Finite Group Databasebeta

← #10696035

Subgroups — #10696035

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#82492C211yesmaximal, characteristic, complemented
2#82442C211yesmaximal, characteristic, complemented
3#79297C211yesmaximal, characteristic
4C3×#1487C211yesmaximal, characteristic, complemented
5#70513C211yesmaximal, characteristic, complemented
6#70496C211yesmaximal, characteristic, complemented
7#67737C211yesmaximal, characteristic, complemented
8#10013—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10#5477C2211yescharacteristic
11C2×C4×D12C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×#423C2211yescharacteristic
14#5231—21no—
15#5227—21no—
16C2×#825—22no—
17#5027—21no—
18C4×#215—22no—
19#5020—21no—
20#4992—21no—
21#4991—21no—
22#4986—21no—
23#4968—21no—
24C4×(C2×D3)⋊C4—22no—
25#4932C2211yescharacteristic
26#4925C2211yescharacteristic
27#4846—21no—
28#4843—21no—
29#4736—21no—
30#4732—21no—
31C4×D24—22no—
32C3×Q8⋊C8—22no—
33#4483—22no—
34#3103—31no—
35#3100—31no—
36#2398—31no—
37#1963—31no—
38#1487D311yescharacteristic, complemented
39#1486—31no—
40#1274—31no—
41C22×C3×Q8D411yescharacteristic
42C2×#235—21no—
43C22×D12D411yescharacteristic
44C22×C4×D3—21no—
45C2×C3×M16—21no—
46C3×C4×Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×C4×Q8—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C4⋊C4D411yescharacteristic
52C2×C3×C42—21no—
53C2×C3×C42C2311yescharacteristic
54C2×#215—22no—
55#930—21no—
56#930—21no—
57C2×(C2×D3)⋊C4—22no—
58C2×(C2×D3)⋊C4—22no—
59C2×(C2×D3)⋊C4—21no—
60C2×C4⋊Dic3D411yescharacteristic
61C2×C4×Dic3D412yes—
62C2×D24—22no—
63#875—21no—
64#874D411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—21no—
69C4×D12—21no—
70C4×D12—21no—
71#869D411yescharacteristic
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C4⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C4×C8D412yes—
77#835—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80C3⋊#95—21no—
81C3⋊#95—21no—
82#830—21no—
83#825—44no—
84#821—22no—
85#821—22no—
86#818—22no—
87#809—22no—
88#808—22no—
89C4⋊(C3⋊C8)—21no—
90C4×C3⋊C8D412yes—
91#532—31no—
92C2×C4×Q8C2×D311yescharacteristic
93C2×C4×D4—31no—
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—22no—
115D4⋊C8—62no—
116C23×D3—21no—
117C2×C3×Q8—21no—
118C2×C3×Q8—21no—
119C2×C3×Q8—42no—
120C22×C3×C4—21no—
121C22×C3×C4—21no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C2×#67—41no—
126C22×Dic3—22no—
127C22×Dic3—21no—
128#235—42no—
129#235—42no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12—21no—
135C2×D12—42no—
136C2×C4×D3—41no—
137C2×C4×D3—41no—
138C2×C4×D3—41no—
139C2×C4×D3—42no—
140C2×C4×D3—42no—
141C2×Dic6—21no—
142C3×M16—42no—
143C2×C3×C8—22no—
144C2×C3×C8—41no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—41no—
148C3×C4⋊C4—41no—
149C3×C42—41no—
150C3×C42—41no—
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42#4111yescharacteristic
155#215—44no—
156(C2×D3)⋊C4—44no—
157(C2×D3)⋊C4—44no—
158(C2×D3)⋊C4—41no—
159(C2×D3)⋊C4—41no—
160(C2×D3)⋊C4—41no—
161(C2×D3)⋊C4—41no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165C4⋊Dic3—21no—
166C4×Dic3—22no—
167C4×Dic3—22no—
168C2×C3⋊C8—22no—
169D24—44no—
170C2×#41—61no—
171C22×Q8D1211yescharacteristic
172C22×D4—31no—
173C23×C4—61no—
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—21no—
183C4×Q8—21no—
184C4×Q8—21no—
185C4×Q8—21no—
186C4×D4—61no—
187C4×D4—61no—
188C4×D4—61no—
189C4×D4—61no—
190C4×D4—61no—
191C4×D4—61no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4—31no—
194C2×C4⋊C4D1211yescharacteristic
195C2×C22⋊C4—62no—
196C2×C22⋊C4—62no—
197C2×C22⋊C4—61no—
198C2×C42—21no—
199C2×C42—32no—
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#12011yescharacteristic
221C22×D3—22no—
222C22×D3—42no—
223C22×D3—41no—
224C22×D3—41no—
225C22×D3—42no—
226C22×D3—41no—
227C3×Q8—22no—
228C3×Q8—22no—
229C3×Q8—41no—
230C3×Q8—41no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—41no—
234C2×C3×C4—41no—
235C2×C3×C4—41no—
236C2×C3×C4—42no—
237C2×C3×C4—41no—
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—42no—
248C2×Dic3—42no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C2×Dic3—42no—
252C2×Dic3—42no—
253D12—22no—
254D12—22no—
255D12—41no—
256D12—41no—
257D12—42no—
258D12—42no—
259D12—42no—
260C4×D3—82no—
261C4×D3—81no—
262C4×D3—81no—
263C4×D3—81no—
264C4×D3—81no—
265Dic6—42no—
266C3×C8—42no—
267C3×C8—81no—
268C3⋊C8—42no—
269C24—61no—
270#41—122no—
271#41—122no—
272C2×Q8—21no—
273C2×Q8—21no—
274C2×Q8—42no—
275C2×Q8—61no—
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—61no—
280C2×D4—61no—
281C2×D4—122no—
282C2×D4—121no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—62no—
286C22×C4—61no—
287C22×C4—121no—
288C22×C4—122no—
289C22×C4—122no—
290C22×C4—121no—
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—41no—
304C4⋊C4—41no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C22⋊C4—124no—
310C22⋊C4—124no—
311C22⋊C4—121no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C22⋊C4—121no—
315C42—41no—
316C42—41no—
317C42—62no—
318C42—62no—
319C42#23511yescharacteristic
320C42D3×D411yescharacteristic
321C42D3×D411yescharacteristic
322C42C2×#6711yescharacteristic
323C22×C3—21no—
324C22×C3—21no—
325C22×C3#39211yescharacteristic
326C22×C3#44912yes—
327C2×D3—44no—
328C2×D3—42no—
329C2×D3—82no—
330C2×D3—81no—
331C2×D3—81no—
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—81no—
341C3×C4#46312yes—
342C3×C4#49511yescharacteristic
343C3×C4#49611yescharacteristic
344Dic3—82no—
345Dic3—81no—
346Dic3—81no—
347C23—62no—
348C23—122no—
349C23—121no—
350C23—121no—
351C23—122no—
352C23—121no—
353C23#88211yescharacteristic
354Q8—22no—
355Q8—22no—
356Q8—41no—
357Q8—41no—
358Q8—122no—
359D4—62no—
360D4—62no—
361D4—121no—
362D4—121no—
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—41no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—121no—
377C2×C4—121no—
378C2×C4—122no—
379C2×C4—122no—
380C2×C4—241no—
381C2×C4—241no—
382C2×C4—241no—
383C2×C4—241no—
384C2×C4—242no—
385C2×C4#88211yescharacteristic
386C2×C4#88411yescharacteristic
387C2×C4#88411yescharacteristic
388C2×C4#91412yes—
389C2×C4#91912yes—
390C2×C4#93011yescharacteristic
391C2×C4#93711yescharacteristic
392C2×C4#94011yescharacteristic
393C8—42no—
394C8—81no—
395C8—122no—
396C2×C3—22no—
397C2×C3#201011yescharacteristic
398C2×C3#201011yescharacteristic
399C2×C3#201211yescharacteristic
400D3—82no—
401D3—161no—
402C22—21no—
403C22—21no—
404C22—124no—
405C22—122no—
406C22—242no—
407C22—241no—
408C22—241no—
409C22—241no—
410C22—241no—
411C22—241no—
412C22#481012yes—
413C22#496011yescharacteristic
414C4—21no—
415C4—21no—
416C4—21no—
417C4—21no—
418C4—42no—
419C4—81no—
420C4—242no—
421C4—241no—
422C4—241no—
423C4#481912yes—
424C4#517611yescharacteristic
425C4#519111yescharacteristic
426C3#1001311yescharacteristic, complemented, Sylow
427C2—22no—
428C2—242no—
429C2—481no—
430C2#6999311yescharacteristic
431C2#7027711yescharacteristic
432C2#7040611yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1487C2
socleC22×C3#392
prime radical#79297C2
hypercenter#1487D3
hypocenterC3#10013
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1487

Complements

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

NH ≅ G/Ntype
1. #82492C2semidirect
2. #82442C2semidirect
4. C3×#1487C2semidirect
5. #70513C2semidirect
6. #70496C2semidirect
7. #67737C2semidirect
38. #1487D3semidirect
426. C3#10013semidirect

Maximal subgroups

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

subgroupindex
#824922
#824422
#792972
C3×#14872
#705132
#704962
#677372
#100133