Finite Group Databasebeta

← #10690075

Subgroups — #10690075

Order 768.

530 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#79049C211yesmaximal, characteristic, complemented
2#79021C211yesmaximal, characteristic
3#70758C211yesmaximal, characteristic, complemented
4#70754C211yesmaximal, characteristic, complemented
5#70498C211yesmaximal, characteristic, complemented
6C3×#1301C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#8995—31nomaximal, Sylow
9C2×#907—21no—
10C2×#904—21no—
11#5477C2211yescharacteristic
12C4×#235C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C3×#423C2211yescharacteristic
15C2×C3×C4⋊C8C2211yescharacteristic
16#5119—21no—
17#5113—21no—
18C2×C4⋊(C3⋊C8)C2211yescharacteristic
19C2×C4×C3⋊C8C2211yescharacteristic
20#4849C411yescharacteristic
21#4848C411yescharacteristic
22#4845—21no—
23#4844—21no—
24#4843C411yescharacteristic
25#4841C411yescharacteristic
26#4839—22no—
27#4837—21no—
28D3×C4⋊C8—21no—
29#4484—21no—
30#4483—21no—
31#4483—22no—
32#4459—21no—
33#4459—21no—
34#3035—31no—
35#2906—31no—
36#1492—31no—
37#1480—31no—
38#1476—31no—
39#1301D311yescharacteristic, complemented
40#1277—31no—
41C2×#235—21no—
42C2×#235D411yescharacteristic
43C2×C3×M16—21no—
44C22×C3×C8—21no—
45C2×C3×C42C2311yescharacteristic
46#930—21no—
47#930—21no—
48C4×#67—21no—
49C4×#67—21no—
50#926D411yescharacteristic
51C22×C3⋊C8—21no—
52C22×C3⋊C8—21no—
53#907—42no—
54#907—42no—
55C2×Dic12—21no—
56#904—42no—
57#904—42no—
58C2×D24—21no—
59C2×#204—21no—
60C2×#204—21no—
61C2×ΓL(1,𝔽25)—22no—
62C2×D3×C8—22no—
63#875—21no—
64#874C2×C411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67C4×D12C2×C411yescharacteristic
68C4×D12D411yescharacteristic
69#872—21no—
70C42×D3—21no—
71#869C2×C411yescharacteristic
72C4×Dic6C2×C411yescharacteristic
73C4×Dic6D411yescharacteristic
74C3×C4⋊C8—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8—21no—
77C3×C4⋊C8C2×C411yescharacteristic
78C3×C4⋊C8C2×C411yescharacteristic
79C3×C4⋊C8D412yes—
80C3×C22⋊C8—21no—
81C3×C4×C8—21no—
82#821—21no—
83#821—21no—
84#816—21no—
85#816—21no—
86Dic3⋊C8—22no—
87Dic3⋊C8—22no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)D411yescharacteristic
90C4⋊(C3⋊C8)D411yescharacteristic
91C4×C3⋊C8—21no—
92C4×C3⋊C8—21no—
93C2×#135—61no—
94C2×#131—61no—
95#532—31no—
96C4×#41—31no—
97#464—61no—
98#463—61no—
99#462—31no—
100#461—31no—
101#460—31no—
102#459—31no—
103#431C2×D311yescharacteristic
104#424—61no—
105#424—62no—
106#423—61no—
107#423C2×D311yescharacteristic
108C2×C4⋊C8—31no—
109C2×C4⋊C8—61no—
110C2×C4⋊C8C2×D311yescharacteristic
111#416—61no—
112C2×C4×C8—31no—
113Q8⋊C8—61no—
114D4⋊C8—61no—
115D4⋊C8—61no—
116D4⋊C8—61no—
117D4⋊C8—62no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4#4111yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×#67—21no—
122C2×#67—41no—
123#235—21no—
124#235—21no—
125#235—41no—
126#235—41no—
127#235—41no—
128#235—41no—
129#235D812yes—
130#235SD1612yes—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12C22⋊C411yescharacteristic
135C2×C4×D3—21no—
136C2×C4×D3—21no—
137C2×C4×D3—21no—
138C2×C4×D3—41no—
139C2×Dic6—21no—
140C2×Dic6C22⋊C411yescharacteristic
141C3×M16—41no—
142C3×M16—41no—
143C2×C3×C8—21no—
144C2×C3×C8—21no—
145C2×C3×C8—22no—
146C2×C3×C8—41no—
147C2×C3×C8—41no—
148C2×C3×C8—41no—
149C3×C42C22×C411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153#217—21no—
154(C2×D3)⋊C4—21no—
155(C2×D3)⋊C4—21no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—41no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3C22⋊C411yescharacteristic
161C4⋊Dic3C22⋊C411yescharacteristic
162Dic3⋊C4—21no—
163Dic3⋊C4—21no—
164C4×Dic3—21no—
165C4×Dic3—21no—
166C4×Dic3—21no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—21no—
169C2×C3⋊C8—21no—
170C2×C3⋊C8—41no—
171C2×C3⋊C8—41no—
172C2×C3⋊C8—41no—
173C2×C3⋊C8—41no—
174Dic12—42no—
175D24—42no—
176#204—42no—
177#204—42no—
178ΓL(1,𝔽25)—44no—
179D3×C8—44no—
180C2×#41—31no—
181C2×#41—61no—
182#135—122no—
183#135—122no—
184C2×Q16—61no—
185C2×SD16—61no—
186C2×SD16—61no—
187C2×D8—61no—
188#131—122no—
189#131—122no—
190C2×M16—21no—
191C2×M16—62no—
192C22×C8—21no—
193C22×C8—62no—
194C22×C8—61no—
195C22×C8—61no—
196C4⋊Q8—31no—
197#127—31no—
198#124—61no—
199#121—61no—
200#121—61no—
201C4×Q8—31no—
202C4×Q8—31no—
203C4×D4—31no—
204C4×D4—31no—
205C4×D4—61no—
206C4×D4—61no—
207C4×D4—61no—
208C4×D4—61no—
209#117—31no—
210#117—61no—
211C2×C42—61no—
212C2×C42C22×D311yescharacteristic
213C4⋊C8—21no—
214C4⋊C8—21no—
215C4⋊C8—21no—
216C4⋊C8—31no—
217C4⋊C8—31no—
218C4⋊C8—62no—
219C4⋊C8—62no—
220C4⋊C8—61no—
221C4⋊C8C4×D311yescharacteristic
222C4⋊C8C4×D311yescharacteristic
223C4⋊C8D1212yes—
224#103—61no—
225#103—61no—
226#102—61no—
227#102—61no—
228C22⋊C8—21no—
229C4×C8—21no—
230C4×C8—61no—
231C4×C8—61no—
232C23×C3#12311yescharacteristic
233C22×D3—21no—
234C22×D3—41no—
235C2×C3×C4—21no—
236C2×C3×C4—21no—
237C2×C3×C4—41no—
238C2×C3×C4C2×C22⋊C411yescharacteristic
239C2×C3×C4C4×D411yescharacteristic
240C2×C3×C4C4×D411yescharacteristic
241C2×C3×C4#12011yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12111yescharacteristic
244C2×C3×C4C2×D811yescharacteristic
245C2×C3×C4C2×SD1611yescharacteristic
246C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
247C2×C3×C4#13711yescharacteristic
248#67—22no—
249#67—22no—
250#67—81no—
251#67—81no—
252C2×Dic3—21no—
253C2×Dic3—21no—
254C2×Dic3—21no—
255C2×Dic3—41no—
256D12—21no—
257D12—42no—
258D12—41no—
259D12—41no—
260D12—41no—
261D12—41no—
262D12#10212yes—
263C4×D3—22no—
264C4×D3—22no—
265C4×D3—41no—
266C4×D3—41no—
267C4×D3—81no—
268C4×D3—81no—
269C4×D3—81no—
270Dic6—21no—
271Dic6—41no—
272Dic6—41no—
273Dic6#10212yes—
274C3×C8—41no—
275C3×C8—41no—
276C3×C8—42no—
277C3×C8—81no—
278C3⋊C8—42no—
279C3⋊C8—41no—
280C3⋊C8—41no—
281#41—32no—
282#41—32no—
283#41—61no—
284#41—61no—
285#41—121no—
286#41—121no—
287#41—121no—
288#41—121no—
289C2×Q8—31no—
290C2×Q8—61no—
291C2×D4—31no—
292C2×D4—61no—
293C2×D4—61no—
294C2×D4—61no—
295C2×D4—61no—
296C2×D4—121no—
297C22×C4—61no—
298C22×C4—61no—
299C22×C4—61no—
300C22×C4—121no—
301C22×C4#23511yescharacteristic
302C22×C4D3×D411yescharacteristic
303C22×C4#23911yescharacteristic
304Q16—122no—
305SD16—122no—
306SD16—122no—
307D8—122no—
308M16—41no—
309M16—41no—
310M16—124no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—22no—
314C2×C8—41no—
315C2×C8—41no—
316C2×C8—41no—
317C2×C8—62no—
318C2×C8—61no—
319C2×C8—61no—
320C2×C8—124no—
321C2×C8—121no—
322C2×C8—121no—
323C2×C8—121no—
324C2×C8—121no—
325C4⋊C4—31no—
326C4⋊C4—31no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C22⋊C4—61no—
332C22⋊C4—61no—
333C22⋊C4—61no—
334C22⋊C4—121no—
335C22⋊C4—121no—
336C42—61no—
337C42—61no—
338C42—61no—
339C42C2×C4×D311yescharacteristic
340C42C2×D1211yescharacteristic
341C42D3×D411yescharacteristic
342C42D3×D411yescharacteristic
343C22×C3—21no—
344C22×C3—21no—
345C22×C3#38611yescharacteristic
346C22×C3#48011yescharacteristic
347C22×C3#48111yescharacteristic
348C2×D3—22no—
349C2×D3—41no—
350C2×D3—41no—
351C2×D3—81no—
352C2×D3—81no—
353C2×D3—81no—
354C3×C4—21no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4C2×#10211yescharacteristic
358C3×C4#41711yescharacteristic
359C3×C4#44311yescharacteristic
360C3×C4#44411yescharacteristic
361C3×C4#44711yescharacteristic
362C3×C4#44811yescharacteristic
363Dic3—22no—
364Dic3—41no—
365Dic3—41no—
366Dic3—81no—
367C23—61no—
368C23—121no—
369C23#89411yescharacteristic
370Q8—32no—
371Q8—61no—
372Q8—121no—
373Q8—121no—
374D4—32no—
375D4—62no—
376D4—62no—
377D4—61no—
378D4—122no—
379D4—121no—
380D4—121no—
381D4—121no—
382D4—121no—
383D4—241no—
384D4—241no—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—41no—
388C2×C4—61no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—61no—
392C2×C4—61no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—241no—
397C2×C4—241no—
398C2×C4—241no—
399C2×C4C4×D1211yescharacteristic
400C2×C4D3×C22⋊C411yescharacteristic
401C2×C4#88211yescharacteristic
402C2×C4#88411yescharacteristic
403C2×C4#89311yescharacteristic
404C2×C4#89511yescharacteristic
405C2×C4D3×D811yescharacteristic
406C2×C4D3×SD1611yescharacteristic
407C2×C4#91411yescharacteristic
408C2×C4#91811yescharacteristic
409C8—41no—
410C8—41no—
411C8—42no—
412C8—81no—
413C8—122no—
414C8—121no—
415C8—121no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3#187411yescharacteristic
419C2×C3#187611yescharacteristic
420C2×C3#188111yescharacteristic
421D3—42no—
422D3—161no—
423C22—21no—
424C22—21no—
425C22—62no—
426C22—121no—
427C22—121no—
428C22—241no—
429C22—241no—
430C22—241no—
431C22#466511yescharacteristic
432C22#486311yescharacteristic
433C22#488211yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—62no—
438C4—121no—
439C4—121no—
440C4—241no—
441C4D3×#10211yescharacteristic
442C4#477311yescharacteristic
443C4#480311yescharacteristic
444C4#480611yescharacteristic
445C4#489411yescharacteristic
446C4#489511yescharacteristic
447C3#899511yescharacteristic, complemented, Sylow
448C2—21no—
449C2—21no—
450C2—122no—
451C2—481no—
452C2#6888011yescharacteristic
453C2#6894611yescharacteristic
454C2#6903511yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1301C2
socleC22×C3#386
prime radical#79021C2
cosocleC2×C3×C42C23
hypercenter#1301D3
hypocenterC3#8995
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1301

Complements

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

NH ≅ G/Ntype
1. #79049C2semidirect
3. #70758C2semidirect
4. #70754C2semidirect
5. #70498C2semidirect
6. C3×#1301C2semidirect
7. #67703C2semidirect
39. #1301D3semidirect
447. C3#8995semidirect

Maximal subgroups

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

subgroupindex
#790492
#790212
#707582
#707542
#704982
C3×#13012
#677032
#89953