124no—
429C22—122no—
430C22C4×D3×D411yescharacteristic
431C22#562711yescharacteristic
432C22#569111yescharacteristic
433C4—42no—
434C4—44no—
435C4—62no—
436C4—122no—
437C4—122no—
438C4C2×#89311yescharacteristic
439C4C4×#23911yescharacteristic
440C4D3×#12111yescharacteristic
441C4#572511yescharacteristic
442C3#2154811yescharacteristic, complemented, Sylow
443C2—21no—
444C2—21no—
445C2—122no—
446C2—242no—
447C2#7596411yescharacteristic
448C2#7989712yes—
subgroupG/H
centerC2×C4C2×D3×D4
commutatorC2×C3×C4C23×C4
FrattiniC22×C4C23×D3
FittingC3×#1946C2
socleC22×C3C2×C4×D4
prime radical#79181C2
cosocleC22×C3×C4C24
hypercenter#1946D3
hypocenterC3#21548
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1946

Complements

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

NH ≅ G/Ntype
1. C2×#4895C2semidirect
3. #79073C2semidirect
4. C2×#4708C2semidirect
5. C3×#1946C2semidirect
6. #70712C2semidirect
7. #70659C2semidirect
8. #70536C2semidirect
9. #69845C2semidirect
10. #68757C2semidirect
11. #68739C2semidirect
34. #4886C4semidirect
35. #4867C4semidirect
36. #4865C4semidirect
48. #1946D3semidirect
101. C8×Dic3D4semidirect
442. C3#21548semidirect

Maximal subgroups

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

subgroupindex
C2×#48952
#791812
#790732
C2×#47082
C3×#19462
#707122
#706592
#705362
#698452
#687572
#687392
#215483