Finite Group Databasebeta

← #10696036

Subgroups — #10696036

Order 768.

598 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 95 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#82348C211yesmaximal, characteristic, complemented
2#82308C211yesmaximal, characteristic
3#79309C211yesmaximal, characteristic, complemented
4C3×#1486C211yesmaximal, characteristic, complemented
5#70513C211yesmaximal, characteristic, complemented
6#70500C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#10013—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×C4×Dic6C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5222—21no—
15#5217—21no—
16C4×#217—22no—
17C2×#825—22no—
18#5015—21no—
19C4×#213—22no—
20#5007—21no—
21#4985—21no—
22#4967—21no—
23#4953—21no—
24#4947—21no—
25#4932C2211yescharacteristic
26#4925C2211yescharacteristic
27#4851—21no—
28#4848—21no—
29#4738—21no—
30#4735—21no—
31C4×#204—22no—
32C3×D4⋊C8—22no—
33#4484—22no—
34#3103—31no—
35#3100—31no—
36#2398—31no—
37#1963—31no—
38#1487—31no—
39#1486D311yescharacteristic, complemented
40#1274—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×#235—21no—
44C22×Dic6D411yescharacteristic
45C2×C3×M16—21no—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#217—22no—
54C2×#217—22no—
55C2×#213—22no—
56#930—21no—
57#930—21no—
58C2×C4⋊Dic3D411yescharacteristic
59C2×Dic3⋊C4—21no—
60C2×C4×Dic3—21no—
61C2×C4×Dic3D412yes—
62C2×#204—22no—
63#875—21no—
64#874D411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67#869D411yescharacteristic
68C4×Dic6—21no—
69C4×Dic6—21no—
70C4×Dic6—21no—
71C4×Dic6—21no—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C4×C8D412yes—
77#832—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82#830—21no—
83#825—44no—
84#821—22no—
85#817—22no—
86#816—22no—
87#809—22no—
88#807—22no—
89C4⋊(C3⋊C8)—21no—
90C4×C3⋊C8D412yes—
91#532—31no—
92C2×C4×Q8—31no—
93C2×C4×D4C2×D311yescharacteristic
94#481—61no—
95#480—61no—
96#475—61no—
97#474—61no—
98#460—61no—
99#459—61no—
100#448—61no—
101#447—61no—
102C4×SD16—62no—
103C4×D8—62no—
104#431—31no—
105#431C2×D311yescharacteristic
106#423C2×D311yescharacteristic
107C2×#104—62no—
108#400—61no—
109#397—61no—
110#394—61no—
111#393—61no—
112C4×C22⋊C4—62no—
113#376—31no—
114Q8⋊C8—62no—
115D4⋊C8—22no—
116C24×C3—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—42no—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—42no—
124C22×C3×C4—42no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4C2×D411yescharacteristic
128C2×#67—41no—
129C22×Dic3—21no—
130C22×Dic3—22no—
131C22×Dic3—21no—
132#235—42no—
133#235—42no—
134C2×D12—21no—
135C2×D12—21no—
136C2×D12—21no—
137C2×C4×D3—41no—
138C2×Dic6—21no—
139C2×Dic6—21no—
140C2×Dic6—21no—
141C2×Dic6—42no—
142C3×M16—42no—
143C2×C3×C8—22no—
144C2×C3×C8—41no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42C2×D411yescharacteristic
152C3×C42#4111yescharacteristic
153#217—44no—
154#217—44no—
155#213—44no—
156(C2×D3)⋊C4—41no—
157(C2×D3)⋊C4—41no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162Dic3⋊C4—41no—
163Dic3⋊C4—41no—
164C4×Dic3—22no—
165C4×Dic3—22no—
166C4×Dic3—41no—
167C4×Dic3—41no—
168C2×C3⋊C8—22no—
169#204—44no—
170C2×#41—61no—
171C22×Q8—31no—
172C22×D4D1211yescharacteristic
173C23×C4—21no—
174C2×SD16—62no—
175C2×D8—62no—
176C2×M16—21no—
177C4⋊Q8—31no—
178#127—31no—
179#124—61no—
180#121—61no—
181#121—61no—
182C4×Q8—61no—
183C4×Q8—61no—
184C4×Q8—61no—
185C4×Q8—61no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—61no—
191C4×D4—61no—
192C2×C4⋊C4—31no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4D1211yescharacteristic
195C2×C22⋊C4—21no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—62no—
198C2×C42—32no—
199C2×C42—61no—
200C2×C42C22×D311yescharacteristic
201#107—62no—
202#106—62no—
203C4⋊C8—21no—
204C4⋊C8—21no—
205C4⋊C8—21no—
206C4⋊C8—61no—
207#104—124no—
208#103—62no—
209#102—62no—
210#102—62no—
211#102—62no—
212C22⋊C8—21no—
213C22⋊C8—61no—
214C4×C8—32no—
215C4×C8#6712yes—
216#95—62no—
217#95—62no—
218#95—61no—
219#95—61no—
220C23×C3—22no—
221C23×C3—42no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—42no—
225C23×C3#12011yescharacteristic
226C22×D3—41no—
227C3×D4—22no—
228C3×D4—22no—
229C3×D4—41no—
230C3×D4—41no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—42no—
235C2×C3×C4—82no—
236C2×C3×C4—81no—
237C2×C3×C4—81no—
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12011yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12111yescharacteristic
243C2×C3×C4#12711yescharacteristic
244C2×C3×C4#13512yes—
245C2×C3×C4AGL(1,ℤ/8)12yes—
246#67—82no—
247C2×Dic3—22no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253C2×Dic3—41no—
254C2×Dic3—41no—
255C2×Dic3—42no—
256D12—42no—
257D12—42no—
258D12—42no—
259C4×D3—81no—
260C4×D3—81no—
261Dic6—22no—
262Dic6—22no—
263Dic6—41no—
264Dic6—41no—
265Dic6—42no—
266C3×C8—42no—
267C3×C8—81no—
268C3⋊C8—42no—
269C24—21no—
270#41—122no—
271#41—122no—
272C2×Q8—61no—
273C2×Q8—61no—
274C2×Q8—61no—
275C2×Q8—122no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—42no—
279C2×D4—61no—
280C2×D4—61no—
281C2×D4—61no—
282C2×D4—121no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—41no—
288C22×C4—61no—
289C22×C4—62no—
290C22×C4—61no—
291C22×C4—121no—
292C22×C4C2×D1211yescharacteristic
293C22×C4D3×D411yescharacteristic
294C22×C4D3×D411yescharacteristic
295SD16—124no—
296D8—124no—
297M16—42no—
298C2×C8—22no—
299C2×C8—41no—
300C2×C8—62no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—124no—
312C22⋊C4—124no—
313C22⋊C4—121no—
314C22⋊C4—121no—
315C42—62no—
316C42—62no—
317C42—121no—
318C42—121no—
319C42#23511yescharacteristic
320C42D3×D411yescharacteristic
321C42D3×D411yescharacteristic
322C42C2×#6711yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3—44no—
326C22×C3—42no—
327C22×C3—82no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3#39211yescharacteristic
331C22×C3#44912yes—
332C2×D3—81no—
333C2×D3—81no—
334C2×D3—81no—
335C3×C4—21no—
336C3×C4—21no—
337C3×C4—21no—
338C3×C4—21no—
339C3×C4—81no—
340C3×C4#46312yes—
341C3×C4#49511yescharacteristic
342C3×C4#49611yescharacteristic
343Dic3—42no—
344Dic3—82no—
345Dic3—81no—
346Dic3—81no—
347C23—22no—
348C23—42no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23—121no—
353C23#88211yescharacteristic
354Q8—62no—
355Q8—62no—
356Q8—121no—
357Q8—121no—
358Q8—122no—
359D4—22no—
360D4—22no—
361D4—41no—
362D4—41no—
363D4—122no—
364D4—122no—
365D4—122no—
366D4—242no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—62no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—82no—
375C2×C4—121no—
376C2×C4—121no—
377C2×C4—122no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—121no—
381C2×C4—121no—
382C2×C4—122no—
383C2×C4—241no—
384C2×C4—241no—
385C2×C4#88211yescharacteristic
386C2×C4#88411yescharacteristic
387C2×C4#88411yescharacteristic
388C2×C4#91112yes—
389C2×C4#91612yes—
390C2×C4#93011yescharacteristic
391C2×C4#93711yescharacteristic
392C2×C4#94011yescharacteristic
393C8—42no—
394C8—81no—
395C8—122no—
396C2×C3—22no—
397C2×C3—82no—
398C2×C3#201011yescharacteristic
399C2×C3#201011yescharacteristic
400C2×C3#201211yescharacteristic
401D3—161no—
402C22—21no—
403C22—21no—
404C22—44no—
405C22—42no—
406C22—82no—
407C22—81no—
408C22—81no—
409C22—241no—
410C22—241no—
411C22—241no—
412C22#478112yes—
413C22#496011yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—81no—
419C4—122no—
420C4—242no—
421C4—241no—
422C4—241no—
423C4#478712yes—
424C4#515411yescharacteristic
425C4#516811yescharacteristic
426C3#1001311yescharacteristic, complemented, Sylow
427C2—22no—
428C2—82no—
429C2—481no—
430C2#6999311yescharacteristic
431C2#7021411yescharacteristic
432C2#7040611yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1486C2
socleC22×C3#392
prime radical#82308C2
hypercenter#1486D3
hypocenterC3#10013
LCSG ▷ C2×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. #82348C2semidirect
3. #79309C2semidirect
4. C3×#1486C2semidirect
5. #70513C2semidirect
6. #70500C2semidirect
7. #67737C2semidirect
39. #1486D3semidirect
426. C3#10013semidirect

Maximal subgroups

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

subgroupindex
#823482
#823082
#793092
C3×#14862
#705132
#705002
#677372
#100133