Finite Group Databasebeta

← #10697560

Subgroups — #10697560

Order 768.

658 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 101 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#83317C211yesmaximal, characteristic, complemented
2#83297C211yesmaximal, characteristic
3#73800C211yesmaximal, characteristic, complemented
4C3×#1502C211yesmaximal, characteristic, complemented
5#70779C211yesmaximal, characteristic, complemented
6#70757C211yesmaximal, characteristic, complemented
7#67620C211yesmaximal, characteristic, complemented
8#10240—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#949—21no—
11C2×#897—22no—
12C2×C4×D12C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16#5232—21no—
17#5093—21no—
18#5083—21no—
19#5076—21no—
20#5072—21no—
21#5057—22no—
22#5040—22no—
23#5019—22no—
24#4939—21no—
25#4935—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4677—22no—
29C3×#327—22no—
30#4481—22no—
31#4459—22no—
32#3206—31no—
33#3036—31no—
34#2542—31no—
35#1548—31no—
36#1502D311yescharacteristic, complemented
37#1478—31no—
38#1303—31no—
39C2×C3×#41—21no—
40C22×D12D411yescharacteristic
41C22×C4×D3—21no—
42C22×Dic6D412yes—
43C3×C4⋊Q8D411yescharacteristic
44C3×#127D411yescharacteristic
45C3×#124—21no—
46C3×#121—21no—
47C3×#121—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53#949—42no—
54#949—42no—
55C2×#215—22no—
56C2×#214—24no—
57C2×#213—22no—
58C2×(C2×D3)⋊C4—24no—
59C2×(C2×D3)⋊C4—21no—
60C2×C4⋊Dic3—24no—
61C2×C4⋊Dic3—21no—
62C2×C4⋊Dic3—21no—
63C2×C4⋊Dic3D411yescharacteristic
64C2×#208—21no—
65#897—48no—
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—21no—
69C4×D12—21no—
70#869—21no—
71#869—21no—
72#869—21no—
73#869—21no—
74C3×C4⋊C8—21no—
75C3×C22⋊C8—21no—
76C3×C4×C8—21no—
77C3×#95D412yes—
78#835—22no—
79#832—22no—
80#830—21no—
81#810—24no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)D412yes—
84C4⋊(C3⋊C8)D412yes—
85C4×C3⋊C8—21no—
86C2×AGL(1,ℤ/8)—61no—
87#532C2×D311yescharacteristic
88C2×C4⋊Q8—31no—
89C2×C22⋊Q8—62no—
90C2×C4×D4—31no—
91#492—61no—
92#487—61no—
93#463—62no—
94#461—62no—
95#461—61no—
96#459—61no—
97#448—62no—
98#431—31no—
99#431C2×D311yescharacteristic
100#423—31no—
101#417—61no—
102#399—62no—
103#390—61no—
104#384—61no—
105C4×C4⋊C4C2×D311yescharacteristic
106#333—62no—
107#327—22no—
108D4⋊C8—62no—
109C23×D3—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—41no—
119C22×C3×C4—22no—
120C22×C3×C4—41no—
121C22×C3×C4C2×D411yescharacteristic
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×Dic3—21no—
125C22×Dic3—22no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12SD1612yescomplemented
132C2×C4×D3—44no—
133C2×C4×D3—41no—
134C2×C4×D3—41no—
135C2×Dic6—22no—
136C2×Dic6—22no—
137C2×Dic6—44no—
138C2×C3×C8—41no—
139C3×C4⋊C4—24no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C22⋊C4—41no—
143C3×C22⋊C4—41no—
144C3×C42—41no—
145C3×C42—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42#4111yescharacteristic
150#215—44no—
151#214—84no—
152#213—44no—
153(C2×D3)⋊C4—48no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—48no—
159C4⋊Dic3—41no—
160C4⋊Dic3—41no—
161C4⋊Dic3—41no—
162C4⋊Dic3—41no—
163#208—41no—
164#208—41no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×#41—21no—
168C22×Q8—32no—
169C22×D4—31no—
170C23×C4—61no—
171AGL(1,ℤ/8)—122no—
172AGL(1,ℤ/8)—122no—
173C2×SD16—62no—
174C2×SD16—64no—
175C2×D8—62no—
176C2×M16—61no—
177C4⋊Q8—61no—
178C4⋊Q8—61no—
179C4⋊Q8—61no—
180C4⋊Q8—61no—
181C4⋊Q8D1211yescharacteristic
182#127D1211yescharacteristic
183#124—21no—
184C22⋊Q8—128no—
185#121—21no—
186#121—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—61no—
190C4×D4—61no—
191C4×D4—61no—
192C4×D4—61no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—31no—
195C2×C4⋊C4—64no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—61no—
198C2×C22⋊C4—64no—
199C2×C22⋊C4—61no—
200C2×C42—21no—
201C2×C42C22×D311yescharacteristic
202C4⋊C8—21no—
203C4⋊C8—32no—
204C4⋊C8—32no—
205C4⋊C8—61no—
206#103—64no—
207#103—62no—
208#102—62no—
209C22⋊C8—21no—
210C22⋊C8—61no—
211C4×C8—21no—
212C4×C8—61no—
213#95#6712yes—
214C23×C3—41no—
215C23×C3#12011yescharacteristic
216C22×D3—22no—
217C22×D3—42no—
218C22×D3—42no—
219C22×D3—41no—
220C22×D3—41no—
221C3×Q8—41no—
222C3×Q8—41no—
223C3×D4—41no—
224C3×D4—41no—
225C3×D4—42no—
226C3×D4—81no—
227C3×D4—81no—
228C3×D4—81no—
229C2×C3×C4—22no—
230C2×C3×C4—24no—
231C2×C3×C4—41no—
232C2×C3×C4—42no—
233C2×C3×C4—42no—
234C2×C3×C4—41no—
235C2×C3×C4—81no—
236C2×C3×C4—81no—
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12011yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12111yescharacteristic
241C2×C3×C4#12111yescharacteristic
242C2×C3×C4#12711yescharacteristic
243C2×C3×C4C2×SD1611yescharacteristic
244C2×C3×C4#13511yescharacteristic
245C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
246C2×C3×C4#13711yescharacteristic
247C2×Dic3—42no—
248C2×Dic3—44no—
249C2×Dic3—41no—
250C2×Dic3—42no—
251D12—22no—
252D12—22no—
253D12—41no—
254D12—41no—
255C4×D3—82no—
256C4×D3—82no—
257Dic6—44no—
258Dic6—82no—
259C3×C8—81no—
260C3⋊C8—42no—
261C3⋊C8—42no—
262C24—61no—
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—32no—
275C2×D4—41no—
276C2×D4—61no—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—61no—
280C2×D4—61no—
281C22×C4—22no—
282C22×C4—41no—
283C22×C4—62no—
284C22×C4—61no—
285C22×C4—124no—
286C22×C4—121no—
287C22×C4—121no—
288C22×C4D3×D411yescharacteristic
289C22×C4D3×D411yescharacteristic
290C22×C4C2×#6711yescharacteristic
291SD16—124no—
292SD16—244no—
293D8—124no—
294M16—121no—
295M16—121no—
296C2×C8—41no—
297C2×C8—62no—
298C2×C8—62no—
299C4⋊C4—24no—
300C4⋊C4—21no—
301C4⋊C4—21no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—128no—
305C4⋊C4—121no—
306C4⋊C4—121no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—128no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C42—41no—
315C42—41no—
316C42C2×D1211yescharacteristic
317C42#23511yescharacteristic
318C42D3×D411yescharacteristic
319C42D3×D411yescharacteristic
320C22×C3—21no—
321C22×C3—21no—
322C22×C3—81no—
323C22×C3—81no—
324C22×C3—81no—
325C22×C3#39211yescharacteristic
326C22×C3#44811yescharacteristic
327C22×C3#44911yescharacteristic
328C2×D3—44no—
329C2×D3—42no—
330C2×D3—82no—
331C2×D3—81no—
332C2×D3—81no—
333C3×C4—21no—
334C3×C4—21no—
335C3×C4—21no—
336C3×C4—44no—
337C3×C4—81no—
338C3×C4#45011yescharacteristic
339C3×C4#45211yescharacteristic
340C3×C4#45312yes—
341C3×C4#46511yescharacteristic
342C3×C4#46811yescharacteristic
343Dic3—82no—
344Dic3—81no—
345C23—41no—
346C23—62no—
347C23—122no—
348C23—122no—
349C23—121no—
350C23—121no—
351C23#93711yescharacteristic
352Q8—41no—
353Q8—41no—
354Q8—124no—
355Q8—242no—
356D4—41no—
357D4—41no—
358D4—42no—
359D4—62no—
360D4—62no—
361D4—81no—
362D4—81no—
363D4—81no—
364D4—121no—
365D4—121no—
366C2×C4—22no—
367C2×C4—24no—
368C2×C4—41no—
369C2×C4—42no—
370C2×C4—42no—
371C2×C4—41no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—124no—
375C2×C4—122no—
376C2×C4—121no—
377C2×C4—122no—
378C2×C4—242no—
379C2×C4—242no—
380C2×C4#88211yescharacteristic
381C2×C4#88211yescharacteristic
382C2×C4#88411yescharacteristic
383C2×C4#88411yescharacteristic
384C2×C4#91111yescharacteristic
385C2×C4#91211yescharacteristic
386C2×C4D3×SD1611yescharacteristic
387C2×C4#91511yescharacteristic
388C2×C4#93011yescharacteristic
389C2×C4#94011yescharacteristic
390C8—81no—
391C8—122no—
392C8—122no—
393C2×C3—21no—
394C2×C3—21no—
395C2×C3—161no—
396C2×C3#200011yescharacteristic
397C2×C3#200211yescharacteristic
398C2×C3#200911yescharacteristic
399D3—82no—
400C22—21no—
401C22—21no—
402C22—81no—
403C22—81no—
404C22—81no—
405C22—124no—
406C22—122no—
407C22—242no—
408C22—241no—
409C22—241no—
410C22#496011yescharacteristic
411C22#515811yescharacteristic
412C22#517011yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—44no—
417C4—81no—
418C4—241no—
419C4—242no—
420C4#477611yescharacteristic
421C4#477811yescharacteristic
422C4#478011yescharacteristic
423C4#478311yescharacteristic
424C4#482212yes—
425C3#1024011yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—161no—
429C2—242no—
430C2#7005311yescharacteristic
431C2#7020511yescharacteristic
432C2#7038511yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1502C2
socleC22×C3#392
prime radical#83297C2
hypercenter#1502D3
hypocenterC3#10240
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1502

Complements

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

NH ≅ G/Ntype
1. #83317C2semidirect
3. #73800C2semidirect
4. C3×#1502C2semidirect
5. #70779C2semidirect
6. #70757C2semidirect
7. #67620C2semidirect
36. #1502D3semidirect
131. C2×D12SD16semidirect
425. C3#10240semidirect

Maximal subgroups

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

subgroupindex
#833172
#832972
#738002
C3×#15022
#707792
#707572
#676202
#102403