Finite Group Databasebeta

← #10690043

Subgroups — #10690043

Order 768.

578 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#83298C211yesmaximal, characteristic
2#79043C211yesmaximal, characteristic, complemented
3C3×#1492C211yesmaximal, characteristic, complemented
4#71818C211yesmaximal, characteristic, complemented
5C4×#821C211yesmaximal, characteristic, complemented
6#67715C211yesmaximal, characteristic, complemented
7#67656C211yesmaximal, characteristic, complemented
8#8991—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×D3×M16—21no—
11C2×C4×D12C2211yescharacteristic
12C3×#423C2211yescharacteristic
13C2×C3×C4×C8C2211yescharacteristic
14C2×#837—21no—
15C2×#837—21no—
16#5126—21no—
17C2×#821—21no—
18#5087C411yescharacteristic, complemented
19#5087C411yescharacteristic, complemented
20#5072C411yescharacteristic, complemented
21#5072C411yescharacteristic, complemented
22#5036—21no—
23#5033—21no—
24C4×#213—21no—
25C4×#213—21no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28C4×#208C2211yescharacteristic
29#4838—21no—
30#4836—22no—
31#4834—21no—
32C3×D4⋊C8—22no—
33#4551—21no—
34#3100—31no—
35D4×M16—31no—
36#1980—31no—
37C4×#102—31no—
38#1492D311yescharacteristic, complemented
39#1303—31no—
40#1278—31no—
41C2×C3×#41—21no—
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C2×C3×M16—21no—
45C22×C3×C8—21no—
46C3×C4⋊Q8C2×C411yescharacteristic
47C3×#127C2×C411yescharacteristic
48C3×#124—21no—
49C3×#121—21no—
50C3×#121—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#213—21no—
55C2×#213—21no—
56C2×(C2×D3)⋊C4—21no—
57C2×C4⋊Dic3—21no—
58C2×C4⋊Dic3D411yescharacteristic
59C2×C4×Dic3—21no—
60C2×#208D411yescharacteristic
61C2×#208D411yescharacteristic
62D3×M16—42no—
63D3×M16—42no—
64C2×ΓL(1,𝔽25)—22no—
65C2×D3×C8—22no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12C2×C411yescharacteristic
69C4×D12C2×C411yescharacteristic
70C4×D12D411yescharacteristic
71C4×D12D411yescharacteristic
72C3×C4⋊C8—21no—
73C3×C4⋊C8D412yes—
74C3×C4×C8—21no—
75C3×C4×C8—21no—
76#837—42no—
77#837—42no—
78#832—21no—
79#832—21no—
80C3⋊#95—21no—
81#830—21no—
82#821—24no—
83(C2×D3)⋊C8—22no—
84(C2×D3)⋊C8—22no—
85#809—21no—
86#809—21no—
87#807—21no—
88#807—21no—
89C4⋊(C3⋊C8)C2×C411yescharacteristic
90C4⋊(C3⋊C8)C2×C411yescharacteristic
91#803—21no—
92C4×C3⋊C8—21no—
93C4×C3⋊C8—21no—
94C22×M16—61no—
95#532C2×D311yescharacteristic
96C2×C4×D4—31no—
97#480—61no—
98#474—31no—
99#474—31no—
100#459—31no—
101#459—31no—
102#447—61no—
103C4×D8—61no—
104C4×D8—61no—
105#436—61no—
106#435—62no—
107C8×D4—61no—
108#431—31no—
109#423C2×D311yescharacteristic
110C2×#104—61no—
111C2×#104—61no—
112C2×#102—61no—
113#407—61no—
114C4×M16—31no—
115C2×C4×C8C2×D311yescharacteristic
116C4×C4⋊C4—31no—
117#340—61no—
118D4⋊C8—22no—
119C23×D3—21no—
120C3×#41—41no—
121C3×#41—41no—
122C3×#41—41no—
123C3×#41—41no—
124C2×C3×Q8—21no—
125C2×C3×D4—21no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—41no—
129C22×C3×C4—41no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4#4111yescharacteristic
133C22×Dic3—21no—
134C22×Dic3—41no—
135C2×D12—22no—
136C2×D12—21no—
137C2×D12—41no—
138C2×D12C22⋊C411yescharacteristic
139C2×D12C22⋊C411yescharacteristic
140C2×C4×D3—21no—
141C2×C4×D3—21no—
142C2×C4×D3—21no—
143C2×C4×D3—21no—
144C2×C4×D3—41no—
145C2×C4×D3—41no—
146C2×C4×D3—41no—
147C2×C4×D3—41no—
148C3×M16—41no—
149C3×M16—41no—
150C2×C3×C8—22no—
151C2×C3×C8—22no—
152C2×C3×C8—41no—
153C2×C3×C8—41no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—21no—
156C3×C22⋊C4—41no—
157C3×C22⋊C4—41no—
158C3×C42C22×C411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42C2×D411yescharacteristic
161C3×C42#4111yescharacteristic
162#213—42no—
163#213—42no—
164(C2×D3)⋊C4—21no—
165(C2×D3)⋊C4—21no—
166(C2×D3)⋊C4—21no—
167(C2×D3)⋊C4—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—22no—
170C4⋊Dic3—41no—
171C4⋊Dic3C22⋊C411yescharacteristic
172C4⋊Dic3C22⋊C411yescharacteristic
173C4×Dic3—41no—
174C4×Dic3—41no—
175#208—21no—
176#208—21no—
177#208—21no—
178#208D812yescomplemented
179C2×C3⋊C8—21no—
180C2×C3⋊C8—21no—
181C2×C3⋊C8—21no—
182C2×C3⋊C8—21no—
183ΓL(1,𝔽25)—44no—
184D3×C8—44no—
185C2×#41—21no—
186C22×D4—31no—
187C23×C4—61no—
188C2×D8—61no—
189C2×D8—61no—
190C2×M16—21no—
191C2×M16—31no—
192C2×M16—31no—
193C2×M16—62no—
194C2×M16—122no—
195C2×M16—122no—
196C22×C8—21no—
197C22×C8—62no—
198C4⋊Q8C4×D311yescharacteristic
199#127C4×D311yescharacteristic
200#124—21no—
201#121—21no—
202#121—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—31no—
206C4×D4—31no—
207C4×D4—31no—
208C4×D4—31no—
209C4×D4—61no—
210C4×D4—61no—
211C2×C4⋊C4—31no—
212C2×C4⋊C4—61no—
213C2×C22⋊C4—61no—
214C2×C42—61no—
215C2×C42C22×D311yescharacteristic
216#107—61no—
217#107—61no—
218C4⋊C8—21no—
219C4⋊C8—31no—
220C4⋊C8—31no—
221C4⋊C8#6712yes—
222#104—122no—
223#104—122no—
224#102—61no—
225#102—61no—
226#102—61no—
227#102—61no—
228#102—64no—
229C22⋊C8—62no—
230C22⋊C8—62no—
231C22⋊C8—61no—
232#97—61no—
233C4×C8—21no—
234C4×C8—21no—
235C4×C8—61no—
236C4×C8—61no—
237#95—61no—
238C23×C3—41no—
239C23×C3#12111yescharacteristic
240C22×D3—21no—
241C22×D3—21no—
242C22×D3—42no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—81no—
246C3×Q8—41no—
247C3×Q8—41no—
248C3×D4—42no—
249C3×D4—41no—
250C3×D4—41no—
251C3×D4—41no—
252C3×D4—41no—
253C3×D4—81no—
254C3×D4—81no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—81no—
260C2×C3×C4—81no—
261C2×C3×C4C2×C22⋊C411yescharacteristic
262C2×C3×C4C4×D411yescharacteristic
263C2×C3×C4C4×D411yescharacteristic
264C2×C3×C4#12011yescharacteristic
265C2×C3×C4#12111yescharacteristic
266C2×C3×C4#12311yescharacteristic
267C2×C3×C4C2×D811yescharacteristic
268C2×C3×C4#13511yescharacteristic
269C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
270C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
271C2×Dic3—21no—
272C2×Dic3—21no—
273C2×Dic3—41no—
274C2×Dic3—41no—
275C2×Dic3—42no—
276C2×Dic3—81no—
277C2×Dic3—81no—
278D12—21no—
279D12—21no—
280D12—21no—
281D12—42no—
282D12#10412yescomplemented
283C4×D3—41no—
284C4×D3—41no—
285C4×D3—41no—
286C4×D3—41no—
287C4×D3—41no—
288C4×D3—41no—
289C4×D3—41no—
290C4×D3—41no—
291C3×C8—42no—
292C3×C8—42no—
293C3⋊C8—22no—
294C3⋊C8—41no—
295C3⋊C8—41no—
296C3⋊C8—41no—
297C24—61no—
298#41—41no—
299#41—41no—
300#41—41no—
301#41—41no—
302C2×Q8—21no—
303C2×D4—21no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—31no—
307C2×D4—31no—
308C2×D4—41no—
309C2×D4—62no—
310C2×D4—61no—
311C2×D4—121no—
312C22×C4—41no—
313C22×C4—61no—
314C22×C4—61no—
315C22×C4—61no—
316C22×C4—61no—
317C22×C4—61no—
318C22×C4—121no—
319C22×C4—121no—
320C22×C4—121no—
321C22×C4—121no—
322C22×C4—121no—
323C22×C4#23511yescharacteristic
324C22×C4D3×D411yescharacteristic
325C22×C4D3×D411yescharacteristic
326D8—122no—
327D8—122no—
328M16—32no—
329M16—41no—
330M16—41no—
331M16—61no—
332M16—61no—
333M16—61no—
334M16—124no—
335C2×C8—22no—
336C2×C8—22no—
337C2×C8—41no—
338C2×C8—41no—
339C2×C8—61no—
340C2×C8—61no—
341C2×C8—61no—
342C2×C8—61no—
343C2×C8—124no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—31no—
347C4⋊C4—31no—
348C4⋊C4—62no—
349C4⋊C4—61no—
350C4⋊C4—121no—
351C22⋊C4—41no—
352C22⋊C4—41no—
353C22⋊C4—61no—
354C22⋊C4—61no—
355C22⋊C4—61no—
356C22⋊C4—61no—
357C42—121no—
358C42—121no—
359C42C2×C4×D311yescharacteristic
360C42D3×D411yescharacteristic
361C42#23711yescharacteristic
362C42C2×#6711yescharacteristic
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—81no—
366C22×C3—81no—
367C22×C3—81no—
368C22×C3#38611yescharacteristic
369C22×C3#45911yescharacteristic
370C22×C3#46311yescharacteristic
371C2×D3—22no—
372C2×D3—41no—
373C2×D3—41no—
374C2×D3—41no—
375C2×D3—41no—
376C2×D3—41no—
377C2×D3—82no—
378C2×D3—82no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—81no—
383C3×C4C2×#10411yescharacteristic
384C3×C4#42111yescharacteristic
385C3×C4C4×D811yescharacteristic
386C3×C4#44211yescharacteristic
387C3×C4#44711yescharacteristic
388C3×C4#44911yescharacteristic
389Dic3—41no—
390Dic3—41no—
391Dic3—82no—
392C23—41no—
393C23—61no—
394C23—61no—
395C23—122no—
396C23—121no—
397C23—121no—
398C23—241no—
399C23#88411yescharacteristic
400Q8—41no—
401Q8—41no—
402D4—32no—
403D4—42no—
404D4—41no—
405D4—41no—
406D4—41no—
407D4—41no—
408D4—61no—
409D4—61no—
410D4—61no—
411D4—81no—
412D4—81no—
413D4—122no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—61no—
419C2×C4—61no—
420C2×C4—81no—
421C2×C4—81no—
422C2×C4—121no—
423C2×C4—121no—
424C2×C4—121no—
425C2×C4—121no—
426C2×C4—121no—
427C2×C4—121no—
428C2×C4—121no—
429C2×C4—121no—
430C2×C4—121no—
431C2×C4—121no—
432C2×C4—122no—
433C2×C4—241no—
434C2×C4—241no—
435C2×C4D3×C22⋊C411yescharacteristic
436C2×C4#67⋊C411yescharacteristic
437C2×C4#88311yescharacteristic
438C2×C4D3×D811yescharacteristic
439C2×C4#91111yescharacteristic
440C2×C4#91411yescharacteristic
441C2×C4#91611yescharacteristic
442C2×C4C4×#6711yescharacteristic
443C2×C4#93711yescharacteristic
444C2×C4#93811yescharacteristic
445C8—42no—
446C8—42no—
447C8—62no—
448C8—121no—
449C8—121no—
450C8—121no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—161no—
454C2×C3#187111yescharacteristic
455C2×C3#187911yescharacteristic
456C2×C3#188111yescharacteristic
457D3—42no—
458D3—81no—
459C22—21no—
460C22—21no—
461C22—62no—
462C22—81no—
463C22—81no—
464C22—81no—
465C22—121no—
466C22—121no—
467C22—121no—
468C22—121no—
469C22—121no—
470C22—242no—
471C22—242no—
472C22#478611yescharacteristic
473C22#478711yescharacteristic
474C22#495611yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—81no—
479C4—121no—
480C4—121no—
481C4—242no—
482C4#475611yescharacteristic
483C4#478511yescharacteristic
484C4D3×#10411yescharacteristic
485C4#482111yescharacteristic
486C4#515611yescharacteristic
487C4#517211yescharacteristic
488C3#899111yescharacteristic, complemented, Sylow
489C2—21no—
490C2—21no—
491C2—122no—
492C2—161no—
493C2—241no—
494C2#7001211yescharacteristic
495C2#7020811yescharacteristic
496C2#7038111yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1492C2
socleC22×C3#386
prime radical#83298C2
cosocleC2×C3×C42C23
hypercenter#1492D3
hypocenterC3#8991
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. #79043C2semidirect
3. C3×#1492C2semidirect
4. #71818C2semidirect
5. C4×#821C2semidirect
6. #67715C2semidirect
7. #67656C2semidirect
18. #5087C4semidirect
19. #5087C4semidirect
20. #5072C4semidirect
21. #5072C4semidirect
38. #1492D3semidirect
178. #208D8semidirect
282. D12#104semidirect
488. C3#8991semidirect

Maximal subgroups

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

subgroupindex
#832982
#790432
C3×#14922
#718182
C4×#8212
#677152
#676562
#89913