Finite Group Databasebeta

← #10691478

Subgroups — #10691478

Order 768.

588 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82319C211yesmaximal, characteristic, complemented
2#82308C211yesmaximal, characteristic
3C3×#1486C211yesmaximal, characteristic, complemented
4#70789C211yesmaximal, characteristic, complemented
5#70512C211yesmaximal, characteristic, complemented
6#67736C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#9237—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10C2×#926—21no—
11#5477C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5221—21no—
15#5217—21no—
16C4×#217—21no—
17C2×#825—21no—
18C2×#825—21no—
19#5015C412yescomplemented
20C4×#213—22no—
21#5007C412yescomplemented
22#4984—21no—
23#4967—21no—
24#4951—22no—
25C4×C4⋊Dic3C2211yescharacteristic
26C42×Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28C3×D4⋊C8—22no—
29#4470—22no—
30#3100—31no—
31#2293—31no—
32#1962—31no—
33#1502—31no—
34#1486D311yescharacteristic, complemented
35#1303—31no—
36#1276—31no—
37C22×C3×D4D411yescharacteristic
38C23×C3×C4—21no—
39C2×#235—21no—
40C2×C3×M16—21no—
41C3×C4×D4—21no—
42C3×C4×D4—21no—
43C3×C4×D4C2×C412yes—
44C3×C4×D4D412yes—
45C2×C3×C4⋊C4D411yescharacteristic
46C2×C3×C22⋊C4—21no—
47C2×C3×C42C2311yescharacteristic
48C2×#217—22no—
49C2×#217—22no—
50C2×#213—22no—
51#930—21no—
52#930—21no—
53#926—42no—
54#926—42no—
55C2×C4⋊Dic3—22no—
56C2×C4⋊Dic3—21no—
57C2×C4×Dic3—21no—
58C2×C4×Dic3—21no—
59C2×C4×Dic3—21no—
60C2×C4×Dic3D411yescharacteristic
61C2×C4×Dic3D411yescharacteristic
62#875—21no—
63#874C2×C411yescharacteristic
64C4×D12—21no—
65C4×D12—21no—
66#869C2×C411yescharacteristic
67C3×C4⋊C8—21no—
68C3×C4⋊C8—21no—
69C3×C4⋊C8—21no—
70C3×C22⋊C8—21no—
71C3×C4×C8—21no—
72#832—22no—
73C3⋊#95—22no—
74C3⋊#95D412yes—
75#830—21no—
76#825—42no—
77#825—42no—
78#809—22no—
79#807—22no—
80C4⋊(C3⋊C8)C2×C412yes—
81C4×C3⋊C8—21no—
82#532—31no—
83C2×C4×D4C2×D311yescharacteristic
84C2×#117—61no—
85#480—61no—
86#474—32no—
87#459—32no—
88#447—61no—
89C4×D8—62no—
90#431—31no—
91#431C2×D311yescharacteristic
92#423C2×D311yescharacteristic
93C2×#104—61no—
94C2×#104—61no—
95#390—61no—
96#389—61no—
97#388—62no—
98C4×C4⋊C4—31no—
99C4×C22⋊C4—61no—
100C43—31no—
101#327—62no—
102D4⋊C8—22no—
103C24×C3—21no—
104C2×C3×D4—21no—
105C2×C3×D4—42no—
106C2×C3×D4C22⋊C412yes—
107C22×C3×C4—21no—
108C22×C3×C4—22no—
109C22×C3×C4—22no—
110C22×C3×C4—41no—
111C22×C3×C4—41no—
112C22×C3×C4—41no—
113C22×C3×C4—41no—
114C22×C3×C4C2×D411yescharacteristic
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C2×#67—41no—
118C22×Dic3—21no—
119C22×Dic3—21no—
120C22×Dic3—22no—
121#235—41no—
122#235—41no—
123#235—41no—
124#235—41no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×C4×D3—41no—
129C2×Dic6—21no—
130C3×M16—41no—
131C3×M16—41no—
132C2×C3×C8—41no—
133C2×C3×C8—41no—
134C3×C4⋊C4—21no—
135C3×C4⋊C4C22⋊C412yes—
136C3×C22⋊C4—22no—
137C3×C22⋊C4—22no—
138C3×C42C22×C411yescharacteristic
139C3×C42C2×D411yescharacteristic
140C3×C42#4111yescharacteristic
141C3×C42#4111yescharacteristic
142#217—44no—
143#217—44no—
144#213—44no—
145(C2×D3)⋊C4—41no—
146(C2×D3)⋊C4—41no—
147C4⋊Dic3—21no—
148C4⋊Dic3—21no—
149C4⋊Dic3—44no—
150C4⋊Dic3—42no—
151C4×Dic3—21no—
152C4×Dic3—21no—
153C4×Dic3—21no—
154C4×Dic3—42no—
155C4×Dic3—42no—
156C4×Dic3—41no—
157C4×Dic3—41no—
158C4×Dic3D812yescomplemented
159C2×C3⋊C8—22no—
160C2×#41—61no—
161C22×D4D1211yescharacteristic
162C23×C4—21no—
163C2×D8—62no—
164C2×M16—21no—
165C4⋊Q8—31no—
166#127—31no—
167#124—61no—
168#121—61no—
169#121—61no—
170C4×D4—21no—
171C4×D4—21no—
172C4×D4—61no—
173C4×D4—61no—
174C4×D4C4×D312yes—
175C4×D4#6712yes—
176#117—122no—
177#117—122no—
178C2×C4⋊C4—62no—
179C2×C4⋊C4—61no—
180C2×C4⋊C4D1211yescharacteristic
181C2×C22⋊C4—21no—
182C2×C22⋊C4—62no—
183C2×C22⋊C4—62no—
184C2×C42—31no—
185C2×C42—31no—
186C2×C42—61no—
187C2×C42—61no—
188C2×C42—61no—
189C2×C42C22×D311yescharacteristic
190#107—62no—
191C4⋊C8—21no—
192C4⋊C8—21no—
193C4⋊C8—21no—
194C4⋊C8—32no—
195#104—122no—
196#104—122no—
197#102—62no—
198#102—62no—
199C22⋊C8—21no—
200C22⋊C8—61no—
201C4×C8—21no—
202C4×C8—61no—
203#95—32no—
204#95—62no—
205C23×C3—22no—
206C23×C3—42no—
207C23×C3—42no—
208C23×C3—41no—
209C23×C3—41no—
210C23×C3#12011yescharacteristic
211C22×D3—41no—
212C3×D4—22no—
213C3×D4—22no—
214C3×D4—42no—
215C2×C3×C4—22no—
216C2×C3×C4—22no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4—42no—
221C2×C3×C4—42no—
222C2×C3×C4—42no—
223C2×C3×C4—42no—
224C2×C3×C4C2×C22⋊C411yescharacteristic
225C2×C3×C4C4×D411yescharacteristic
226C2×C3×C4C4×D411yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4#12311yescharacteristic
230C2×C3×C4C2×D811yescharacteristic
231C2×C3×C4#13511yescharacteristic
232C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
233C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
234#67—81no—
235#67—81no—
236C2×Dic3—22no—
237C2×Dic3—22no—
238C2×Dic3—41no—
239C2×Dic3—41no—
240C2×Dic3—41no—
241C2×Dic3—41no—
242C2×Dic3—42no—
243C2×Dic3—42no—
244C2×Dic3—41no—
245C2×Dic3—42no—
246D12—42no—
247D12—41no—
248D12—41no—
249D12—81no—
250C4×D3—81no—
251C4×D3—81no—
252Dic6—41no—
253Dic6—41no—
254C3×C8—81no—
255C3×C8—81no—
256C3⋊C8—42no—
257C24—21no—
258#41—121no—
259#41—121no—
260#41—121no—
261#41—121no—
262C2×Q8—61no—
263C2×D4—21no—
264C2×D4—42no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—121no—
269C2×D4(C2×D3)⋊C412yes—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—41no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4—61no—
278C22×C4—61no—
279C22×C4—62no—
280C22×C4—121no—
281C22×C4C2×D1211yescharacteristic
282C22×C4D3×D411yescharacteristic
283C22×C4D3×D411yescharacteristic
284D8—124no—
285M16—41no—
286M16—41no—
287C2×C8—41no—
288C2×C8—41no—
289C2×C8—62no—
290C4⋊C4—21no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—122no—
294C4⋊C4—124no—
295C4⋊C4(C2×D3)⋊C412yes—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—124no—
299C22⋊C4—124no—
300C22⋊C4—121no—
301C22⋊C4—121no—
302C42—32no—
303C42—61no—
304C42—61no—
305C42—61no—
306C42—122no—
307C42—122no—
308C42—121no—
309C42—121no—
310C42C2×C4×D311yescharacteristic
311C42#23711yescharacteristic
312C42#23711yescharacteristic
313C42C2×#6711yescharacteristic
314C22×C3—21no—
315C22×C3—21no—
316C22×C3—42no—
317C22×C3—42no—
318C22×C3—42no—
319C22×C3—82no—
320C22×C3—82no—
321C22×C3#38611yescharacteristic
322C22×C3#44711yescharacteristic
323C22×C3#44911yescharacteristic
324C2×D3—81no—
325C2×D3—81no—
326C2×D3—81no—
327C3×C4—21no—
328C3×C4—21no—
329C3×C4—21no—
330C3×C4—42no—
331C3×C4#42112yes—
332C3×C4C4×D811yescharacteristic
333C3×C4#44211yescharacteristic
334C3×C4#45911yescharacteristic
335C3×C4#46311yescharacteristic
336Dic3—42no—
337Dic3—42no—
338Dic3—82no—
339Dic3—81no—
340C23—22no—
341C23—42no—
342C23—42no—
343C23—41no—
344C23—41no—
345C23—121no—
346C23#88211yescharacteristic
347Q8—121no—
348Q8—121no—
349D4—22no—
350D4—22no—
351D4—42no—
352D4—122no—
353D4—121no—
354D4—121no—
355D4—241no—
356D4—241no—
357D4—241no—
358C2×C4—22no—
359C2×C4—22no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—42no—
367C2×C4—62no—
368C2×C4—62no—
369C2×C4—121no—
370C2×C4—121no—
371C2×C4—121no—
372C2×C4—121no—
373C2×C4—122no—
374C2×C4—122no—
375C2×C4—121no—
376C2×C4—122no—
377C2×C4—241no—
378C2×C4—241no—
379C2×C4#67⋊C411yescharacteristic
380C2×C4#67⋊C411yescharacteristic
381C2×C4#88611yescharacteristic
382C2×C4D3×D811yescharacteristic
383C2×C4#91111yescharacteristic
384C2×C4#91411yescharacteristic
385C2×C4#91611yescharacteristic
386C2×C4C2×(C2×D3)⋊C411yescharacteristic
387C2×C4#93911yescharacteristic
388C2×C4#93911yescharacteristic
389C8—81no—
390C8—81no—
391C8—122no—
392C2×C3—21no—
393C2×C3—21no—
394C2×C3—82no—
395C2×C3#187911yescharacteristic
396C2×C3#189711yescharacteristic
397C2×C3#190611yescharacteristic
398D3—161no—
399C22—21no—
400C22—21no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—82no—
405C22—82no—
406C22—241no—
407C22—241no—
408C22—241no—
409C22#477311yescharacteristic
410C22#478111yescharacteristic
411C22#495811yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—122no—
417C4—122no—
418C4—242no—
419C4—241no—
420C4#475611yescharacteristic
421C4#478511yescharacteristic
422C4#513912yes—
423C4#515011yescharacteristic
424C4#516511yescharacteristic
425C3#923711yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—82no—
429C2—481no—
430C2#6996611yescharacteristic
431C2#7017211yescharacteristic
432C2#7039511yescharacteristic
subgroupG/H
centerC2×C4#939
commutatorC3×C42C22×C4
FrattiniC2×C42C22×D3
FittingC3×#1486C2
socleC22×C3#386
prime radical#82308C2
cosocleC2×C3×C42C23
hypercenter#1486D3
hypocenterC3#9237
LCSG ▷ C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1486

Complements

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

NH ≅ G/Ntype
1. #82319C2semidirect
3. C3×#1486C2semidirect
4. #70789C2semidirect
5. #70512C2semidirect
6. #67736C2semidirect
7. #67715C2semidirect
19. #5015C4semidirect
21. #5007C4semidirect
34. #1486D3semidirect
158. C4×Dic3D8semidirect
425. C3#9237semidirect

Maximal subgroups

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

subgroupindex
#823192
#823082
C3×#14862
#707892
#705122
#677362
#677152
#92373