Finite Group Databasebeta

← #10690048

Subgroups — #10690048

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×#3100C211yesmaximal, characteristic
2D4×#208C211yesmaximal, characteristic, complemented
3#71818C211yesmaximal, characteristic, complemented
4#71802C211yesmaximal, characteristic, complemented
5C4×#832C211yesmaximal, characteristic, complemented
6#67746C211yesmaximal, characteristic, complemented
7#67739C211yesmaximal, characteristic, complemented
8#8991—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11C22×#208—21no—
12C3×#480—21no—
13C3×#474C411yescharacteristic, complemented
14C3×#474C411yescharacteristic, complemented
15C3×#459C411yescharacteristic, complemented
16C3×#459C411yescharacteristic, complemented
17C3×#447—21no—
18C3×C4×D8—21no—
19C3×C4×D8—21no—
20C3×#431C2211yescharacteristic
21C2×#837—21no—
22C2×#837—21no—
23C2×#832—21no—
24#5207—21no—
25#5012—21no—
26#5011—22no—
27D4×C3⋊C8—21no—
28C4×C4⋊Dic3C2211yescharacteristic
29#4924C2211yescharacteristic
30C4×#208C2211yescharacteristic
31C2×C4×C3⋊C8C2211yescharacteristic
32D4⋊(C3⋊C8)—22no—
33#4529—21no—
34#3100D311yescharacteristic
35D4×M16—31no—
36#1980—31no—
37C4×#102—31no—
38#1492—31no—
39#1303—31no—
40#1278—31no—
41C2×C3×#41—21no—
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C2×C3×D8—21no—
45C2×C3×D8—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—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4C2×C411yescharacteristic
56C3×C4×D4C2×C411yescharacteristic
57C3×C4×D4D411yescharacteristic
58C3×C4×D4D411yescharacteristic
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×C4⋊Dic3—21no—
63C2×C4×Dic3—21no—
64C2×#208—22no—
65C2×#208—21no—
66C2×#208—42no—
67C2×#208—42no—
68C2×#208D411yescharacteristic
69C2×#208D411yescharacteristic
70C22×C3⋊C8—22no—
71C22×C3⋊C8—21no—
72C3×#107—21no—
73C3×#107—21no—
74C3×C4⋊C8C2×C411yescharacteristic
75C3×C4⋊C8C2×C411yescharacteristic
76C3×#102—21no—
77C3×#102—21no—
78C3×#102—21no—
79C3×#102—21no—
80C3×C22⋊C8—21no—
81C3×C4×C8—21no—
82#837—42no—
83#837—42no—
84#832—24no—
85C3⋊#95—21no—
86#830—22no—
87#830—22no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)D412yes—
90#803—21no—
91C4×C3⋊C8—21no—
92C4×C3⋊C8—21no—
93C4×C3⋊C8—21no—
94C22×M16—61no—
95#532C2×D311yescharacteristic
96C2×C4×D4C2×D311yescharacteristic
97#480—21no—
98#474Dic311yescharacteristic, complemented
99#474Dic311yescharacteristic, complemented
100#459Dic311yescharacteristic, complemented
101#459Dic311yescharacteristic, complemented
102#447—21no—
103C4×D8—21no—
104C4×D8—21no—
105#436—61no—
106#435—62no—
107C8×D4—61no—
108#431C2×D311yescharacteristic
109#423—31no—
110C2×#104—61no—
111C2×#104—61no—
112C2×#102—61no—
113#407—61no—
114C4×M16—31no—
115C2×C4×C8—31no—
116C4×C4⋊C4—31no—
117#340—61no—
118D4⋊C8—62no—
119C24×C3—21no—
120C3×#41—41no—
121C3×#41—41no—
122C3×#41—41no—
123C3×#41—41no—
124C2×C3×Q8—21no—
125C2×C3×D4—22no—
126C2×C3×D4—21no—
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—41no—
131C2×C3×D4—41no—
132C2×C3×D4C22⋊C411yescharacteristic
133C2×C3×D4C22⋊C411yescharacteristic
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—21no—
138C22×C3×C4—21no—
139C22×C3×C4—41no—
140C22×C3×C4—41no—
141C22×C3×C4—41no—
142C22×C3×C4—41no—
143C22×C3×C4—41no—
144C22×C3×C4C2×D411yescharacteristic
145C22×C3×C4C2×D411yescharacteristic
146C22×C3×C4#4111yescharacteristic
147C22×Dic3—41no—
148C3×D8—42no—
149C3×D8—42no—
150C2×C3×C8—21no—
151C2×C3×C8—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4—21no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4C22⋊C411yescharacteristic
156C3×C4⋊C4C22⋊C411yescharacteristic
157C3×C22⋊C4—21no—
158C3×C22⋊C4—21no—
159C3×C22⋊C4—21no—
160C3×C22⋊C4—21no—
161C3×C22⋊C4—41no—
162C3×C22⋊C4—41no—
163C3×C42C22×C411yescharacteristic
164C3×C42C2×D411yescharacteristic
165C3×C42C2×D411yescharacteristic
166C3×C42#4111yescharacteristic
167C4⋊Dic3—22no—
168C4⋊Dic3—41no—
169C4×Dic3—41no—
170C4×Dic3—41no—
171#208—21no—
172#208—21no—
173#208—21no—
174#208—44no—
175#208—41no—
176#208—41no—
177#208D812yescomplemented
178C2×C3⋊C8—21no—
179C2×C3⋊C8—22no—
180C2×C3⋊C8—21no—
181C2×C3⋊C8—22no—
182C2×C3⋊C8—44no—
183C2×C3⋊C8—41no—
184C2×C3⋊C8—41no—
185C2×#41—21no—
186C22×D4#6711yescharacteristic
187C23×C4—21no—
188C2×D8—21no—
189C2×D8—21no—
190C2×M16—31no—
191C2×M16—31no—
192C2×M16—62no—
193C2×M16—61no—
194C2×M16—122no—
195C2×M16—122no—
196C22×C8—62no—
197C22×C8—61no—
198C4⋊Q8C2×Dic311yescharacteristic
199#127C2×Dic311yescharacteristic
200#124—21no—
201#121—21no—
202#121—21no—
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4C2×Dic311yescharacteristic
208C4×D4C2×Dic311yescharacteristic
209C4×D4#6711yescharacteristic
210C4×D4#6711yescharacteristic
211C2×C4⋊C4—61no—
212C2×C4⋊C4#6711yescharacteristic
213C2×C22⋊C4—21no—
214C2×C42—61no—
215C2×C42C22×D311yescharacteristic
216#107—21no—
217#107—21no—
218C4⋊C8—32no—
219C4⋊C8—61no—
220C4⋊C8C2×Dic311yescharacteristic
221C4⋊C8C2×Dic311yescharacteristic
222#104—122no—
223#104—122no—
224#102—21no—
225#102—21no—
226#102—21no—
227#102—21no—
228#102—64no—
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—61no—
238C23×C3—21no—
239C23×C3—21no—
240C23×C3—42no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3—41no—
244C23×C3—81no—
245C23×C3#12111yescharacteristic
246C3×Q8—41no—
247C3×Q8—41no—
248C3×D4—21no—
249C3×D4—21no—
250C3×D4—21no—
251C3×D4—42no—
252C3×D4—42no—
253C3×D4—41no—
254C3×D4—41no—
255C3×D4—41no—
256C3×D4—41no—
257C3×D4—81no—
258C3×D4—81no—
259C3×D4#10412yescomplemented
260C2×C3×C4—21no—
261C2×C3×C4—21no—
262C2×C3×C4—21no—
263C2×C3×C4—21no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4—41no—
269C2×C3×C4—41no—
270C2×C3×C4—41no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—81no—
277C2×C3×C4—81no—
278C2×C3×C4C2×C22⋊C411yescharacteristic
279C2×C3×C4C4×D411yescharacteristic
280C2×C3×C4C4×D411yescharacteristic
281C2×C3×C4#12011yescharacteristic
282C2×C3×C4#12111yescharacteristic
283C2×C3×C4#12311yescharacteristic
284C2×C3×C4C2×D811yescharacteristic
285C2×C3×C4#13511yescharacteristic
286C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
287C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
288C2×Dic3—42no—
289C2×Dic3—81no—
290C2×Dic3—81no—
291C3×C8—41no—
292C3×C8—41no—
293C3⋊C8—22no—
294C3⋊C8—42no—
295C3⋊C8—42no—
296C3⋊C8—41no—
297C24—21no—
298#41—41no—
299#41—41no—
300#41—41no—
301#41—41no—
302C2×Q8—21no—
303C2×D4—22no—
304C2×D4—21no—
305C2×D4—21no—
306C2×D4—21no—
307C2×D4—21no—
308C2×D4—41no—
309C2×D4—41no—
310C2×D4#21711yescharacteristic
311C2×D4#21711yescharacteristic
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—42no—
327D8—42no—
328M16—32no—
329M16—61no—
330M16—61no—
331M16—61no—
332M16—124no—
333M16—121no—
334M16—121no—
335C2×C8—21no—
336C2×C8—21no—
337C2×C8—61no—
338C2×C8—62no—
339C2×C8—61no—
340C2×C8—62no—
341C2×C8—124no—
342C2×C8—121no—
343C2×C8—121no—
344C4⋊C4—21no—
345C4⋊C4—21no—
346C4⋊C4—21no—
347C4⋊C4—62no—
348C4⋊C4—121no—
349C4⋊C4#21711yescharacteristic
350C4⋊C4#21711yescharacteristic
351C22⋊C4—21no—
352C22⋊C4—21no—
353C22⋊C4—21no—
354C22⋊C4—21no—
355C22⋊C4—41no—
356C22⋊C4—41no—
357C42—121no—
358C42—121no—
359C42D3×D411yescharacteristic
360C42#23711yescharacteristic
361C42C22×Dic311yescharacteristic
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—81no—
374C22×C3—81no—
375C22×C3—81no—
376C22×C3#38611yescharacteristic
377C22×C3#45911yescharacteristic
378C22×C3#46311yescharacteristic
379C3×C4—21no—
380C3×C4—21no—
381C3×C4—21no—
382C3×C4—41no—
383C3×C4—41no—
384C3×C4—81no—
385C3×C4C2×#10411yescharacteristic
386C3×C4#42111yescharacteristic
387C3×C4C4×D811yescharacteristic
388C3×C4#44211yescharacteristic
389C3×C4#44711yescharacteristic
390C3×C4#44911yescharacteristic
391Dic3—82no—
392C23—21no—
393C23—21no—
394C23—42no—
395C23—41no—
396C23—41no—
397C23—41no—
398C23—81no—
399C23#93911yescharacteristic
400Q8—41no—
401Q8—41no—
402D4—21no—
403D4—21no—
404D4—21no—
405D4—42no—
406D4—42no—
407D4—41no—
408D4—41no—
409D4—41no—
410D4—41no—
411D4—81no—
412D4—81no—
413D4#83712yescomplemented
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—41no—
430C2×C4—81no—
431C2×C4—81no—
432C2×C4—122no—
433C2×C4—241no—
434C2×C4—241no—
435C2×C4D3×D811yescharacteristic
436C2×C4#91111yescharacteristic
437C2×C4#91111yescharacteristic
438C2×C4#91211yescharacteristic
439C2×C4D4×Dic311yescharacteristic
440C2×C4D4×Dic311yescharacteristic
441C2×C4#93511yescharacteristic
442C2×C4#93711yescharacteristic
443C2×C4#93911yescharacteristic
444C2×C4C2×#21711yescharacteristic
445C8—41no—
446C8—41no—
447C8—62no—
448C8—122no—
449C8—122no—
450C8—121no—
451C2×C3—21no—
452C2×C3—21no—
453C2×C3—42no—
454C2×C3—81no—
455C2×C3—161no—
456C2×C3#187111yescharacteristic
457C2×C3#187911yescharacteristic
458C2×C3#188111yescharacteristic
459C22—21no—
460C22—21no—
461C22—22no—
462C22—41no—
463C22—41no—
464C22—41no—
465C22—41no—
466C22—41no—
467C22—82no—
468C22—82no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22#515011yescharacteristic
473C22#515311yescharacteristic
474C22#522011yescharacteristic
475C4—21no—
476C4—21no—
477C4—21no—
478C4—41no—
479C4—41no—
480C4—81no—
481C4—242no—
482C4Dic3×D811yescharacteristic
483C4#515211yescharacteristic
484C4#515611yescharacteristic
485C4#515811yescharacteristic
486C4C2×#83711yescharacteristic
487C4#523611yescharacteristic
488C3#899111yescharacteristic, complemented, Sylow
489C2—21no—
490C2—21no—
491C2—42no—
492C2—81no—
493C2—161no—
494C2#7171311yescharacteristic
495C2#7203711yescharacteristic
496C2#7211611yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3100C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3100D3
hypocenterC3#8991
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3100

Complements

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

NH ≅ G/Ntype
2. D4×#208C2semidirect
3. #71818C2semidirect
4. #71802C2semidirect
5. C4×#832C2semidirect
6. #67746C2semidirect
7. #67739C2semidirect
13. C3×#474C4semidirect
14. C3×#474C4semidirect
15. C3×#459C4semidirect
16. C3×#459C4semidirect
98. #474Dic3semidirect
99. #474Dic3semidirect
100. #459Dic3semidirect
101. #459Dic3semidirect
177. #208D8semidirect
259. C3×D4#104semidirect
413. D4#837semidirect
488. C3#8991semidirect

Maximal subgroups

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

subgroupindex
C3×#31002
D4×#2082
#718182
#718022
C4×#8322
#677462
#677392
#89913