Finite Group Databasebeta

← #10690050

Subgroups — #10690050

Order 768.

500 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83315C211yesmaximal, characteristic
2#79046C211yesmaximal, characteristic, complemented
3C3×#1492C211yesmaximal, characteristic
4#71818C211yesmaximal, characteristic
5C4×#816C211yesmaximal, characteristic, complemented
6#67715C211yesmaximal, characteristic, complemented
7#67656C211yesmaximal, characteristic, complemented
8#8992—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C4×Dic6C2211yescharacteristic
11C3×#423C2211yescharacteristic
12C2×C3×C4×C8C2211yescharacteristic
13C2×#837—21no—
14C2×#837—21no—
15#5118—21no—
16Dic3×M16—21no—
17C2×#816—21no—
18#5094C411yescharacteristic, complemented
19#5094C411yescharacteristic, complemented
20#5075C411yescharacteristic, complemented
21#5075C411yescharacteristic, complemented
22#5040—21no—
23#5034—21no—
24C4×#214—21no—
25C4×#214—21no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4831—21no—
30#4830—21no—
31#4828—22no—
32C3×D4⋊C8—22no—
33#4551—21no—
34#3103—31no—
35Q8×M16—31no—
36#1980—31no—
37C4×#103—31no—
38#1492D311yescharacteristic
39#1303—31no—
40#1278—31no—
41C2×C3×#41—21no—
42C22×Dic6D411yescharacteristic
43C2×C3×M16—21no—
44C22×C3×C8—21no—
45C3×C4⋊Q8C2×C411yescharacteristic
46C3×#127C2×C411yescharacteristic
47C3×#124—21no—
48C3×#121—21no—
49C3×#121—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#214—21no—
54C2×#214—21no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3D411yescharacteristic
57C2×Dic3⋊C4—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—21no—
60C2×#208D411yescharacteristic
61C2×#208D411yescharacteristic
62C4×Dic6—21no—
63C4×Dic6—21no—
64C4×Dic6C2×C411yescharacteristic
65C4×Dic6C2×C411yescharacteristic
66C4×Dic6D411yescharacteristic
67C4×Dic6D411yescharacteristic
68C3×C4⋊C8—21no—
69C3×C4⋊C8D412yes—
70C3×C4×C8—21no—
71C3×C4×C8—21no—
72#837—42no—
73#837—42no—
74#832—21no—
75#832—21no—
76C3⋊#95—21no—
77#830—21no—
78#816—24no—
79#815—22no—
80Dic3⋊C8—22no—
81Dic3⋊C8—22no—
82C8×Dic3—22no—
83#810—21no—
84#810—21no—
85#808—21no—
86#808—21no—
87C4⋊(C3⋊C8)C2×C411yescharacteristic
88C4⋊(C3⋊C8)C2×C411yescharacteristic
89#803—21no—
90C4×C3⋊C8—21no—
91C4×C3⋊C8—21no—
92#532C2×D311yescharacteristic
93C2×C4×Q8—31no—
94#481—61no—
95#475—31no—
96#475—31no—
97#460—31no—
98#460—31no—
99#448—61no—
100#446—61no—
101#446—62no—
102C8×Q8—61no—
103C4×SD16—61no—
104C4×SD16—61no—
105#431—31no—
106#423—61no—
107#423C2×D311yescharacteristic
108C2×#104—61no—
109C2×#104—61no—
110C2×#103—61no—
111C4×M16—31no—
112C4×M16—61no—
113C2×C4×C8C2×D311yescharacteristic
114C4×C4⋊C4—31no—
115#340—61no—
116D4⋊C8—22no—
117C3×#41—41no—
118C3×#41—41no—
119C3×#41—41no—
120C3×#41—41no—
121C2×C3×Q8—21no—
122C2×C3×D4—21no—
123C2×C3×D4—21no—
124C2×C3×D4—21no—
125C2×C3×D4—41no—
126C22×C3×C4—41no—
127C22×C3×C4C2×D411yescharacteristic
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C22×Dic3—21no—
131C22×Dic3—21no—
132C22×Dic3—41no—
133C2×Dic6—22no—
134C2×Dic6—21no—
135C2×Dic6—41no—
136C2×Dic6C22⋊C411yescharacteristic
137C2×Dic6C22⋊C411yescharacteristic
138C3×M16—41no—
139C3×M16—41no—
140C2×C3×C8—22no—
141C2×C3×C8—22no—
142C2×C3×C8—41no—
143C2×C3×C8—41no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42C22×C411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152#214—42no—
153#214—42no—
154C4⋊Dic3—21no—
155C4⋊Dic3—22no—
156C4⋊Dic3—41no—
157C4⋊Dic3C22⋊C411yescharacteristic
158C4⋊Dic3C22⋊C411yescharacteristic
159Dic3⋊C4—21no—
160Dic3⋊C4—21no—
161Dic3⋊C4—21no—
162Dic3⋊C4—21no—
163C4×Dic3—21no—
164C4×Dic3—21no—
165C4×Dic3—21no—
166C4×Dic3—21no—
167C4×Dic3—41no—
168C4×Dic3—41no—
169#208—21no—
170#208—21no—
171#208—21no—
172#208SD1612yescomplemented
173C2×C3⋊C8—21no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—21no—
177C2×#41—21no—
178C22×Q8—31no—
179C2×SD16—61no—
180C2×SD16—61no—
181C2×M16—21no—
182C2×M16—31no—
183C2×M16—31no—
184C22×C8—21no—
185C4⋊Q8C4×D311yescharacteristic
186#127C4×D311yescharacteristic
187#124—21no—
188#121—21no—
189#121—21no—
190C4×Q8—31no—
191C4×Q8—31no—
192C4×Q8—31no—
193C4×Q8—31no—
194C4×Q8—61no—
195C4×Q8—61no—
196C4×D4—21no—
197C4×D4—21no—
198C2×C4⋊C4—31no—
199C2×C4⋊C4—61no—
200C2×C4⋊C4—61no—
201C2×C42—61no—
202C2×C42—61no—
203C2×C42C22×D311yescharacteristic
204#106—61no—
205#106—61no—
206C4⋊C8—21no—
207C4⋊C8—31no—
208C4⋊C8—31no—
209C4⋊C8—62no—
210C4⋊C8—62no—
211C4⋊C8#6712yes—
212#104—122no—
213#104—122no—
214#103—61no—
215#103—61no—
216#103—64no—
217#102—61no—
218#102—61no—
219C22⋊C8—61no—
220#97—62no—
221#97—61no—
222C4×C8—21no—
223C4×C8—21no—
224C4×C8—62no—
225C4×C8—61no—
226C4×C8—61no—
227#95—61no—
228C23×C3—41no—
229C23×C3#12111yescharacteristic
230C3×Q8—41no—
231C3×Q8—41no—
232C3×D4—42no—
233C3×D4—41no—
234C3×D4—41no—
235C3×D4—41no—
236C3×D4—41no—
237C3×D4—81no—
238C3×D4—81no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—41no—
242C2×C3×C4—41no—
243C2×C3×C4—81no—
244C2×C3×C4—81no—
245C2×C3×C4C2×C22⋊C411yescharacteristic
246C2×C3×C4C4×D411yescharacteristic
247C2×C3×C4C4×D411yescharacteristic
248C2×C3×C4#12011yescharacteristic
249C2×C3×C4#12111yescharacteristic
250C2×C3×C4#12311yescharacteristic
251C2×C3×C4C2×SD1611yescharacteristic
252C2×C3×C4#13511yescharacteristic
253C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
254C2×C3×C4#13711yescharacteristic
255C2×Dic3—21no—
256C2×Dic3—21no—
257C2×Dic3—21no—
258C2×Dic3—21no—
259C2×Dic3—42no—
260C2×Dic3—42no—
261C2×Dic3—81no—
262C2×Dic3—81no—
263C2×Dic3—81no—
264Dic6—21no—
265Dic6—21no—
266Dic6—21no—
267Dic6—42no—
268Dic6#10412yescomplemented
269C3×C8—42no—
270C3×C8—42no—
271C3⋊C8—22no—
272C3⋊C8—41no—
273C3⋊C8—41no—
274C3⋊C8—41no—
275#41—41no—
276#41—41no—
277#41—41no—
278#41—41no—
279C2×Q8—21no—
280C2×Q8—31no—
281C2×Q8—31no—
282C2×Q8—62no—
283C2×Q8—61no—
284C2×Q8—121no—
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C22×C4—41no—
290C22×C4—61no—
291C22×C4—61no—
292C22×C4—121no—
293C22×C4#23511yescharacteristic
294C22×C4D3×D411yescharacteristic
295C22×C4D3×D411yescharacteristic
296SD16—122no—
297SD16—122no—
298M16—32no—
299M16—41no—
300M16—41no—
301M16—61no—
302M16—61no—
303M16—61no—
304C2×C8—22no—
305C2×C8—22no—
306C2×C8—41no—
307C2×C8—41no—
308C2×C8—61no—
309C2×C8—61no—
310C2×C8—61no—
311C2×C8—61no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—31no—
315C4⋊C4—31no—
316C4⋊C4—62no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—61no—
320C4⋊C4—61no—
321C4⋊C4—61no—
322C4⋊C4—121no—
323C22⋊C4—41no—
324C22⋊C4—41no—
325C42—61no—
326C42—61no—
327C42—61no—
328C42—61no—
329C42—121no—
330C42—121no—
331C42C2×C4×D311yescharacteristic
332C42D3×D411yescharacteristic
333C42#23711yescharacteristic
334C42C2×#6711yescharacteristic
335C22×C3—21no—
336C22×C3—21no—
337C22×C3—81no—
338C22×C3—81no—
339C22×C3—81no—
340C22×C3#38611yescharacteristic
341C22×C3#46011yescharacteristic
342C22×C3#46411yescharacteristic
343C3×C4—21no—
344C3×C4—21no—
345C3×C4—21no—
346C3×C4—81no—
347C3×C4C2×#10411yescharacteristic
348C3×C4#42111yescharacteristic
349C3×C4C4×SD1611yescharacteristic
350C3×C4#44011yescharacteristic
351C3×C4#44811yescharacteristic
352C3×C4#44911yescharacteristic
353Dic3—22no—
354Dic3—41no—
355Dic3—41no—
356Dic3—41no—
357Dic3—82no—
358C23—41no—
359C23#88411yescharacteristic
360Q8—32no—
361Q8—41no—
362Q8—41no—
363Q8—61no—
364Q8—61no—
365Q8—61no—
366Q8—122no—
367D4—42no—
368D4—41no—
369D4—41no—
370D4—41no—
371D4—41no—
372D4—81no—
373D4—81no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—81no—
383C2×C4—81no—
384C2×C4—122no—
385C2×C4—122no—
386C2×C4—241no—
387C2×C4—241no—
388C2×C4—241no—
389C2×C4D3×C22⋊C411yescharacteristic
390C2×C4#67⋊C411yescharacteristic
391C2×C4#88311yescharacteristic
392C2×C4#91111yescharacteristic
393C2×C4#91211yescharacteristic
394C2×C4D3×SD1611yescharacteristic
395C2×C4#91511yescharacteristic
396C2×C4C4×#6711yescharacteristic
397C2×C4#93711yescharacteristic
398C2×C4#93811yescharacteristic
399C8—42no—
400C8—42no—
401C8—62no—
402C8—121no—
403C8—121no—
404C8—121no—
405C2×C3—21no—
406C2×C3—21no—
407C2×C3—161no—
408C2×C3#187111yescharacteristic
409C2×C3#188011yescharacteristic
410C2×C3#188111yescharacteristic
411C22—21no—
412C22—21no—
413C22—81no—
414C22—81no—
415C22—81no—
416C22#476211yescharacteristic
417C22#476711yescharacteristic
418C22#495611yescharacteristic
419C4—21no—
420C4—21no—
421C4—21no—
422C4—62no—
423C4—81no—
424C4—121no—
425C4—121no—
426C4—121no—
427C4—242no—
428C4#475711yescharacteristic
429C4#475811yescharacteristic
430C4D3×#10411yescharacteristic
431C4#482111yescharacteristic
432C4#515811yescharacteristic
433C4#517011yescharacteristic
434C3#899211yescharacteristic, complemented, Sylow
435C2—21no—
436C2—21no—
437C2—161no—
438C2#7001211yescharacteristic
439C2#7021311yescharacteristic
440C2#7038111yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1492C2
socleC22×C3#386
prime radicalC3×#532C22
cosocleC2×C3×C42C23
hypercenter#1492D3
hypocenterC3#8992
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1492

Complements

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

NH ≅ G/Ntype
2. #79046C2semidirect
5. C4×#816C2semidirect
6. #67715C2semidirect
7. #67656C2semidirect
18. #5094C4semidirect
19. #5094C4semidirect
20. #5075C4semidirect
21. #5075C4semidirect
172. #208SD16semidirect
268. Dic6#104semidirect
434. C3#8992semidirect

Maximal subgroups

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

subgroupindex
#833152
#790462
C3×#14922
#718182
C4×#8162
#677152
#676562
#89923