Finite Group Databasebeta

← #10690046

Subgroups — #10690046

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
1C3×D4×M16C211yesmaximal, characteristic, complemented
2#82308C211yesmaximal, characteristic
3C4×#832C211yesmaximal, characteristic, complemented
4#70498C211yesmaximal, characteristic, complemented
5#69867C211yesmaximal, characteristic, complemented
6#67715C211yesmaximal, characteristic, complemented
7#67667C211yesmaximal, characteristic, complemented
8#8991—31nomaximal, Sylow
9C22×C3×M16—21no—
10C2×C3×C4×D4C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#436—21no—
13C3×#435—22no—
14C3×C8×D4—21no—
15C3×#423C2211yescharacteristic
16C3×#407—21no—
17C3×C4×M16C2211yescharacteristic
18#5217—21no—
19C2×#832—21no—
20C2×#825—21no—
21C2×#825—21no—
22#5015C411yescharacteristic, complemented
23#5015C411yescharacteristic, complemented
24C4×#213—21no—
25C4×#213—21no—
26#5007C411yescharacteristic, complemented
27#5007C411yescharacteristic, complemented
28#4967—21no—
29C4×C4⋊Dic3C2211yescharacteristic
30#4925C2211yescharacteristic
31C2×C4×C3⋊C8C2211yescharacteristic
32#4529—21no—
33#4483—22no—
34#3100—31no—
35D4×M16D311yescharacteristic, complemented
36#1980—31no—
37C4×#102—31no—
38#1492—31no—
39#1303—31no—
40#1278—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×#235—21no—
44C2×C3×M16—22no—
45C2×C3×M16—21no—
46C2×C3×M16—42no—
47C2×C3×M16—42no—
48C2×C3×M16D411yescharacteristic
49C2×C3×M16D411yescharacteristic
50C22×C3×C8—22no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C3×C4×D4C2×C411yescharacteristic
54C3×C4×D4C2×C411yescharacteristic
55C3×C4×D4D411yescharacteristic
56C3×C4×D4D411yescharacteristic
57C2×C3×C4⋊C4D411yescharacteristic
58C2×C3×C22⋊C4—21no—
59C2×C3×C42C2311yescharacteristic
60C2×#213—21no—
61C2×#213—21no—
62#930—21no—
63#930—21no—
64C2×C4⋊Dic3—21no—
65C2×C4×Dic3—21no—
66C22×C3⋊C8—21no—
67#875—21no—
68#874C2×C411yescharacteristic
69C4×D12—21no—
70C4×D12—21no—
71#869C2×C411yescharacteristic
72C3×C4⋊C8—21no—
73C3×C4⋊C8D412yes—
74C3×C22⋊C8—22no—
75C3×C22⋊C8—22no—
76C3×#97—21no—
77C3×C4×C8—21no—
78#832—24no—
79#832—21no—
80#832—21no—
81C3⋊#95—21no—
82#830—21no—
83#825—42no—
84#825—42no—
85#809—21no—
86#809—21no—
87#807—21no—
88#807—21no—
89C4⋊(C3⋊C8)C2×C411yescharacteristic
90C4⋊(C3⋊C8)C2×C411yescharacteristic
91C4×C3⋊C8—21no—
92C4×C3⋊C8—21no—
93C4×C3⋊C8—21no—
94C22×M16—21no—
95#532—31no—
96C2×C4×D4C2×D311yescharacteristic
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—21no—
106#435—22no—
107C8×D4—21no—
108#431—31no—
109#423C2×D311yescharacteristic
110C2×#104—61no—
111C2×#104—61no—
112C2×#102—61no—
113#407—21no—
114C4×M16C2×D311yescharacteristic
115C2×C4×C8—31no—
116C4×C4⋊C4—31no—
117#340—61no—
118D4⋊C8—62no—
119C24×C3—21no—
120C2×C3×D4—22no—
121C2×C3×D4—21no—
122C2×C3×D4—41no—
123C2×C3×D4C22⋊C411yescharacteristic
124C2×C3×D4C22⋊C411yescharacteristic
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4—21no—
129C22×C3×C4—21no—
130C22×C3×C4—41no—
131C22×C3×C4—41no—
132C22×C3×C4—41no—
133C22×C3×C4—41no—
134C22×C3×C4C2×D411yescharacteristic
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4#4111yescharacteristic
137C2×#67—41no—
138C22×Dic3—41no—
139#235—41no—
140#235—41no—
141#235—41no—
142#235—41no—
143C2×D12—21no—
144C2×D12—21no—
145C2×D12—21no—
146C2×C4×D3—41no—
147C2×Dic6—21no—
148C3×M16—21no—
149C3×M16—21no—
150C3×M16—21no—
151C3×M16—44no—
152C3×M16—41no—
153C3×M16—41no—
154C3×M16D812yescomplemented
155C2×C3×C8—21no—
156C2×C3×C8—21no—
157C2×C3×C8—22no—
158C2×C3×C8—44no—
159C3×C4⋊C4—21no—
160C3×C4⋊C4C22⋊C411yescharacteristic
161C3×C4⋊C4C22⋊C411yescharacteristic
162C3×C22⋊C4—21no—
163C3×C22⋊C4—21no—
164C3×C22⋊C4—21no—
165C3×C22⋊C4—21no—
166C3×C42C22×C411yescharacteristic
167C3×C42C2×D411yescharacteristic
168C3×C42C2×D411yescharacteristic
169C3×C42#4111yescharacteristic
170#213—42no—
171#213—42no—
172(C2×D3)⋊C4—41no—
173(C2×D3)⋊C4—41no—
174C4⋊Dic3—22no—
175C4⋊Dic3—21no—
176C4⋊Dic3—21no—
177C4⋊Dic3—41no—
178C4×Dic3—41no—
179C4×Dic3—41no—
180C2×C3⋊C8—22no—
181C2×C3⋊C8—21no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—41no—
184C2×C3⋊C8—41no—
185C2×#41—61no—
186C22×D4D1211yescharacteristic
187C23×C4—21no—
188C2×D8—61no—
189C2×D8—61no—
190C2×M16—22no—
191C2×M16—21no—
192C2×M16—42no—
193C2×M16—42no—
194C2×M16#6711yescharacteristic
195C2×M16#6711yescharacteristic
196C22×C8—22no—
197C22×C8—61no—
198C4⋊Q8—31no—
199#127—31no—
200#124—61no—
201#121—61no—
202#121—61no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—61no—
206C4×D4—61no—
207C4×D4C4×D311yescharacteristic
208C4×D4C4×D311yescharacteristic
209C4×D4#6711yescharacteristic
210C4×D4#6711yescharacteristic
211C2×C4⋊C4—61no—
212C2×C4⋊C4D1211yescharacteristic
213C2×C22⋊C4—21no—
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—64no—
225#102—61no—
226#102—61no—
227#102—61no—
228#102—61no—
229C22⋊C8—22no—
230C22⋊C8—22no—
231C22⋊C8—61no—
232#97—21no—
233C4×C8—21no—
234C4×C8—61no—
235C4×C8—61no—
236C4×C8—61no—
237#95—61no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—42no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—81no—
244C23×C3#12111yescharacteristic
245C22×D3—41no—
246C3×D4—21no—
247C3×D4—21no—
248C3×D4—21no—
249C3×D4—42no—
250C3×D4#10412yescomplemented
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—41no—
256C2×C3×C4—41no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4C2×C22⋊C411yescharacteristic
267C2×C3×C4C4×D411yescharacteristic
268C2×C3×C4C4×D411yescharacteristic
269C2×C3×C4#12011yescharacteristic
270C2×C3×C4#12111yescharacteristic
271C2×C3×C4#12311yescharacteristic
272C2×C3×C4C2×D811yescharacteristic
273C2×C3×C4#13511yescharacteristic
274C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
275C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
276#67—81no—
277#67—81no—
278C2×Dic3—42no—
279C2×Dic3—41no—
280C2×Dic3—81no—
281C2×Dic3—81no—
282D12—42no—
283D12—41no—
284D12—41no—
285D12—41no—
286D12—41no—
287C4×D3—81no—
288C4×D3—81no—
289Dic6—41no—
290Dic6—41no—
291C3×C8—22no—
292C3×C8—42no—
293C3×C8—41no—
294C3⋊C8—42no—
295C3⋊C8—41no—
296C3⋊C8—41no—
297C24—21no—
298#41—121no—
299#41—121no—
300#41—121no—
301#41—121no—
302C2×Q8—61no—
303C2×D4—22no—
304C2×D4—21no—
305C2×D4—41no—
306C2×D4—61no—
307C2×D4—61no—
308C2×D4—61no—
309C2×D4—121no—
310C2×D4(C2×D3)⋊C411yescharacteristic
311C2×D4(C2×D3)⋊C411yescharacteristic
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—21no—
316C22×C4—21no—
317C22×C4—41no—
318C22×C4—41no—
319C22×C4—41no—
320C22×C4—41no—
321C22×C4—121no—
322C22×C4—121no—
323C22×C4C2×D1211yescharacteristic
324C22×C4#23511yescharacteristic
325C22×C4C2×#6711yescharacteristic
326D8—122no—
327D8—122no—
328M16—21no—
329M16—21no—
330M16—21no—
331M16—44no—
332M16—41no—
333M16—41no—
334M16#21312yescomplemented
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—22no—
338C2×C8—44no—
339C2×C8—62no—
340C2×C8—61no—
341C2×C8—61no—
342C2×C8—121no—
343C2×C8—121no—
344C4⋊C4—21no—
345C4⋊C4—62no—
346C4⋊C4—61no—
347C4⋊C4—61no—
348C4⋊C4—121no—
349C4⋊C4(C2×D3)⋊C411yescharacteristic
350C4⋊C4(C2×D3)⋊C411yescharacteristic
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—121no—
356C22⋊C4—121no—
357C42—121no—
358C42—121no—
359C42C2×C4×D311yescharacteristic
360C42#23511yescharacteristic
361C42C2×#6711yescharacteristic
362C42C2×#6711yescharacteristic
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—22no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3—41no—
369C22×C3—41no—
370C22×C3—41no—
371C22×C3—82no—
372C22×C3—82no—
373C22×C3#38611yescharacteristic
374C22×C3#45911yescharacteristic
375C22×C3#46311yescharacteristic
376C2×D3—81no—
377C2×D3—81no—
378C2×D3—81no—
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—41no—
383C3×C4—41no—
384C3×C4C2×#10411yescharacteristic
385C3×C4#42111yescharacteristic
386C3×C4C4×D811yescharacteristic
387C3×C4#44211yescharacteristic
388C3×C4#44711yescharacteristic
389C3×C4#44911yescharacteristic
390Dic3—82no—
391Dic3—81no—
392C23—21no—
393C23—21no—
394C23—42no—
395C23—41no—
396C23—41no—
397C23—81no—
398C23—121no—
399C23#93011yescharacteristic
400Q8—121no—
401Q8—121no—
402D4—21no—
403D4—21no—
404D4—21no—
405D4—42no—
406D4—122no—
407D4—121no—
408D4—121no—
409D4—121no—
410D4—121no—
411D4—241no—
412D4—241no—
413D4#82512yescomplemented
414C2×C4—21no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—41no—
428C2×C4—41no—
429C2×C4—122no—
430C2×C4—121no—
431C2×C4—241no—
432C2×C4—241no—
433C2×C4—241no—
434C2×C4—241no—
435C2×C4C2×(C2×D3)⋊C411yescharacteristic
436C2×C4C4×#6711yescharacteristic
437C2×C4C4×#6711yescharacteristic
438C2×C4#92911yescharacteristic
439C2×C4#93011yescharacteristic
440C2×C4C2×#21311yescharacteristic
441C2×C4#93211yescharacteristic
442C2×C4#94911yescharacteristic
443C2×C4#95011yescharacteristic
444C2×C4#95311yescharacteristic
445C8—22no—
446C8—42no—
447C8—41no—
448C8—122no—
449C8—121no—
450C8—121no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—42no—
454C2×C3—81no—
455C2×C3#187111yescharacteristic
456C2×C3#187911yescharacteristic
457C2×C3#188111yescharacteristic
458D3—161no—
459C22—21no—
460C22—21no—
461C22—22no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—82no—
468C22—82no—
469C22—241no—
470C22—241no—
471C22—241no—
472C22#501511yescharacteristic
473C22#501611yescharacteristic
474C22#521311yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—41no—
479C4—41no—
480C4—242no—
481C4—241no—
482C4C4×#21311yescharacteristic
483C4#501411yescharacteristic
484C4C2×#82511yescharacteristic
485C4#513911yescharacteristic
486C4#521711yescharacteristic
487C4#523811yescharacteristic
488C3#899111yescharacteristic, complemented, Sylow
489C2—21no—
490C2—21no—
491C2—42no—
492C2—81no—
493C2—481no—
494C2#7161311yescharacteristic
495C2#7198411yescharacteristic
496C2#7202011yescharacteristic
subgroupG/H
centerC2×C4#953
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×D4×M16C2
socleC22×C3#386
prime radical#82308C2
cosocleC2×C3×C42C23
hypercenterD4×M16D3
hypocenterC3#8991
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ D4×M16

Complements

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

NH ≅ G/Ntype
1. C3×D4×M16C2semidirect
3. C4×#832C2semidirect
4. #70498C2semidirect
5. #69867C2semidirect
6. #67715C2semidirect
7. #67667C2semidirect
22. #5015C4semidirect
23. #5015C4semidirect
26. #5007C4semidirect
27. #5007C4semidirect
35. D4×M16D3semidirect
154. C3×M16D8semidirect
250. C3×D4#104semidirect
334. M16#213semidirect
413. D4#825semidirect
488. C3#8991semidirect

Maximal subgroups

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

subgroupindex
C3×D4×M162
#823082
C4×#8322
#704982
#698672
#677152
#676672
#89913