Finite Group Databasebeta

← #10690044

Subgroups — #10690044

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#82308C211yesmaximal, characteristic
2D4×#208C211yesmaximal, characteristic, complemented
3C3×C4×#102C211yesmaximal, characteristic, complemented
4#69735C211yesmaximal, characteristic, complemented
5#69492C211yesmaximal, characteristic, complemented
6#67656C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#8991—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C22×#208—21no—
11#5477C2211yescharacteristic
12C2×C3×#102—21no—
13C2×C3×C4×C8C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5217—21no—
16#5207—21no—
17#5015C411yescharacteristic, complemented
18#5015C411yescharacteristic, complemented
19C4×#213—21no—
20C4×#213—21no—
21#5012—21no—
22#5011—22no—
23D4×C3⋊C8—21no—
24#5007C411yescharacteristic, complemented
25#5007C411yescharacteristic, complemented
26#4967—21no—
27C2×#805—21no—
28C2×#805—21no—
29#4925C2211yescharacteristic
30#4924C2211yescharacteristic
31C4×#208C2211yescharacteristic
32C3×#340—21no—
33#4459—22no—
34#3100—31no—
35D4×M16—31no—
36#1980—31no—
37C4×#102D311yescharacteristic, complemented
38#1492—31no—
39#1303—31no—
40#1278—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×#235—21no—
44C22×C3×C8—21no—
45C3×C4×D4—21no—
46C3×C4×D4—21no—
47C3×C4×D4C2×C411yescharacteristic
48C3×C4×D4C2×C411yescharacteristic
49C3×C4×D4D411yescharacteristic
50C3×C4×D4D411yescharacteristic
51C2×C3×C4⋊C4—21no—
52C2×C3×C4⋊C4D411yescharacteristic
53C2×C3×C22⋊C4—21no—
54C2×C3×C42—21no—
55C2×C3×C42C2311yescharacteristic
56C2×#213—21no—
57C2×#213—21no—
58#930—21no—
59#930—21no—
60C2×#208—22no—
61C2×#208—21no—
62C2×#208—42no—
63C2×#208—42no—
64C2×#208D411yescharacteristic
65C2×#208D411yescharacteristic
66C22×C3⋊C8—22no—
67#875—21no—
68#874C2×C411yescharacteristic
69C4×D12—21no—
70C4×D12—21no—
71#869C2×C411yescharacteristic
72C3×#102—24no—
73C3×C4×C8—21no—
74C3×C4×C8—21no—
75C3×#95—21no—
76#832—21no—
77#832—21no—
78#830—22no—
79#830—22no—
80#830—21no—
81#809—21no—
82#809—21no—
83#807—21no—
84#807—21no—
85#805—42no—
86#805—42no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)C2×C411yescharacteristic
89C4⋊(C3⋊C8)C2×C411yescharacteristic
90C4⋊(C3⋊C8)D412yes—
91#803—21no—
92C4×C3⋊C8—21no—
93C4×C3⋊C8—21no—
94C22×M16—61no—
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—61no—
106#435—62no—
107C8×D4—61no—
108#431—31no—
109#423—31no—
110C2×#104—61no—
111C2×#104—61no—
112C2×#102—21no—
113#407—61no—
114C4×M16—31no—
115C2×C4×C8C2×D311yescharacteristic
116C4×C4⋊C4C2×D311yescharacteristic
117#340—21no—
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×C4—41no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C2×#67—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—
148C2×C3×C8—22no—
149C2×C3×C8—41no—
150C2×C3×C8—41no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—22no—
153C3×C4⋊C4—41no—
154C3×C4⋊C4C22⋊C411yescharacteristic
155C3×C4⋊C4C22⋊C411yescharacteristic
156C3×C22⋊C4—21no—
157C3×C22⋊C4—21no—
158C3×C22⋊C4—21no—
159C3×C22⋊C4—21no—
160C3×C42—41no—
161C3×C42—41no—
162C3×C42C22×C411yescharacteristic
163C3×C42C2×D411yescharacteristic
164C3×C42C2×D411yescharacteristic
165C3×C42#4111yescharacteristic
166#213—42no—
167#213—42no—
168(C2×D3)⋊C4—41no—
169(C2×D3)⋊C4—41no—
170C4⋊Dic3—21no—
171C4⋊Dic3—21no—
172#208—21no—
173#208—21no—
174#208—21no—
175#208—44no—
176#208—41no—
177#208—41no—
178#208D812yescomplemented
179C2×C3⋊C8—21no—
180C2×C3⋊C8—21no—
181C2×C3⋊C8—21no—
182C2×C3⋊C8—21no—
183C2×C3⋊C8—22no—
184C2×C3⋊C8—44no—
185C2×#41—61no—
186C22×D4#6711yescharacteristic
187C23×C4—21no—
188C2×D8—61no—
189C2×D8—61no—
190C2×M16—31no—
191C2×M16—31no—
192C2×M16—62no—
193C2×M16—61no—
194C2×M16—122no—
195C2×M16—122no—
196C22×C8—21no—
197C22×C8—62no—
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×D4D1211yescharacteristic
210C4×D4D1211yescharacteristic
211C2×C4⋊C4—21no—
212C2×C4⋊C4#6711yescharacteristic
213C2×C22⋊C4—21no—
214C2×C42—21no—
215C2×C42C22×D311yescharacteristic
216#107—61no—
217#107—61no—
218C4⋊C8—31no—
219C4⋊C8—31no—
220C4⋊C8—32no—
221C4⋊C8—61no—
222#104—122no—
223#104—122no—
224#102—24no—
225#102—61no—
226#102—61no—
227#102—61no—
228#102—61no—
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—21no—
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×C4—42no—
267C2×C3×C4—81no—
268C2×C3×C4—81no—
269C2×C3×C4C2×C22⋊C411yescharacteristic
270C2×C3×C4C4×D411yescharacteristic
271C2×C3×C4C4×D411yescharacteristic
272C2×C3×C4#12011yescharacteristic
273C2×C3×C4#12111yescharacteristic
274C2×C3×C4#12311yescharacteristic
275C2×C3×C4C2×D811yescharacteristic
276C2×C3×C4#13511yescharacteristic
277C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
278C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
279#67—81no—
280#67—81no—
281C2×Dic3—41no—
282D12—42no—
283D12—41no—
284D12—41no—
285D12—41no—
286D12—41no—
287C4×D3—81no—
288C4×D3—81no—
289Dic6—41no—
290Dic6—41no—
291C3×C8—42no—
292C3⋊C8—22no—
293C3⋊C8—41no—
294C3⋊C8—41no—
295C3⋊C8—42no—
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—41no—
322C22×C4—121no—
323C22×C4D3×D411yescharacteristic
324C22×C4#23711yescharacteristic
325C22×C4C2×#6711yescharacteristic
326D8—122no—
327D8—122no—
328M16—32no—
329M16—61no—
330M16—61no—
331M16—61no—
332M16—124no—
333M16—121no—
334M16—121no—
335C2×C8—22no—
336C2×C8—41no—
337C2×C8—41no—
338C2×C8—61no—
339C2×C8—61no—
340C2×C8—62no—
341C2×C8—61no—
342C2×C8—61no—
343C2×C8—124no—
344C4⋊C4—21no—
345C4⋊C4—22no—
346C4⋊C4—41no—
347C4⋊C4—61no—
348C4⋊C4—61no—
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—41no—
358C42—41no—
359C42C2×C4×D311yescharacteristic
360C42C2×D1211yescharacteristic
361C42D3×D411yescharacteristic
362C42#23711yescharacteristic
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×C4—82no—
385C3×C4C2×#10411yescharacteristic
386C3×C4#42111yescharacteristic
387C3×C4C4×D811yescharacteristic
388C3×C4#44211yescharacteristic
389C3×C4#44711yescharacteristic
390C3×C4#44911yescharacteristic
391Dic3—81no—
392C23—21no—
393C23—21no—
394C23—42no—
395C23—41no—
396C23—41no—
397C23—81no—
398C23—121no—
399C23#93911yescharacteristic
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#80512yescomplemented
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—42no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—121no—
433C2×C4—241no—
434C2×C4—241no—
435C2×C4#67⋊C411yescharacteristic
436C2×C4#67⋊C411yescharacteristic
437C2×C4#88211yescharacteristic
438C2×C4#88611yescharacteristic
439C2×C4D3×D811yescharacteristic
440C2×C4#91111yescharacteristic
441C2×C4#91411yescharacteristic
442C2×C4#91611yescharacteristic
443C2×C4C2×(C2×D3)⋊C411yescharacteristic
444C2×C4#93911yescharacteristic
445C8—42no—
446C8—62no—
447C8—121no—
448C8—121no—
449C8—122no—
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#495811yescharacteristic
473C22#515011yescharacteristic
474C22#516511yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—41no—
479C4—41no—
480C4—82no—
481C4—241no—
482C4#475611yescharacteristic
483C4#477311yescharacteristic
484C4#478111yescharacteristic
485C4#478511yescharacteristic
486C4C2×#80511yescharacteristic
487C4#500511yescharacteristic
488C3#899111yescharacteristic, complemented, Sylow
489C2—21no—
490C2—21no—
491C2—42no—
492C2—81no—
493C2—481no—
494C2#6945311yescharacteristic
495C2#7017211yescharacteristic
496C2#7029311yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×C4×#102C2
socleC22×C3#386
prime radical#82308C2
cosocleC2×C3×C42C23
hypercenterC4×#102D3
hypocenterC3#8991
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ C4×#102

Complements

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

NH ≅ G/Ntype
2. D4×#208C2semidirect
3. C3×C4×#102C2semidirect
4. #69735C2semidirect
5. #69492C2semidirect
6. #67656C2semidirect
7. #67620C2semidirect
17. #5015C4semidirect
18. #5015C4semidirect
24. #5007C4semidirect
25. #5007C4semidirect
37. C4×#102D3semidirect
178. #208D8semidirect
250. C3×D4#104semidirect
413. D4#805semidirect
488. C3#8991semidirect

Maximal subgroups

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

subgroupindex
#823082
D4×#2082
C3×C4×#1022
#697352
#694922
#676562
#676202
#89913