Finite Group Databasebeta

← #10690076

Subgroups — #10690076

Order 768.

530 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#3035C211yesmaximal, characteristic
2#82512C211yesmaximal, characteristic, complemented
3#71802C211yesmaximal, characteristic, complemented
4#71800C211yesmaximal, characteristic, complemented
5#71638C211yesmaximal, characteristic, complemented
6#67707C211yesmaximal, characteristic, complemented
7#67703C211yesmaximal, characteristic, complemented
8#8995—31nomaximal, Sylow
9C2×C3×#135—21no—
10C3×#532C2211yescharacteristic
11C3×C4×#41C2211yescharacteristic
12C2×#948—21no—
13C3×#464—21no—
14C3×#463—21no—
15C3×#462C411yescharacteristic
16C3×#461C411yescharacteristic
17C3×#460C411yescharacteristic
18C3×#459C411yescharacteristic
19C2×C3×C4⋊C8C2211yescharacteristic
20C3×#416—21no—
21#4999—21no—
22#4999—22no—
23#4925C2211yescharacteristic
24#4924—21no—
25#4924C2211yescharacteristic
26C2×C4⋊(C3⋊C8)—21no—
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28C2×C4×C3⋊C8C2211yescharacteristic
29#4535—21no—
30D4⋊(C3⋊C8)—21no—
31D4⋊(C3⋊C8)—21no—
32D4⋊(C3⋊C8)—21no—
33D4⋊(C3⋊C8)—22no—
34#3035D311yescharacteristic
35#2906—31no—
36#1492—31no—
37#1480—31no—
38#1476—31no—
39#1301—31no—
40#1277—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
67#948—42no—
68#948—42no—
69C2×#208—22no—
70C2×#208—21no—
71C22×C3⋊C8—22no—
72C22×C3⋊C8—21no—
73C22×C3⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8C2×C411yescharacteristic
76C3×C4⋊C8C2×C411yescharacteristic
77C3×#103—21no—
78C3×#103—21no—
79C3×#102—21no—
80C3×#102—21no—
81#830—21no—
82C4⋊(C3⋊C8)—22no—
83C4⋊(C3⋊C8)—22no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D411yescharacteristic
88C4⋊(C3⋊C8)D411yescharacteristic
89C4⋊(C3⋊C8)D412yes—
90C4×C3⋊C8—21no—
91C4×C3⋊C8—21no—
92C4×C3⋊C8—21no—
93C2×#135—21no—
94C2×#131—61no—
95#532C2×D311yescharacteristic
96C4×#41C2×D311yescharacteristic
97#464—21no—
98#463—21no—
99#462Dic311yescharacteristic
100#461Dic311yescharacteristic
101#460Dic311yescharacteristic
102#459Dic311yescharacteristic
103#431—31no—
104#424—62no—
105#424—61no—
106#423—31no—
107#423—61no—
108C2×C4⋊C8—31no—
109C2×C4⋊C8—61no—
110C2×C4⋊C8C2×D311yescharacteristic
111#416—21no—
112C2×C4×C8—31no—
113Q8⋊C8—61no—
114D4⋊C8—61no—
115D4⋊C8—61no—
116D4⋊C8—62no—
117D4⋊C8—61no—
118C3×#41—21no—
119C3×#41—21no—
120C3×#41—41no—
121C3×#41—41no—
122C3×#41—41no—
123C3×#41—41no—
124C3×#41D812yes—
125C3×#41SD1612yes—
126C2×C3×Q8—21no—
127C2×C3×Q8C22⋊C411yescharacteristic
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—21no—
132C2×C3×D4—41no—
133C2×C3×D4C22⋊C411yescharacteristic
134C22×C3×C4—21no—
135C22×C3×C4—21no—
136C22×C3×C4—21no—
137C22×C3×C4—41no—
138C22×C3×C4C2×D411yescharacteristic
139C22×C3×C4#4111yescharacteristic
140C22×C3×C4#4111yescharacteristic
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×C42C2×D411yescharacteristic
167#208—44no—
168#208—41no—
169#208—41no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—21no—
174C2×C3⋊C8—41no—
175C2×C3⋊C8—44no—
176C2×C3⋊C8—41no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—41no—
179C2×C3⋊C8—41no—
180C2×#41—21no—
181C2×#41#6711yescharacteristic
182#135—42no—
183#135—42no—
184C2×Q16—21no—
185C2×SD16—21no—
186C2×SD16—21no—
187C2×D8—21no—
188#131—122no—
189#131—122no—
190C2×M16—62no—
191C2×M16—61no—
192C22×C8—21no—
193C22×C8—62no—
194C22×C8—61no—
195C22×C8—61no—
196C4⋊Q8C2×Dic311yescharacteristic
197#127C2×Dic311yescharacteristic
198#124—21no—
199#121—21no—
200#121—21no—
201C4×Q8C2×Dic311yescharacteristic
202C4×Q8#6711yescharacteristic
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—21no—
206C4×D4—21no—
207C4×D4C2×Dic311yescharacteristic
208C4×D4#6711yescharacteristic
209#117—21no—
210#117#6711yescharacteristic
211C2×C42—21no—
212C2×C42C22×D311yescharacteristic
213C4⋊C8—21no—
214C4⋊C8—31no—
215C4⋊C8—31no—
216C4⋊C8—32no—
217C4⋊C8—62no—
218C4⋊C8—62no—
219C4⋊C8—61no—
220C4⋊C8—61no—
221C4⋊C8—61no—
222C4⋊C8C2×Dic311yescharacteristic
223C4⋊C8C2×Dic311yescharacteristic
224#103—21no—
225#103—21no—
226#102—21no—
227#102—21no—
228C22⋊C8—61no—
229C4×C8—61no—
230C4×C8—61no—
231C4×C8—61no—
232C23×C3—21no—
233C23×C3—41no—
234C23×C3#12311yescharacteristic
235C3×Q8—21no—
236C3×Q8—41no—
237C3×Q8—41no—
238C3×Q8#10212yes—
239C3×D4—22no—
240C3×D4—22no—
241C3×D4—21no—
242C3×D4—42no—
243C3×D4—41no—
244C3×D4—41no—
245C3×D4—41no—
246C3×D4—41no—
247C3×D4—81no—
248C3×D4—81no—
249C3×D4#10212yes—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—21no—
253C2×C3×C4—22no—
254C2×C3×C4—22no—
255C2×C3×C4—21no—
256C2×C3×C4—21no—
257C2×C3×C4—41no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—81no—
262C2×C3×C4—81no—
263C2×C3×C4—81no—
264C2×C3×C4C2×C22⋊C411yescharacteristic
265C2×C3×C4C4×D411yescharacteristic
266C2×C3×C4C4×D411yescharacteristic
267C2×C3×C4#12011yescharacteristic
268C2×C3×C4#12111yescharacteristic
269C2×C3×C4#12111yescharacteristic
270C2×C3×C4C2×D811yescharacteristic
271C2×C3×C4C2×SD1611yescharacteristic
272C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
273C2×C3×C4#13711yescharacteristic
274C3×C8—41no—
275C3×C8—41no—
276C3⋊C8—42no—
277C3⋊C8—41no—
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#23711yescharacteristic
302C22×C4#23711yescharacteristic
303C22×C4C2×#6711yescharacteristic
304Q16—42no—
305SD16—42no—
306SD16—42no—
307D8—42no—
308M16—124no—
309M16—121no—
310M16—121no—
311C2×C8—21no—
312C2×C8—21no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—62no—
316C2×C8—61no—
317C2×C8—61no—
318C2×C8—62no—
319C2×C8—124no—
320C2×C8—121no—
321C2×C8—121no—
322C2×C8—121no—
323C2×C8—121no—
324C2×C8—121no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—21no—
329C4⋊C4#21711yescharacteristic
330C4⋊C4#21711yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42D3×D411yescharacteristic
340C42D3×D411yescharacteristic
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#48011yescharacteristic
353C22×C3#48111yescharacteristic
354C3×C4—22no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4—81no—
361C3×C4C2×#10211yescharacteristic
362C3×C4#41711yescharacteristic
363C3×C4#44311yescharacteristic
364C3×C4#44411yescharacteristic
365C3×C4#44711yescharacteristic
366C3×C4#44811yescharacteristic
367C23—21no—
368C23—41no—
369C23#93511yescharacteristic
370Q8—21no—
371Q8—41no—
372Q8—41no—
373Q8#83212yes—
374D4—22no—
375D4—22no—
376D4—21no—
377D4—42no—
378D4—41no—
379D4—41no—
380D4—41no—
381D4—41no—
382D4—81no—
383D4—81no—
384D4#83212yes—
385C2×C4—21no—
386C2×C4—21no—
387C2×C4—21no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—41no—
396C2×C4—81no—
397C2×C4—81no—
398C2×C4—81no—
399C2×C4C2×#21311yescharacteristic
400C2×C4C2×#21411yescharacteristic
401C2×C4D4×Dic311yescharacteristic
402C2×C4D4×Dic311yescharacteristic
403C2×C4#93711yescharacteristic
404C2×C4#93911yescharacteristic
405C2×C4#93911yescharacteristic
406C2×C4#94911yescharacteristic
407C2×C4#95111yescharacteristic
408C2×C4C2×#21711yescharacteristic
409C8—41no—
410C8—41no—
411C8—122no—
412C8—121no—
413C8—122no—
414C8—121no—
415C8—241no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3—42no—
419C2×C3—161no—
420C2×C3#187411yescharacteristic
421C2×C3#187611yescharacteristic
422C2×C3#188111yescharacteristic
423C22—21no—
424C22—21no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—81no—
429C22—81no—
430C22—81no—
431C22#503311yescharacteristic
432C22#503411yescharacteristic
433C22#522011yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—22no—
438C4—41no—
439C4—41no—
440C4—81no—
441C4#503611yescharacteristic
442C4#504011yescharacteristic
443C4#519611yescharacteristic
444C4#519711yescharacteristic
445C4C2×#83211yescharacteristic
446C4#523211yescharacteristic
447C3#899511yescharacteristic, complemented, Sylow
448C2—21no—
449C2—21no—
450C2—42no—
451C2—161no—
452C2#7167711yescharacteristic
453C2#7208711yescharacteristic
454C2#7211611yescharacteristic
subgroupG/H
centerC2×C4#937
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3035C2
socleC22×C3#386
cosocleC2×C3×C42C23
hypercenter#3035D3
hypocenterC3#8995
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. #82512C2semidirect
3. #71802C2semidirect
4. #71800C2semidirect
5. #71638C2semidirect
6. #67707C2semidirect
7. #67703C2semidirect
447. C3#8995semidirect

Maximal subgroups

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

subgroupindex
C3×#30352
#825122
#718022
#718002
#716382
#677072
#677032
#89953