Finite Group Databasebeta

← #10690072

Subgroups — #10690072

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
1#82525C211yesmaximal, characteristic
2#82512C211yesmaximal, characteristic, complemented
3C3×#1476C211yesmaximal, characteristic, complemented
4#70758C211yesmaximal, characteristic, complemented
5#69735C211yesmaximal, characteristic, complemented
6#67745C211yesmaximal, characteristic, complemented
7#67654C211yesmaximal, characteristic, complemented
8#8995—31nomaximal, Sylow
9C3×C4×#41C2211yescharacteristic
10C2×#950—21no—
11C2×#948—21no—
12#5477C2211yescharacteristic
13C3×#431C2211yescharacteristic
14C2×C3×C4×C8C2211yescharacteristic
15#5031C411yescharacteristic
16#5028—21no—
17#5027C411yescharacteristic
18#5019C411yescharacteristic
19#5016—21no—
20#5015C411yescharacteristic
21#5002—21no—
22#4999—21no—
23#4999—22no—
24#4924—21no—
25#4924C2211yescharacteristic
26C2×C4⋊(C3⋊C8)—21no—
27C2×C4⋊(C3⋊C8)C2211yescharacteristic
28C2×C4⋊(C3⋊C8)C2211yescharacteristic
29C3×Q8⋊C8—21no—
30C3×D4⋊C8—21no—
31#4459—21no—
32#4459—21no—
33#4459—22no—
34#3035—31no—
35#2906—31no—
36#1492—31no—
37#1480—31no—
38#1476D311yescharacteristic, complemented
39#1301—31no—
40#1277—31no—
41C2×C3×#41D411yescharacteristic
42C2×#235—21no—
43C22×C3×C8—21no—
44C3×C4×Q8C2×C411yescharacteristic
45C3×C4×Q8D411yescharacteristic
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4C2×C411yescharacteristic
49C3×C4×D4D411yescharacteristic
50C3×#117—21no—
51C3×#117D411yescharacteristic
52C2×C3×C42—21no—
53C2×C3×C42C2311yescharacteristic
54#950—42no—
55#950—42no—
56#948—42no—
57#948—42no—
58C2×#216—21no—
59C2×#215—21no—
60C2×#214—21no—
61C2×#213—21no—
62#930—21no—
63#930—21no—
64C2×#208—22no—
65C2×#208—21no—
66C22×C3⋊C8—22no—
67C22×C3⋊C8—21no—
68C22×C3⋊C8—21no—
69#875—21no—
70#874C2×C411yescharacteristic
71C4×D12—21no—
72C4×D12—21no—
73#869C2×C411yescharacteristic
74C3×C4⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C4×C8—21no—
77C3×C4×C8—21no—
78C3×C4×C8—21no—
79#810—21no—
80#810—21no—
81#809—21no—
82#809—21no—
83C4⋊(C3⋊C8)—22no—
84C4⋊(C3⋊C8)—22no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)C2×C411yescharacteristic
89C4⋊(C3⋊C8)C2×C411yescharacteristic
90C4⋊(C3⋊C8)D411yescharacteristic
91C4⋊(C3⋊C8)D411yescharacteristic
92C4⋊(C3⋊C8)D412yes—
93C2×#135—61no—
94C2×#131—61no—
95#532—31no—
96C4×#41C2×D311yescharacteristic
97#464—61no—
98#463—61no—
99#462—31no—
100#461—31no—
101#460—31no—
102#459—31no—
103#431C2×D311yescharacteristic
104#424—62no—
105#424—61no—
106#423—31no—
107#423—61no—
108C2×C4⋊C8—31no—
109C2×C4⋊C8—31no—
110C2×C4⋊C8—61no—
111#416—61no—
112C2×C4×C8C2×D311yescharacteristic
113Q8⋊C8—21no—
114D4⋊C8—21no—
115D4⋊C8—61no—
116D4⋊C8—61no—
117D4⋊C8—62no—
118C3×#41—21no—
119C3×#41—21no—
120C3×#41D812yes—
121C3×#41SD1612yes—
122C2×C3×Q8C22⋊C411yescharacteristic
123C2×C3×D4—21no—
124C2×C3×D4C22⋊C411yescharacteristic
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—21no—
128C22×C3×C4C2×D411yescharacteristic
129C22×C3×C4#4111yescharacteristic
130C22×C3×C4#4111yescharacteristic
131C2×#67—41no—
132#235—41no—
133#235—41no—
134#235—41no—
135#235—41no—
136C2×D12—21no—
137C2×D12—21no—
138C2×D12—21no—
139C2×C4×D3—41no—
140C2×Dic6—21no—
141C2×C3×C8—22no—
142C2×C3×C8—41no—
143C2×C3×C8—41no—
144C2×C3×C8—41no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4C22⋊C411yescharacteristic
148C3×C4⋊C4C22⋊C411yescharacteristic
149C3×C22⋊C4—21no—
150C3×C22⋊C4—21no—
151C3×C22⋊C4—21no—
152C3×C42—21no—
153C3×C42—21no—
154C3×C42—21no—
155C3×C42C22×C411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159#216—42no—
160#215—42no—
161#214—42no—
162#213—42no—
163(C2×D3)⋊C4—41no—
164(C2×D3)⋊C4—41no—
165C4⋊Dic3—21no—
166C4⋊Dic3—21no—
167#208—44no—
168#208—41no—
169#208—41no—
170C2×C3⋊C8—21no—
171C2×C3⋊C8—21no—
172C2×C3⋊C8—21no—
173C2×C3⋊C8—22no—
174C2×C3⋊C8—21no—
175C2×C3⋊C8—44no—
176C2×C3⋊C8—41no—
177C2×C3⋊C8—41no—
178C2×C3⋊C8—41no—
179C2×C3⋊C8—41no—
180C2×#41—61no—
181C2×#41D1211yescharacteristic
182#135—122no—
183#135—122no—
184C2×Q16—61no—
185C2×SD16—61no—
186C2×SD16—61no—
187C2×D8—61no—
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⋊Q8—31no—
197#127—31no—
198#124—61no—
199#121—61no—
200#121—61no—
201C4×Q8C4×D311yescharacteristic
202C4×Q8#6711yescharacteristic
203C4×D4—21no—
204C4×D4—21no—
205C4×D4—61no—
206C4×D4—61no—
207C4×D4C4×D311yescharacteristic
208C4×D4#6711yescharacteristic
209#117—21no—
210#117D1211yescharacteristic
211C2×C42—21no—
212C2×C42C22×D311yescharacteristic
213C4⋊C8—21no—
214C4⋊C8—31no—
215C4⋊C8—31no—
216C4⋊C8—31no—
217C4⋊C8—31no—
218C4⋊C8—32no—
219C4⋊C8—62no—
220C4⋊C8—62no—
221C4⋊C8—61no—
222C4⋊C8—61no—
223C4⋊C8—61no—
224#103—61no—
225#103—61no—
226#102—61no—
227#102—61no—
228C22⋊C8—21no—
229C4×C8—21no—
230C4×C8—21no—
231C4×C8—21no—
232C23×C3—21no—
233C23×C3#12311yescharacteristic
234C22×D3—41no—
235C3×Q8—21no—
236C3×Q8#10212yes—
237C3×D4—22no—
238C3×D4—22no—
239C3×D4—21no—
240C3×D4#10212yes—
241C2×C3×C4—21no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—22no—
245C2×C3×C4—22no—
246C2×C3×C4—21no—
247C2×C3×C4—21no—
248C2×C3×C4—41no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—81no—
252C2×C3×C4C2×C22⋊C411yescharacteristic
253C2×C3×C4C4×D411yescharacteristic
254C2×C3×C4C4×D411yescharacteristic
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4C2×D811yescharacteristic
259C2×C3×C4C2×SD1611yescharacteristic
260C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
261C2×C3×C4#13711yescharacteristic
262#67—81no—
263#67—81no—
264C2×Dic3—41no—
265D12—42no—
266D12—41no—
267D12—41no—
268D12—41no—
269D12—41no—
270C4×D3—81no—
271C4×D3—81no—
272Dic6—41no—
273Dic6—41no—
274C3×C8—42no—
275C3×C8—81no—
276C3⋊C8—41no—
277C3⋊C8—41no—
278C3⋊C8—41no—
279C3⋊C8—42no—
280C3⋊C8—41no—
281#41—21no—
282#41—21no—
283#41—121no—
284#41—121no—
285#41—121no—
286#41—121no—
287#41#20412yes—
288#41D2412yes—
289C2×Q8—61no—
290C2×Q8(C2×D3)⋊C411yescharacteristic
291C2×D4—21no—
292C2×D4—61no—
293C2×D4—61no—
294C2×D4—61no—
295C2×D4—121no—
296C2×D4(C2×D3)⋊C411yescharacteristic
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—21no—
300C22×C4—121no—
301C22×C4C2×D1211yescharacteristic
302C22×C4#23711yescharacteristic
303C22×C4#23711yescharacteristic
304Q16—122no—
305SD16—122no—
306SD16—122no—
307D8—122no—
308M16—124no—
309M16—121no—
310M16—121no—
311C2×C8—22no—
312C2×C8—41no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—61no—
316C2×C8—61no—
317C2×C8—62no—
318C2×C8—61no—
319C2×C8—61no—
320C2×C8—124no—
321C2×C8—121no—
322C2×C8—121no—
323C2×C8—121no—
324C2×C8—121no—
325C4⋊C4—21no—
326C4⋊C4—21no—
327C4⋊C4—61no—
328C4⋊C4—61no—
329C4⋊C4(C2×D3)⋊C411yescharacteristic
330C4⋊C4(C2×D3)⋊C411yescharacteristic
331C22⋊C4—21no—
332C22⋊C4—21no—
333C22⋊C4—21no—
334C22⋊C4—121no—
335C22⋊C4—121no—
336C42—21no—
337C42—21no—
338C42—21no—
339C42C2×C4×D311yescharacteristic
340C42D3×D411yescharacteristic
341C42D3×D411yescharacteristic
342C42C2×#6711yescharacteristic
343C22×C3—21no—
344C22×C3—21no—
345C22×C3—22no—
346C22×C3—41no—
347C22×C3—41no—
348C22×C3#38611yescharacteristic
349C22×C3#48011yescharacteristic
350C22×C3#48111yescharacteristic
351C2×D3—81no—
352C2×D3—81no—
353C2×D3—81no—
354C3×C4—22no—
355C3×C4—21no—
356C3×C4—21no—
357C3×C4—21no—
358C3×C4—41no—
359C3×C4—41no—
360C3×C4C2×#10211yescharacteristic
361C3×C4#41711yescharacteristic
362C3×C4#44311yescharacteristic
363C3×C4#44411yescharacteristic
364C3×C4#44711yescharacteristic
365C3×C4#44811yescharacteristic
366Dic3—81no—
367C23—21no—
368C23—121no—
369C23#88611yescharacteristic
370Q8—21no—
371Q8—121no—
372Q8—121no—
373Q8#82112yes—
374D4—22no—
375D4—22no—
376D4—21no—
377D4—122no—
378D4—121no—
379D4—121no—
380D4—121no—
381D4—121no—
382D4—241no—
383D4—241no—
384D4#82112yes—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—41no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—81no—
396C2×C4—121no—
397C2×C4—241no—
398C2×C4—241no—
399C2×C4#67⋊C411yescharacteristic
400C2×C4#67⋊C411yescharacteristic
401C2×C4#88211yescharacteristic
402C2×C4C2×#20411yescharacteristic
403C2×C4C2×D2411yescharacteristic
404C2×C4#90811yescharacteristic
405C2×C4#90911yescharacteristic
406C2×C4C2×(C2×D3)⋊C411yescharacteristic
407C2×C4#93911yescharacteristic
408C2×C4#93911yescharacteristic
409C8—42no—
410C8—81no—
411C8—121no—
412C8—122no—
413C8—121no—
414C8—121no—
415C8—121no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3—42no—
419C2×C3#187411yescharacteristic
420C2×C3#187611yescharacteristic
421C2×C3#188111yescharacteristic
422D3—161no—
423C22—21no—
424C22—21no—
425C22—22no—
426C22—41no—
427C22—41no—
428C22—241no—
429C22—241no—
430C22—241no—
431C22#473511yescharacteristic
432C22#473611yescharacteristic
433C22#495811yescharacteristic
434C4—22no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—41no—
439C4—41no—
440C4—241no—
441C4#473211yescharacteristic
442C4#473811yescharacteristic
443C4#482311yescharacteristic
444C4#482411yescharacteristic
445C4C2×#82111yescharacteristic
446C4#513311yescharacteristic
447C3#899511yescharacteristic, complemented, Sylow
448C2—21no—
449C2—21no—
450C2—42no—
451C2—481no—
452C2#6993211yescharacteristic
453C2#7029311yescharacteristic
454C2#7037711yescharacteristic
subgroupG/H
centerC2×C4#882
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1476C2
socleC22×C3#386
prime radical#82525C2
cosocleC2×C3×C42C23
hypercenter#1476D3
hypocenterC3#8995
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1476

Complements

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

NH ≅ G/Ntype
2. #82512C2semidirect
3. C3×#1476C2semidirect
4. #70758C2semidirect
5. #69735C2semidirect
6. #67745C2semidirect
7. #67654C2semidirect
38. #1476D3semidirect
447. C3#8995semidirect

Maximal subgroups

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

subgroupindex
#825252
#825122
C3×#14762
#707582
#697352
#677452
#676542
#89953