Finite Group Databasebeta

← #10697558

Subgroups — #10697558

Order 768.

658 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 105 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×#3036C211yesmaximal, characteristic
2#83317C211yesmaximal, characteristic, complemented
3#82379C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic, complemented
5#71597C211yesmaximal, characteristic, complemented
6#70507C211yesmaximal, characteristic, complemented
7#67715C211yesmaximal, characteristic, complemented
8#10240—31nomaximal, Sylow
9C2×C3×AGL(1,ℤ/8)—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×D4C2211yescharacteristic
12C2×#924—22no—
13C2×#869C2211yescharacteristic
14C3×#463—22no—
15C3×#461—21no—
16C3×#459—21no—
17C3×#423C2211yescharacteristic
18C3×#417—21no—
19#5221—21no—
20#5093—21no—
21#5076—21no—
22#5040—22no—
23#5019—22no—
24#4975—21no—
25#4949—22no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28#4925C2211yescharacteristic
29#4537—22no—
30D4⋊(C3⋊C8)—22no—
31#4481—22no—
32#3206—31no—
33#3036D311yescharacteristic
34#2542—31no—
35#1548—31no—
36#1502—31no—
37#1478—31no—
38#1303—31no—
39C2×C3×#41—21no—
40C22×C3×D4D411yescharacteristic
41C23×C3×C4—21no—
42C22×Dic6D412yes—
43C3×AGL(1,ℤ/8)—42no—
44C3×AGL(1,ℤ/8)—42no—
45C2×C3×SD16—22no—
46C2×C3×D8—22no—
47C2×C3×M16—21no—
48C3×C4⋊Q8D411yescharacteristic
49C3×#127D411yescharacteristic
50C3×#124—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C3×C4×D4—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C2×C3×C4⋊C4D411yescharacteristic
60C2×C3×C22⋊C4—21no—
61C2×C3×C42C2311yescharacteristic
62C2×#217—24no—
63C2×#214—24no—
64C2×C4⋊Dic3—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3—21no—
67#924—48no—
68C2×Dic3⋊C4—24no—
69C2×C4×Dic3—21no—
70#869—21no—
71#869—21no—
72#869—21no—
73#869—21no—
74C3×C4⋊C8—21no—
75C3×C4⋊C8D412yes—
76C3×#103—22no—
77C3×#102—22no—
78C3⋊#95D412yes—
79#830—21no—
80#830—21no—
81#810—24no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4×C3⋊C8—21no—
85C4×C3⋊C8—21no—
86C2×AGL(1,ℤ/8)—21no—
87#532C2×D311yescharacteristic
88C2×C4⋊Q8—31no—
89C2×C22⋊Q8—62no—
90C2×C4×D4C2×D311yescharacteristic
91#492—61no—
92#487—61no—
93#463—22no—
94#461—21no—
95#461—62no—
96#459—21no—
97#448—62no—
98#431—31no—
99#431—31no—
100#423C2×D311yescharacteristic
101#417—21no—
102#399—62no—
103#390—61no—
104#384—61no—
105C4×C4⋊C4—31no—
106#333—62no—
107#327—62no—
108D4⋊C8—62no—
109C24×C3—21no—
110C3×#41—41no—
111C3×#41—41no—
112C3×#41—41no—
113C3×#41—41no—
114C2×C3×Q8—21no—
115C2×C3×D4—21no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—21no—
122C2×C3×D4—21no—
123C2×C3×D4—41no—
124C2×C3×D4SD1612yescomplemented
125C22×C3×C4—21no—
126C22×C3×C4—44no—
127C22×C3×C4—41no—
128C22×C3×C4—41no—
129C22×C3×C4—41no—
130C22×C3×C4C2×D411yescharacteristic
131C22×C3×C4C2×D411yescharacteristic
132C22×C3×C4C2×D411yescharacteristic
133C22×Dic3—22no—
134C22×Dic3—22no—
135C2×Dic6—22no—
136C2×Dic6—22no—
137C2×Dic6—44no—
138C3×SD16—44no—
139C3×D8—44no—
140C3×M16—41no—
141C3×M16—41no—
142C2×C3×C8—22no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C22⋊C4—41no—
148C3×C22⋊C4—41no—
149C3×C22⋊C4—41no—
150C3×C22⋊C4—41no—
151C3×C42C2×D411yescharacteristic
152C3×C42C2×D411yescharacteristic
153C3×C42C2×D411yescharacteristic
154C3×C42#4111yescharacteristic
155#217—48no—
156#214—84no—
157C4⋊Dic3—24no—
158C4⋊Dic3—41no—
159C4⋊Dic3—41no—
160C4⋊Dic3—41no—
161C4⋊Dic3—41no—
162Dic3⋊C4—48no—
163C4×Dic3—41no—
164C4×Dic3—41no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167C2×#41—21no—
168C22×Q8—32no—
169C22×D4#6711yescharacteristic
170C23×C4—21no—
171AGL(1,ℤ/8)—42no—
172AGL(1,ℤ/8)—42no—
173C2×SD16—22no—
174C2×SD16—64no—
175C2×D8—22no—
176C2×M16—21no—
177C4⋊Q8—61no—
178C4⋊Q8—61no—
179C4⋊Q8—61no—
180C4⋊Q8—61no—
181C4⋊Q8#6711yescharacteristic
182#127#6711yescharacteristic
183#124—21no—
184C22⋊Q8—128no—
185#121—21no—
186#121—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—21no—
191C4×D4—21no—
192C4×D4—21no—
193C2×C4⋊C4—61no—
194C2×C4⋊C4—64no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4#6711yescharacteristic
198C2×C22⋊C4—21no—
199C2×C22⋊C4—64no—
200C2×C42—61no—
201C2×C42C22×D311yescharacteristic
202C4⋊C8—21no—
203C4⋊C8—32no—
204C4⋊C8—61no—
205C4⋊C8#6712yes—
206#103—22no—
207#103—64no—
208#102—22no—
209C22⋊C8—61no—
210C22⋊C8—61no—
211C4×C8—61no—
212C4×C8—61no—
213#95—32no—
214C23×C3—22no—
215C23×C3—42no—
216C23×C3—42no—
217C23×C3—41no—
218C23×C3—41no—
219C23×C3—41no—
220C23×C3#12011yescharacteristic
221C3×Q8—41no—
222C3×Q8—41no—
223C3×D4—22no—
224C3×D4—22no—
225C3×D4—41no—
226C3×D4—41no—
227C3×D4—41no—
228C3×D4—42no—
229C3×D4—41no—
230C3×D4—81no—
231C3×D4—81no—
232C3×D4—81no—
233C2×C3×C4—22no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
236C2×C3×C4—41no—
237C2×C3×C4—41no—
238C2×C3×C4—82no—
239C2×C3×C4—82no—
240C2×C3×C4—81no—
241C2×C3×C4—81no—
242C2×C3×C4#12011yescharacteristic
243C2×C3×C4#12011yescharacteristic
244C2×C3×C4#12111yescharacteristic
245C2×C3×C4#12111yescharacteristic
246C2×C3×C4#12111yescharacteristic
247C2×C3×C4#12711yescharacteristic
248C2×C3×C4C2×SD1611yescharacteristic
249C2×C3×C4#13511yescharacteristic
250C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
251C2×C3×C4#13711yescharacteristic
252C2×Dic3—24no—
253C2×Dic3—42no—
254C2×Dic3—42no—
255C2×Dic3—44no—
256C2×Dic3—42no—
257Dic6—44no—
258Dic6—82no—
259C3×C8—42no—
260C3⋊C8—42no—
261C3⋊C8—81no—
262C24—21no—
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267C2×Q8—21no—
268C2×Q8—62no—
269C2×Q8—62no—
270C2×Q8—124no—
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—21no—
275C2×D4—21no—
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279C2×D4—41no—
280C2×D4#21412yescomplemented
281C22×C4—21no—
282C22×C4—44no—
283C22×C4—41no—
284C22×C4—41no—
285C22×C4—41no—
286C22×C4—62no—
287C22×C4—62no—
288C22×C4D3×D411yescharacteristic
289C22×C4D3×D411yescharacteristic
290C22×C4C2×#6711yescharacteristic
291SD16—44no—
292SD16—244no—
293D8—44no—
294M16—41no—
295M16—41no—
296C2×C8—22no—
297C2×C8—62no—
298C2×C8—121no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—21no—
303C4⋊C4—64no—
304C4⋊C4—128no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—41no—
312C22⋊C4—41no—
313C22⋊C4—128no—
314C42—121no—
315C42—121no—
316C42D3×D411yescharacteristic
317C42#23711yescharacteristic
318C42C2×#6711yescharacteristic
319C42C2×#6711yescharacteristic
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—44no—
323C22×C3—42no—
324C22×C3—82no—
325C22×C3—81no—
326C22×C3—81no—
327C22×C3—81no—
328C22×C3—81no—
329C22×C3—81no—
330C22×C3#39211yescharacteristic
331C22×C3#44811yescharacteristic
332C22×C3#44911yescharacteristic
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—81no—
337C3×C4—81no—
338C3×C4#45011yescharacteristic
339C3×C4#45211yescharacteristic
340C3×C4#45312yes—
341C3×C4#46511yescharacteristic
342C3×C4#46811yescharacteristic
343Dic3—44no—
344Dic3—82no—
345C23—22no—
346C23—42no—
347C23—42no—
348C23—41no—
349C23—41no—
350C23—41no—
351C23#93711yescharacteristic
352Q8—41no—
353Q8—41no—
354Q8—124no—
355Q8—242no—
356D4—22no—
357D4—22no—
358D4—42no—
359D4—41no—
360D4—41no—
361D4—41no—
362D4—41no—
363D4—81no—
364D4—81no—
365D4—81no—
366C2×C4—22no—
367C2×C4—41no—
368C2×C4—42no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—64no—
372C2×C4—82no—
373C2×C4—82no—
374C2×C4—81no—
375C2×C4—81no—
376C2×C4—122no—
377C2×C4—122no—
378C2×C4—124no—
379C2×C4—122no—
380C2×C4#93211yescharacteristic
381C2×C4C2×#21411yescharacteristic
382C2×C4#93711yescharacteristic
383C2×C4#93811yescharacteristic
384C2×C4#93911yescharacteristic
385C2×C4#93911yescharacteristic
386C2×C4#94011yescharacteristic
387C2×C4#95011yescharacteristic
388C2×C4#95111yescharacteristic
389C2×C4#95311yescharacteristic
390C8—42no—
391C8—122no—
392C8—241no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—82no—
396C2×C3—161no—
397C2×C3#200011yescharacteristic
398C2×C3#200211yescharacteristic
399C2×C3#200911yescharacteristic
400C22—21no—
401C22—21no—
402C22—44no—
403C22—42no—
404C22—82no—
405C22—81no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22—81no—
410C22#503711yescharacteristic
411C22#504011yescharacteristic
412C22#522311yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—81no—
417C4—81no—
418C4—124no—
419C4—242no—
420C4#504111yescharacteristic
421C4#504211yescharacteristic
422C4#519812yes—
423C4#521811yescharacteristic
424C4#523911yescharacteristic
425C3#1024011yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—82no—
429C2—161no—
430C2#7168511yescharacteristic
431C2#7209411yescharacteristic
432C2#7212011yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
Fitting, prime radicalC3×#3036C2
socleC22×C3#392
hypercenter#3036D3
hypocenterC3#10240
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3036

Complements

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

NH ≅ G/Ntype
2. #83317C2semidirect
3. #82379C2semidirect
4. #71811C2semidirect
5. #71597C2semidirect
6. #70507C2semidirect
7. #67715C2semidirect
124. C2×C3×D4SD16semidirect
280. C2×D4#214semidirect
425. C3#10240semidirect

Maximal subgroups

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

subgroupindex
C3×#30362
#833172
#823792
#718112
#715972
#705072
#677152
#102403