Finite Group Databasebeta

← #10697559

Subgroups — #10697559

Order 768.

658 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 658 conjugacy classes of subgroups — too many to draw. Only its 103 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×#3206C211yesmaximal, characteristic, complemented
2#83297C211yesmaximal, characteristic
3#83259C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic, complemented
5#71774C211yesmaximal, characteristic, complemented
6#70756C211yesmaximal, characteristic, complemented
7#67746C211yesmaximal, characteristic, complemented
8#10240—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×C3×C4⋊Q8C2211yescharacteristic
11C2×#949—21no—
12C2×#946—22no—
13C2×C4×D12C2211yescharacteristic
14C3×#492—21no—
15C3×#487—21no—
16C3×#461—22no—
17C3×#448—22no—
18C3×#431C2211yescharacteristic
19#5232—21no—
20#5083—21no—
21#5072—21no—
22#5057—22no—
23#4993—22no—
24#4988—21no—
25#4975—21no—
26C4×C4⋊Dic3C2211yescharacteristic
27#4925C2211yescharacteristic
28#4924C2211yescharacteristic
29#4546—22no—
30#4537—22no—
31#4483—22no—
32#3206D311yescharacteristic, complemented
33#3036—31no—
34#2542—31no—
35#1548—31no—
36#1502—31no—
37#1478—31no—
38#1303—31no—
39C2×C3×#41—21no—
40C22×C3×Q8D412yes—
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C2×C3×SD16—24no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8—21no—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8D411yescharacteristic
49C3×#127D411yescharacteristic
50C3×#124—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4—21no—
57C2×C3×C42C2311yescharacteristic
58#949—42no—
59#949—42no—
60#946—48no—
61C2×#215—22no—
62C2×#213—22no—
63C2×(C2×D3)⋊C4—24no—
64C2×(C2×D3)⋊C4—21no—
65C2×C4⋊Dic3—21no—
66C2×C4⋊Dic3D411yescharacteristic
67C2×Dic3⋊C4—24no—
68C2×C4×Dic3—21no—
69C2×#208—21no—
70C4×D12—21no—
71C4×D12—21no—
72C4×D12—21no—
73C4×D12—21no—
74C3×C4⋊C8D412yes—
75C3×#103—24no—
76C3×C22⋊C8—21no—
77C3×C4×C8—21no—
78#835—22no—
79#832—22no—
80C3⋊#95D412yes—
81#830—21no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)D412yes—
85C4×C3⋊C8—21no—
86C2×AGL(1,ℤ/8)—61no—
87#532C2×D311yescharacteristic
88C2×C4⋊Q8C2×D311yescharacteristic
89C2×C22⋊Q8—62no—
90C2×C4×D4—31no—
91#492—21no—
92#487—21no—
93#463—62no—
94#461—22no—
95#461—61no—
96#459—61no—
97#448—22no—
98#431—31no—
99#431C2×D311yescharacteristic
100#423—31no—
101#417—61no—
102#399—62no—
103#390—61no—
104#384—61no—
105C4×C4⋊C4—31no—
106#333—62no—
107#327—62no—
108D4⋊C8—62no—
109C23×D3—21no—
110C3×#41—41no—
111C3×#41—41no—
112C3×#41—41no—
113C3×#41—41no—
114C2×C3×Q8—21no—
115C2×C3×Q8—22no—
116C2×C3×Q8—22no—
117C2×C3×Q8—44no—
118C2×C3×D4—21no—
119C2×C3×D4—21no—
120C2×C3×D4—21no—
121C2×C3×D4—41no—
122C22×C3×C4—22no—
123C22×C3×C4—41no—
124C22×C3×C4C2×D411yescharacteristic
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×Dic3—22no—
128C22×Dic3—21no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×D12SD1612yescomplemented
135C2×C4×D3—44no—
136C2×C4×D3—41no—
137C2×C4×D3—41no—
138C3×SD16—84no—
139C2×C3×C8—22no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C4⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C22⋊C4—41no—
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151C3×C42#4111yescharacteristic
152#215—44no—
153#213—44no—
154(C2×D3)⋊C4—48no—
155(C2×D3)⋊C4—41no—
156(C2×D3)⋊C4—41no—
157C4⋊Dic3—24no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160Dic3⋊C4—48no—
161C4×Dic3—41no—
162C4×Dic3—41no—
163#208—41no—
164#208—41no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—41no—
167C2×#41—21no—
168C22×Q8#6712yes—
169C22×D4—31no—
170C23×C4—61no—
171AGL(1,ℤ/8)—122no—
172AGL(1,ℤ/8)—122no—
173C2×SD16—24no—
174C2×SD16—62no—
175C2×D8—62no—
176C2×M16—61no—
177C4⋊Q8—21no—
178C4⋊Q8—21no—
179C4⋊Q8—21no—
180C4⋊Q8—21no—
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—61no—
190C4×D4—61no—
191C4×D4—61no—
192C4×D4—61no—
193C2×C4⋊C4—21no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—31no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4—64no—
198C2×C22⋊C4—64no—
199C2×C22⋊C4—61no—
200C2×C42—61no—
201C2×C42C22×D311yescharacteristic
202C4⋊C8—32no—
203C4⋊C8—61no—
204C4⋊C8—61no—
205C4⋊C8#6712yes—
206#103—24no—
207#103—62no—
208#102—62no—
209C22⋊C8—21no—
210C22⋊C8—61no—
211C4×C8—21no—
212C4×C8—61no—
213#95—32no—
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×Q8—44no—
224C3×Q8—82no—
225C3×D4—41no—
226C3×D4—41no—
227C3×D4—42no—
228C3×D4—81no—
229C3×D4—81no—
230C3×D4—81no—
231C2×C3×C4—22no—
232C2×C3×C4—41no—
233C2×C3×C4—44no—
234C2×C3×C4—41no—
235C2×C3×C4—42no—
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×C4C2×SD1611yescharacteristic
245C2×C3×C4#13511yescharacteristic
246C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
247C2×C3×C4#13711yescharacteristic
248C2×Dic3—24no—
249C2×Dic3—42no—
250C2×Dic3—42no—
251C2×Dic3—42no—
252C2×Dic3—41no—
253D12—22no—
254D12—22no—
255D12—41no—
256D12—41no—
257C4×D3—82no—
258C4×D3—82no—
259C3×C8—42no—
260C3⋊C8—42no—
261C3⋊C8—81no—
262C24—61no—
263#41—41no—
264#41—41no—
265#41—41no—
266#41—41no—
267C2×Q8—21no—
268C2×Q8—22no—
269C2×Q8—22no—
270C2×Q8—44no—
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—84no—
292SD16—124no—
293D8—124no—
294M16—121no—
295M16—121no—
296C2×C8—22no—
297C2×C8—62no—
298C2×C8—121no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—64no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—128no—
309C22⋊C4—41no—
310C22⋊C4—41no—
311C22⋊C4—128no—
312C22⋊C4—121no—
313C22⋊C4—121no—
314C42—121no—
315C42—121no—
316C42D3×D411yescharacteristic
317C42#23711yescharacteristic
318C42C2×#6711yescharacteristic
319C42C2×#6711yescharacteristic
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—81no—
337C3×C4—82no—
338C3×C4#45011yescharacteristic
339C3×C4#45211yescharacteristic
340C3×C4#45312yes—
341C3×C4#46511yescharacteristic
342C3×C4#46811yescharacteristic
343Dic3—44no—
344Dic3—81no—
345C23—41no—
346C23—62no—
347C23—122no—
348C23—122no—
349C23—121no—
350C23—121no—
351C23#93711yescharacteristic
352Q8—41no—
353Q8—41no—
354Q8—44no—
355Q8—82no—
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—41no—
368C2×C4—44no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—64no—
372C2×C4—81no—
373C2×C4—81no—
374C2×C4—122no—
375C2×C4—122no—
376C2×C4—122no—
377C2×C4—121no—
378C2×C4—242no—
379C2×C4—242no—
380C2×C4D3×SD1611yescharacteristic
381C2×C4#91411yescharacteristic
382C2×C4#91511yescharacteristic
383C2×C4#91611yescharacteristic
384C2×C4#93711yescharacteristic
385C2×C4#93811yescharacteristic
386C2×C4#93911yescharacteristic
387C2×C4#93911yescharacteristic
388C2×C4#94011yescharacteristic
389C2×C4#95311yescharacteristic
390C8—42no—
391C8—122no—
392C8—241no—
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#517011yescharacteristic
411C22#517211yescharacteristic
412C22#522311yescharacteristic
413C4—21no—
414C4—21no—
415C4—21no—
416C4—81no—
417C4—82no—
418C4—124no—
419C4—241no—
420C4#516911yescharacteristic
421C4#517111yescharacteristic
422C4#517311yescharacteristic
423C4#517411yescharacteristic
424C4#524112yes—
425C3#1024011yescharacteristic, complemented, Sylow
426C2—21no—
427C2—21no—
428C2—161no—
429C2—242no—
430C2#7185611yescharacteristic
431C2#7205411yescharacteristic
432C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3206C2
socleC22×C3#392
prime radical#83297C2
hypercenter#3206D3
hypocenterC3#10240
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3206

Complements

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

NH ≅ G/Ntype
1. C3×#3206C2semidirect
3. #83259C2semidirect
4. #71811C2semidirect
5. #71774C2semidirect
6. #70756C2semidirect
7. #67746C2semidirect
32. #3206D3semidirect
134. 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
C3×#32062
#832972
#832592
#718112
#717742
#707562
#677462
#102403