Finite Group Databasebeta

← #10696034

Subgroups — #10696034

Order 768.

598 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 97 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
1C3×#2398C211yesmaximal, characteristic, complemented
2#82442C211yesmaximal, characteristic, complemented
3#82308C211yesmaximal, characteristic
4#71622C211yesmaximal, characteristic, complemented
5#71598C211yesmaximal, characteristic, complemented
6#69440C211yesmaximal, characteristic, complemented
7#67614C211yesmaximal, characteristic, complemented
8#10013—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C2×C3×C4×D4C2211yescharacteristic
11#5477C2211yescharacteristic
12C3×#400—21no—
13C3×#397—21no—
14C3×#394—21no—
15C3×#393—21no—
16C3×C4×C22⋊C4—22no—
17C3×#376C2211yescharacteristic
18#5227—21no—
19#5217—21no—
20#5027—21no—
21C4×#215—22no—
22#5020—21no—
23#5015—21no—
24C4×#213—22no—
25#5007—21no—
26#4968—21no—
27#4967—21no—
28C2×#805—22no—
29#4925C2211yescharacteristic
30#4925C2211yescharacteristic
31#4924C2211yescharacteristic
32#4535—22no—
33D4⋊(C3⋊C8)—22no—
34#3103—31no—
35#3100—31no—
36#2398D311yescharacteristic, complemented
37#1963—31no—
38#1487—31no—
39#1486—31no—
40#1274—31no—
41C22×C3×Q8D411yescharacteristic
42C22×C3×D4D411yescharacteristic
43C23×C3×C4—21no—
44C2×#235—21no—
45C3×C4×Q8—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×D4—21no—
50C3×C4×D4—21no—
51C3×C4×D4—21no—
52C3×C4×D4—21no—
53C2×C3×C4⋊C4—21no—
54C2×C3×C4⋊C4D411yescharacteristic
55C2×C3×C4⋊C4D411yescharacteristic
56C2×C3×C22⋊C4—22no—
57C2×C3×C22⋊C4—22no—
58C2×C3×C22⋊C4—21no—
59C2×C3×C42—21no—
60C2×C3×C42D412yes—
61C2×C3×C42C2311yescharacteristic
62C2×#215—22no—
63C2×#213—22no—
64#930—21no—
65#930—21no—
66C2×#208—21no—
67#875—21no—
68#874D411yescharacteristic
69C4×D12—21no—
70C4×D12—21no—
71#869D411yescharacteristic
72C3×#95—22no—
73C3×#95—22no—
74C3×#95—21no—
75C3×#95—21no—
76#835—22no—
77#832—22no—
78#830—21no—
79#830—21no—
80#809—22no—
81#809—22no—
82#808—22no—
83#807—22no—
84#805—44no—
85C4⋊(C3⋊C8)—21no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4×C3⋊C8D412yes—
90C4×C3⋊C8D412yes—
91#532—31no—
92C2×C4×Q8C2×D311yescharacteristic
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#431—31no—
106#423—31no—
107C2×#104—62no—
108#400—21no—
109#397—21no—
110#394—21no—
111#393—21no—
112C4×C22⋊C4—22no—
113#376C2×D311yescharacteristic
114Q8⋊C8—62no—
115D4⋊C8—62no—
116C24×C3—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—42no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—42no—
123C22×C3×C4—21no—
124C22×C3×C4—22no—
125C22×C3×C4—21no—
126C22×C3×C4—21no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—42no—
130C22×C3×C4—42no—
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×C3×C4C2×D411yescharacteristic
134C2×#67—41no—
135#235—42no—
136#235—42no—
137C2×D12—21no—
138C2×D12—21no—
139C2×D12—21no—
140C2×C4×D3—41no—
141C2×Dic6—21no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—41no—
147C3×C4⋊C4—41no—
148C3×C22⋊C4—44no—
149C3×C22⋊C4—44no—
150C3×C22⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C42—22no—
153C3×C42—22no—
154C3×C42—41no—
155C3×C42—41no—
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159C3×C42#4111yescharacteristic
160#215—44no—
161#213—44no—
162(C2×D3)⋊C4—41no—
163(C2×D3)⋊C4—41no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166#208—42no—
167C2×C3⋊C8—22no—
168C2×C3⋊C8—22no—
169C2×C3⋊C8—41no—
170C2×#41—61no—
171C22×Q8#6711yescharacteristic
172C22×D4#6711yescharacteristic
173C23×C4—21no—
174C2×SD16—62no—
175C2×D8—62no—
176C2×M16—61no—
177C4⋊Q8—31no—
178#127—31no—
179#124—61no—
180#121—61no—
181#121—61no—
182C4×Q8—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—61no—
191C4×D4—61no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4#6711yescharacteristic
194C2×C4⋊C4#6711yescharacteristic
195C2×C22⋊C4—22no—
196C2×C22⋊C4—22no—
197C2×C22⋊C4—21no—
198C2×C42—21no—
199C2×C42#6712yes—
200C2×C42C22×D311yescharacteristic
201#107—62no—
202#106—62no—
203C4⋊C8—61no—
204C4⋊C8—61no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207#104—124no—
208#103—62no—
209#102—62no—
210#102—62no—
211#102—62no—
212C22⋊C8—61no—
213C22⋊C8—61no—
214C4×C8—32no—
215C4×C8—32no—
216#95—22no—
217#95—22no—
218#95—21no—
219#95—21no—
220C23×C3—22no—
221C23×C3—42no—
222C23×C3—41no—
223C23×C3—41no—
224C23×C3—42no—
225C23×C3#12011yescharacteristic
226C22×D3—41no—
227C3×Q8—22no—
228C3×Q8—22no—
229C3×Q8—41no—
230C3×Q8—41no—
231C3×D4—22no—
232C3×D4—22no—
233C3×D4—41no—
234C3×D4—41no—
235C2×C3×C4—22no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—41no—
239C2×C3×C4—41no—
240C2×C3×C4—42no—
241C2×C3×C4—42no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4—41no—
245C2×C3×C4—42no—
246C2×C3×C4—42no—
247C2×C3×C4—82no—
248C2×C3×C4—81no—
249C2×C3×C4—81no—
250C2×C3×C4#12011yescharacteristic
251C2×C3×C4#12011yescharacteristic
252C2×C3×C4#12111yescharacteristic
253C2×C3×C4#12111yescharacteristic
254C2×C3×C4#12111yescharacteristic
255C2×C3×C4#12711yescharacteristic
256C2×C3×C4#13512yes—
257C2×C3×C4AGL(1,ℤ/8)12yes—
258#67—82no—
259C2×Dic3—41no—
260D12—42no—
261D12—42no—
262D12—42no—
263C4×D3—81no—
264C4×D3—81no—
265Dic6—42no—
266C3⋊C8—42no—
267C3⋊C8—42no—
268C3⋊C8—81no—
269C24—21no—
270#41—122no—
271#41—122no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—42no—
275C2×Q8—61no—
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—22no—
285C22×C4—21no—
286C22×C4—21no—
287C22×C4—41no—
288C22×C4—41no—
289C22×C4—42no—
290C22×C4—42no—
291C22×C4—121no—
292C22×C4C2×#6711yescharacteristic
293C22×C4C2×#6711yescharacteristic
294C22×C4C2×#6711yescharacteristic
295SD16—124no—
296D8—124no—
297M16—122no—
298C2×C8—62no—
299C2×C8—62no—
300C2×C8—121no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—41no—
306C4⋊C4—41no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C22⋊C4—44no—
310C22⋊C4—44no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—121no—
314C22⋊C4—121no—
315C42—22no—
316C42—22no—
317C42—41no—
318C42—41no—
319C42D3×D411yescharacteristic
320C42D3×D411yescharacteristic
321C42D3×D411yescharacteristic
322C42#23711yescharacteristic
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—42no—
340C3×C4—82no—
341C3×C4—81no—
342C3×C4—81no—
343C3×C4#46312yes—
344C3×C4#49511yescharacteristic
345C3×C4#49611yescharacteristic
346Dic3—81no—
347C23—22no—
348C23—42no—
349C23—41no—
350C23—41no—
351C23—42no—
352C23—121no—
353C23#95311yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—41no—
357Q8—41no—
358Q8—122no—
359D4—22no—
360D4—22no—
361D4—41no—
362D4—41no—
363D4—122no—
364D4—122no—
365D4—122no—
366D4—242no—
367C2×C4—22no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—42no—
379C2×C4—82no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—121no—
383C2×C4—241no—
384C2×C4—241no—
385C2×C4#93711yescharacteristic
386C2×C4#93711yescharacteristic
387C2×C4#93811yescharacteristic
388C2×C4#93911yescharacteristic
389C2×C4#93911yescharacteristic
390C2×C4#94011yescharacteristic
391C2×C4#94912yes—
392C2×C4#95012yes—
393C8—122no—
394C8—122no—
395C8—241no—
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#522311yescharacteristic
413C22#523812yes—
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—42no—
419C4—82no—
420C4—81no—
421C4—81no—
422C4—241no—
423C4#505712yes—
424C4#505811yescharacteristic
425C4#505911yescharacteristic
426C3#1001311yescharacteristic, complemented, Sylow
427C2—22no—
428C2—82no—
429C2—481no—
430C2#7171511yescharacteristic
431C2#7171511yescharacteristic
432C2#7172811yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#2398C2
socleC22×C3#392
prime radical#82308C2
hypercenter#2398D3
hypocenterC3#10013
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2398

Complements

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

NH ≅ G/Ntype
1. C3×#2398C2semidirect
2. #82442C2semidirect
4. #71622C2semidirect
5. #71598C2semidirect
6. #69440C2semidirect
7. #67614C2semidirect
36. #2398D3semidirect
426. C3#10013semidirect

Maximal subgroups

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

subgroupindex
C3×#23982
#824422
#823082
#716222
#715982
#694402
#676142
#100133