Finite Group Databasebeta

← #10691279

Subgroups — #10691279

Order 768.

532 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3035C211yesmaximal, characteristic
2#41×C3⋊C8C211yesmaximal, characteristic, complemented
3#71811C211yesmaximal, characteristic, complemented
4#71802C211yesmaximal, characteristic, complemented
5#71646C211yesmaximal, characteristic, complemented
6#67707C211yesmaximal, characteristic, complemented
7#67704C211yesmaximal, characteristic, complemented
8#9208—31nomaximal, Sylow
9C2×C3×#135—21no—
10C3×#532C2211yescharacteristic
11C3×C4×#41C2211yescharacteristic
12C3×#464—21no—
13C3×#463—21no—
14C3×#462C411yescharacteristic, complemented
15C3×#461C411yescharacteristic, complemented
16C3×#460C411yescharacteristic, complemented
17C3×#459C411yescharacteristic, complemented
18C2×C3×C4⋊C8C2211yescharacteristic
19C3×#416—21no—
20C2×#837—21no—
21Q8×C3⋊C8—21no—
22D4×C3⋊C8—21no—
23D4×C3⋊C8—22no—
24C4×C4⋊Dic3C2211yescharacteristic
25#4925—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28C2×C4×C3⋊C8—21no—
29C2×C4×C3⋊C8C2211yescharacteristic
30#4537—22no—
31D4⋊(C3⋊C8)—21no—
32D4⋊(C3⋊C8)—21no—
33#4532—21no—
34#3035D311yescharacteristic
35C8×#41—31no—
36#1806—31no—
37#1502—31no—
38#1492—31no—
39#1301—31no—
40#1279—31no—
41C2×C3×#41—21no—
42C2×C3×#41D411yescharacteristic
43C3×#135—42no—
44C3×#135—42no—
45C2×C3×Q16—21no—
46C2×C3×SD16—21no—
47C2×C3×SD16—21no—
48C2×C3×D8—21no—
49C22×C3×C8—21no—
50C3×C4⋊Q8C2×C411yescharacteristic
51C3×#127C2×C411yescharacteristic
52C3×#124—21no—
53C3×#121—21no—
54C3×#121—21no—
55C3×C4×Q8C2×C411yescharacteristic
56C3×C4×Q8D411yescharacteristic
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C3×C4×D4—21no—
60C3×C4×D4—21no—
61C3×C4×D4C2×C411yescharacteristic
62C3×C4×D4D411yescharacteristic
63C3×#117—21no—
64C3×#117D411yescharacteristic
65C2×C3×C42—21no—
66C2×C3×C42C2311yescharacteristic
67C2×C4⋊Dic3—21no—
68C2×C4×Dic3—21no—
69C2×#208—21no—
70C22×C3⋊C8—22no—
71C22×C3⋊C8—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8C2×C411yescharacteristic
74C3×C4⋊C8C2×C411yescharacteristic
75C3×#103—21no—
76C3×#103—21no—
77C3×#102—21no—
78C3×#102—21no—
79#837—42no—
80C3⋊#95—21no—
81#830—22no—
82#830D412yes—
83C4⋊(C3⋊C8)—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4×C3⋊C8—22no—
88C4×C3⋊C8—21no—
89C4×C3⋊C8—21no—
90C4×C3⋊C8D411yescharacteristic
91C4×C3⋊C8D411yescharacteristic
92C2×#135—21no—
93#532C2×D311yescharacteristic
94C4×#41C2×D311yescharacteristic
95#464—21no—
96#463—21no—
97#462Dic311yescharacteristic, complemented
98#461Dic311yescharacteristic, complemented
99#460Dic311yescharacteristic, complemented
100#459Dic311yescharacteristic, complemented
101C8×Q8—61no—
102C8×D4—62no—
103C8×D4—61no—
104#431—31no—
105#431—61no—
106#423—31no—
107C2×C4⋊C8C2×D311yescharacteristic
108C2×#104—61no—
109#416—21no—
110C2×C4×C8—31no—
111C2×C4×C8—61no—
112C4×C4⋊C4—31no—
113#339—61no—
114#327—62no—
115D4⋊C8—61no—
116D4⋊C8—61no—
117C3×#41—21no—
118C3×#41—21no—
119C3×#41—41no—
120C3×#41—41no—
121C3×#41—41no—
122C3×#41—41no—
123C3×#41D812yes—
124C3×#41SD1612yes—
125C2×C3×Q8—21no—
126C2×C3×Q8C22⋊C411yescharacteristic
127C2×C3×D4—21no—
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—41no—
132C2×C3×D4C22⋊C411yescharacteristic
133C22×C3×C4—21no—
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—41no—
137C22×C3×C4C2×D411yescharacteristic
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×Dic3—41no—
141C3×Q16—42no—
142C3×SD16—42no—
143C3×SD16—42no—
144C3×D8—42no—
145C2×C3×C8—21no—
146C2×C3×C8—21no—
147C2×C3×C8—41no—
148C2×C3×C8—41no—
149C3×C4⋊C4—21no—
150C3×C4⋊C4—21no—
151C3×C4⋊C4—21no—
152C3×C4⋊C4—21no—
153C3×C4⋊C4C22⋊C411yescharacteristic
154C3×C4⋊C4C22⋊C411yescharacteristic
155C3×C22⋊C4—21no—
156C3×C22⋊C4—21no—
157C3×C22⋊C4—21no—
158C3×C22⋊C4—41no—
159C3×C22⋊C4—41no—
160C3×C42—21no—
161C3×C42—21no—
162C3×C42—21no—
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—41no—
172#208—41no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—22no—
175C2×C3⋊C8—21no—
176C2×C3⋊C8—42no—
177C2×C3⋊C8—44no—
178C2×C3⋊C8—41no—
179C2×#41—21no—
180C2×#41#6711yescharacteristic
181#135—42no—
182#135—42no—
183C2×Q16—21no—
184C2×SD16—21no—
185C2×SD16—21no—
186C2×D8—21no—
187C2×M16—61no—
188C22×C8—21no—
189C22×C8—62no—
190C22×C8—61no—
191C4⋊Q8C2×Dic311yescharacteristic
192#127C2×Dic311yescharacteristic
193#124—21no—
194#121—21no—
195#121—21no—
196C4×Q8C2×Dic311yescharacteristic
197C4×Q8#6711yescharacteristic
198C4×D4—21no—
199C4×D4—21no—
200C4×D4—21no—
201C4×D4—21no—
202C4×D4C2×Dic311yescharacteristic
203C4×D4#6711yescharacteristic
204#117—21no—
205#117#6711yescharacteristic
206C2×C4⋊C4—61no—
207C2×C42—21no—
208C2×C42—61no—
209C2×C42C22×D311yescharacteristic
210C4⋊C8—21no—
211C4⋊C8—62no—
212C4⋊C8—61no—
213C4⋊C8—61no—
214C4⋊C8—61no—
215C4⋊C8C2×Dic311yescharacteristic
216C4⋊C8C2×Dic311yescharacteristic
217#104—122no—
218#103—21no—
219#103—21no—
220#102—21no—
221#102—21no—
222C22⋊C8—32no—
223C22⋊C8—62no—
224C4×C8—31no—
225C4×C8—31no—
226C4×C8—62no—
227C4×C8—61no—
228C4×C8—61no—
229#95—61no—
230C23×C3—21no—
231C23×C3—41no—
232C23×C3#12111yescharacteristic
233C3×Q8—21no—
234C3×Q8—41no—
235C3×Q8—41no—
236C3×Q8#10212yes—
237C3×D4—22no—
238C3×D4—22no—
239C3×D4—21no—
240C3×D4—41no—
241C3×D4—41no—
242C3×D4—41no—
243C3×D4—41no—
244C3×D4—81no—
245C3×D4—81no—
246C3×D4—81no—
247C3×D4#10212yes—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—41no—
256C2×C3×C4—42no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—81no—
261C2×C3×C4—81no—
262C2×C3×C4C2×C22⋊C411yescharacteristic
263C2×C3×C4C4×D411yescharacteristic
264C2×C3×C4C4×D411yescharacteristic
265C2×C3×C4#12011yescharacteristic
266C2×C3×C4#12111yescharacteristic
267C2×C3×C4#12311yescharacteristic
268C2×C3×C4C2×D811yescharacteristic
269C2×C3×C4C2×SD1611yescharacteristic
270C2×C3×C4#13511yescharacteristic
271C2×C3×C4#13511yescharacteristic
272C2×Dic3—42no—
273C2×Dic3—81no—
274C2×Dic3—81no—
275C3×C8—41no—
276C3×C8—41no—
277C3⋊C8—22no—
278C3⋊C8—42no—
279C3⋊C8—41no—
280C3⋊C8—81no—
281#41—21no—
282#41—21no—
283#41—41no—
284#41—41no—
285#41—41no—
286#41—41no—
287#41#21312yes—
288#41#21412yes—
289C2×Q8—21no—
290C2×Q8#21711yescharacteristic
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—21no—
295C2×D4—41no—
296C2×D4#21711yescharacteristic
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—41no—
301C22×C4—121no—
302C22×C4D3×D411yescharacteristic
303C22×C4#23711yescharacteristic
304C22×C4C2×#6711yescharacteristic
305Q16—42no—
306SD16—42no—
307SD16—42no—
308D8—42no—
309M16—121no—
310M16—121no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—61no—
316C2×C8—61no—
317C2×C8—62no—
318C2×C8—122no—
319C2×C8—124no—
320C2×C8—121no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—62no—
326C4⋊C4—121no—
327C4⋊C4#21711yescharacteristic
328C4⋊C4#21711yescharacteristic
329C22⋊C4—21no—
330C22⋊C4—21no—
331C22⋊C4—21no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C42—21no—
335C42—21no—
336C42—21no—
337C42—121no—
338C42—121no—
339C42D3×D411yescharacteristic
340C42#23711yescharacteristic
341C42C22×Dic311yescharacteristic
342C42C2×#6711yescharacteristic
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3—81no—
349C22×C3—81no—
350C22×C3—81no—
351C22×C3#38611yescharacteristic
352C22×C3#46511yescharacteristic
353C22×C3#46611yescharacteristic
354C3×C4—21no—
355C3×C4—22no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—81no—
361C3×C4C2×#10211yescharacteristic
362C3×C4#41611yescharacteristic
363C3×C4#44311yescharacteristic
364C3×C4#44411yescharacteristic
365C3×C4#46511yescharacteristic
366C3×C4#46611yescharacteristic
367Dic3—82no—
368C23—21no—
369C23—41no—
370C23#93911yescharacteristic
371Q8—21no—
372Q8—41no—
373Q8—41no—
374Q8#83212yes—
375D4—22no—
376D4—22no—
377D4—21no—
378D4—41no—
379D4—41no—
380D4—41no—
381D4—41no—
382D4—81no—
383D4—81no—
384D4—81no—
385D4#83212yes—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—21no—
389C2×C4—22no—
390C2×C4—22no—
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—41no—
394C2×C4—42no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—81no—
399C2×C4—81no—
400C2×C4—122no—
401C2×C4—241no—
402C2×C4—241no—
403C2×C4C2×#21311yescharacteristic
404C2×C4C2×#21411yescharacteristic
405C2×C4D4×Dic311yescharacteristic
406C2×C4D4×Dic311yescharacteristic
407C2×C4#93511yescharacteristic
408C2×C4#93711yescharacteristic
409C2×C4#93911yescharacteristic
410C2×C4#95011yescharacteristic
411C2×C4#95011yescharacteristic
412C2×C4C2×#21711yescharacteristic
413C8—41no—
414C8—41no—
415C8—62no—
416C8—121no—
417C8—122no—
418C8—241no—
419C2×C3—21no—
420C2×C3—21no—
421C2×C3—42no—
422C2×C3—161no—
423C2×C3#187611yescharacteristic
424C2×C3#189311yescharacteristic
425C2×C3#190611yescharacteristic
426C22—21no—
427C22—21no—
428C22—22no—
429C22—41no—
430C22—41no—
431C22—81no—
432C22—81no—
433C22—81no—
434C22#503811yescharacteristic
435C22#504111yescharacteristic
436C22#522011yescharacteristic
437C4—22no—
438C4—21no—
439C4—21no—
440C4—21no—
441C4—41no—
442C4—41no—
443C4—81no—
444C4—242no—
445C4#503811yescharacteristic
446C4#504111yescharacteristic
447C4#519611yescharacteristic
448C4#519711yescharacteristic
449C4C2×#83211yescharacteristic
450C4#523311yescharacteristic
451C3#920811yescharacteristic, complemented, Sylow
452C2—21no—
453C2—21no—
454C2—42no—
455C2—161no—
456C2#7168111yescharacteristic
457C2#7208711yescharacteristic
458C2#7211411yescharacteristic
subgroupG/H
centerC2×C4#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3035C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3035D3
hypocenterC3#9208
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3035

Complements

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

NH ≅ G/Ntype
2. #41×C3⋊C8C2semidirect
3. #71811C2semidirect
4. #71802C2semidirect
5. #71646C2semidirect
6. #67707C2semidirect
7. #67704C2semidirect
14. C3×#462C4semidirect
15. C3×#461C4semidirect
16. C3×#460C4semidirect
17. C3×#459C4semidirect
97. #462Dic3semidirect
98. #461Dic3semidirect
99. #460Dic3semidirect
100. #459Dic3semidirect
451. C3#9208semidirect

Maximal subgroups

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

subgroupindex
C3×#30352
#41×C3⋊C82
#718112
#718022
#716462
#677072
#677042
#92083