Finite Group Databasebeta

← #10690047

Subgroups — #10690047

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#79297C211yesmaximal, characteristic
2#79043C211yesmaximal, characteristic, complemented
3#70498C211yesmaximal, characteristic, complemented
4#69492C211yesmaximal, characteristic, complemented
5C4×#809C211yesmaximal, characteristic, complemented
6C3×#1303C211yesmaximal, characteristic, complemented
7#67739C211yesmaximal, characteristic, complemented
8#8991—31nomaximal, Sylow
9C2×D3×M16—21no—
10#5477C2211yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5126—21no—
16C2×#809—21no—
17C2×#805—21no—
18C2×#805—21no—
19C4×#208C2211yescharacteristic
20C2×C4×C3⋊C8C2211yescharacteristic
21#4846C411yescharacteristic, complemented
22#4846C411yescharacteristic, complemented
23#4843C411yescharacteristic, complemented
24#4843C411yescharacteristic, complemented
25#4838—21no—
26#4836—22no—
27#4834—21no—
28#4736—21no—
29#4732—21no—
30C4×D24—21no—
31C4×D24—21no—
32#4529—21no—
33#4483—22no—
34#3100—31no—
35D4×M16—31no—
36#1980—31no—
37C4×#102—31no—
38#1492—31no—
39#1303D311yescharacteristic, complemented
40#1278—31no—
41C2×#235—21no—
42C22×D12D411yescharacteristic
43C22×C4×D3—21no—
44C2×C3×M16—21no—
45C2×C3×C4⋊C4—21no—
46C2×C3×C42—21no—
47C2×C3×C42C2311yescharacteristic
48#930—21no—
49#930—21no—
50C2×(C2×D3)⋊C4—21no—
51C2×C4⋊Dic3D411yescharacteristic
52C2×#208D411yescharacteristic
53C2×#208D411yescharacteristic
54C22×C3⋊C8—21no—
55D3×M16—42no—
56D3×M16—42no—
57C2×D24—21no—
58C2×D24—21no—
59C2×ΓL(1,𝔽25)—22no—
60C2×D3×C8—22no—
61#875—21no—
62#874C2×C411yescharacteristic
63C4×D12—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12C2×C411yescharacteristic
68C4×D12C2×C411yescharacteristic
69C4×D12D411yescharacteristic
70C4×D12D411yescharacteristic
71#869C2×C411yescharacteristic
72C3×C4⋊C8—21no—
73C3×C4⋊C8C2×C411yescharacteristic
74C3×C4⋊C8C2×C411yescharacteristic
75C3×C4⋊C8D412yes—
76C3×C22⋊C8—21no—
77C3×C4×C8—21no—
78C3×#95—21no—
79#821—21no—
80#821—21no—
81#821—21no—
82#821—21no—
83(C2×D3)⋊C8—22no—
84(C2×D3)⋊C8—22no—
85#818—21no—
86#818—21no—
87#809—24no—
88#805—42no—
89#805—42no—
90#803—21no—
91C4×C3⋊C8—21no—
92C4×C3⋊C8—21no—
93C4×C3⋊C8—21no—
94C22×M16—61no—
95#532—31no—
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#431C2×D311yescharacteristic
109#423C2×D311yescharacteristic
110C2×#104—61no—
111C2×#104—61no—
112C2×#102—61no—
113#407—61no—
114C4×M16—31no—
115C2×C4×C8—31no—
116C4×C4⋊C4C2×D311yescharacteristic
117#340—61no—
118D4⋊C8—62no—
119C23×D3—21no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4#4111yescharacteristic
124C2×#67—41no—
125C22×Dic3—21no—
126#235—41no—
127#235—41no—
128#235—41no—
129#235—41no—
130C2×D12—22no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—41no—
136C2×D12C22⋊C411yescharacteristic
137C2×D12C22⋊C411yescharacteristic
138C2×C4×D3—21no—
139C2×C4×D3—21no—
140C2×C4×D3—21no—
141C2×C4×D3—21no—
142C2×C4×D3—41no—
143C2×C4×D3—41no—
144C2×C4×D3—41no—
145C2×C4×D3—41no—
146C2×C4×D3—41no—
147C2×Dic6—21no—
148C3×M16—41no—
149C3×M16—41no—
150C2×C3×C8—21no—
151C2×C3×C8—21no—
152C2×C3×C8—22no—
153C3×C4⋊C4—22no—
154C3×C4⋊C4—41no—
155C3×C42—41no—
156C3×C42—41no—
157C3×C42C22×C411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42C2×D411yescharacteristic
160C3×C42#4111yescharacteristic
161(C2×D3)⋊C4—21no—
162(C2×D3)⋊C4—21no—
163(C2×D3)⋊C4—21no—
164(C2×D3)⋊C4—21no—
165(C2×D3)⋊C4—41no—
166(C2×D3)⋊C4—41no—
167C4⋊Dic3—21no—
168C4⋊Dic3—21no—
169C4⋊Dic3—21no—
170C4⋊Dic3C22⋊C411yescharacteristic
171C4⋊Dic3C22⋊C411yescharacteristic
172#208—21no—
173#208—21no—
174#208—21no—
175#208D812yescomplemented
176C2×C3⋊C8—21no—
177C2×C3⋊C8—22no—
178C2×C3⋊C8—21no—
179C2×C3⋊C8—41no—
180C2×C3⋊C8—41no—
181D24—42no—
182D24—42no—
183ΓL(1,𝔽25)—44no—
184D3×C8—44no—
185C2×#41—61no—
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—62no—
197C22×C8—61no—
198C4⋊Q8—31no—
199#127—31no—
200#124—61no—
201#121—61no—
202#121—61no—
203C4×D4—31no—
204C4×D4—31no—
205C4×D4—31no—
206C4×D4—31no—
207C4×D4—61no—
208C4×D4—61no—
209C4×D4—61no—
210C4×D4—61no—
211C2×C4⋊C4—21no—
212C2×C4⋊C4—31no—
213C2×C22⋊C4—61no—
214C2×C42—21no—
215C2×C42C22×D311yescharacteristic
216#107—61no—
217#107—61no—
218C4⋊C8—21no—
219C4⋊C8C4×D311yescharacteristic
220C4⋊C8C4×D311yescharacteristic
221C4⋊C8D1212yes—
222#104—122no—
223#104—122no—
224#102—64no—
225#102—61no—
226#102—61no—
227#102—61no—
228#102—61no—
229C22⋊C8—21no—
230C22⋊C8—62no—
231C22⋊C8—62no—
232#97—61no—
233C4×C8—21no—
234C4×C8—61no—
235C4×C8—61no—
236C4×C8—61no—
237#95—21no—
238C23×C3#12111yescharacteristic
239C22×D3—21no—
240C22×D3—21no—
241C22×D3—42no—
242C22×D3—41no—
243C22×D3—41no—
244C22×D3—41no—
245C22×D3—81no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—41no—
249C2×C3×C4—42no—
250C2×C3×C4—81no—
251C2×C3×C4—81no—
252C2×C3×C4C2×C22⋊C411yescharacteristic
253C2×C3×C4C4×D411yescharacteristic
254C2×C3×C4C4×D411yescharacteristic
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12311yescharacteristic
258C2×C3×C4C2×D811yescharacteristic
259C2×C3×C4#13511yescharacteristic
260C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
261C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
262#67—81no—
263#67—81no—
264C2×Dic3—21no—
265C2×Dic3—21no—
266C2×Dic3—41no—
267C2×Dic3—41no—
268C2×Dic3—41no—
269D12—21no—
270D12—21no—
271D12—21no—
272D12—42no—
273D12—42no—
274D12—41no—
275D12—41no—
276D12—41no—
277D12—41no—
278D12#10412yescomplemented
279C4×D3—41no—
280C4×D3—41no—
281C4×D3—41no—
282C4×D3—41no—
283C4×D3—41no—
284C4×D3—41no—
285C4×D3—41no—
286C4×D3—41no—
287C4×D3—81no—
288C4×D3—81no—
289Dic6—41no—
290Dic6—41no—
291C3×C8—41no—
292C3×C8—41no—
293C3×C8—42no—
294C3⋊C8—22no—
295C3⋊C8—42no—
296C3⋊C8—41no—
297C24—61no—
298#41—121no—
299#41—121no—
300#41—121no—
301#41—121no—
302C2×Q8—61no—
303C2×D4—31no—
304C2×D4—31no—
305C2×D4—62no—
306C2×D4—61no—
307C2×D4—61no—
308C2×D4—61no—
309C2×D4—61no—
310C2×D4—121no—
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—21no—
336C2×C8—21no—
337C2×C8—22no—
338C2×C8—61no—
339C2×C8—62no—
340C2×C8—61no—
341C2×C8—124no—
342C2×C8—121no—
343C2×C8—121no—
344C4⋊C4—22no—
345C4⋊C4—31no—
346C4⋊C4—31no—
347C4⋊C4—41no—
348C4⋊C4—61no—
349C4⋊C4—61no—
350C4⋊C4—61no—
351C22⋊C4—61no—
352C22⋊C4—61no—
353C22⋊C4—61no—
354C22⋊C4—61no—
355C22⋊C4—121no—
356C22⋊C4—121no—
357C42—41no—
358C42—41no—
359C42C2×C4×D311yescharacteristic
360C42C2×D1211yescharacteristic
361C42D3×D411yescharacteristic
362C42#23911yescharacteristic
363C22×C3—21no—
364C22×C3—21no—
365C22×C3#38611yescharacteristic
366C22×C3#45911yescharacteristic
367C22×C3#46311yescharacteristic
368C2×D3—22no—
369C2×D3—41no—
370C2×D3—41no—
371C2×D3—41no—
372C2×D3—41no—
373C2×D3—41no—
374C2×D3—82no—
375C2×D3—82no—
376C2×D3—81no—
377C2×D3—81no—
378C2×D3—81no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—82no—
383C3×C4C2×#10411yescharacteristic
384C3×C4#42111yescharacteristic
385C3×C4C4×D811yescharacteristic
386C3×C4#44211yescharacteristic
387C3×C4#44711yescharacteristic
388C3×C4#44911yescharacteristic
389Dic3—41no—
390Dic3—41no—
391Dic3—81no—
392C23—61no—
393C23—61no—
394C23—122no—
395C23—121no—
396C23—121no—
397C23—121no—
398C23—241no—
399C23#88411yescharacteristic
400Q8—121no—
401Q8—121no—
402D4—32no—
403D4—61no—
404D4—61no—
405D4—61no—
406D4—122no—
407D4—122no—
408D4—121no—
409D4—121no—
410D4—121no—
411D4—121no—
412D4—241no—
413D4—241no—
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—41no—
417C2×C4—42no—
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—121no—
433C2×C4—241no—
434C2×C4—241no—
435C2×C4C4×D1211yescharacteristic
436C2×C4D3×C22⋊C411yescharacteristic
437C2×C4#88211yescharacteristic
438C2×C4#89311yescharacteristic
439C2×C4#89411yescharacteristic
440C2×C4#89511yescharacteristic
441C2×C4D3×D811yescharacteristic
442C2×C4#91411yescharacteristic
443C2×C4#91411yescharacteristic
444C2×C4#91911yescharacteristic
445C8—42no—
446C8—41no—
447C8—41no—
448C8—62no—
449C8—122no—
450C8—121no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3#187111yescharacteristic
454C2×C3#187911yescharacteristic
455C2×C3#188111yescharacteristic
456D3—42no—
457D3—81no—
458D3—161no—
459C22—21no—
460C22—21no—
461C22—62no—
462C22—121no—
463C22—121no—
464C22—121no—
465C22—121no—
466C22—121no—
467C22—242no—
468C22—242no—
469C22—241no—
470C22—241no—
471C22—241no—
472C22#466511yescharacteristic
473C22#478611yescharacteristic
474C22#481911yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—82no—
479C4—121no—
480C4—121no—
481C4—241no—
482C4#477311yescharacteristic
483C4#481011yescharacteristic
484C4D3×#10411yescharacteristic
485C4#482111yescharacteristic
486C4#486911yescharacteristic
487C4#487211yescharacteristic
488C3#899111yescharacteristic, complemented, Sylow
489C2—21no—
490C2—21no—
491C2—122no—
492C2—241no—
493C2—481no—
494C2#6875511yescharacteristic
495C2#6889711yescharacteristic
496C2#6903511yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1303C2
socleC22×C3#386
prime radical#79297C2
cosocleC2×C3×C42C23
hypercenter#1303D3
hypocenterC3#8991
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1303

Complements

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

NH ≅ G/Ntype
2. #79043C2semidirect
3. #70498C2semidirect
4. #69492C2semidirect
5. C4×#809C2semidirect
6. C3×#1303C2semidirect
7. #67739C2semidirect
21. #4846C4semidirect
22. #4846C4semidirect
23. #4843C4semidirect
24. #4843C4semidirect
39. #1303D3semidirect
175. #208D8semidirect
278. 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
#792972
#790432
#704982
#694922
C4×#8092
C3×#13032
#677392
#89913