Finite Group Databasebeta

← #10697676

Subgroups — #10697676

Order 768.

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

normal subgroups only

This group has 528 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×#3037C211yesmaximal, characteristic
2#83320C211yesmaximal, characteristic
3#82431C211yesmaximal, characteristic, complemented
4#71802C211yesmaximal, characteristic
5#71634C211yesmaximal, characteristic, complemented
6#70501C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#10257—31nomaximal, Sylow
9C2×C3×#137—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12#5470C2211yescharacteristic
13C3×#464—22no—
14C3×#462—21no—
15C3×#460—21no—
16C3×#423C2211yescharacteristic
17C3×#417—21no—
18#5225—21no—
19#5094—21no—
20#5075—21no—
21#5041—22no—
22#5022—22no—
23#5021—21no—
24#5020—21no—
25#5019—21no—
26#5008—21no—
27#4964—21no—
28#4924C2211yescharacteristic
29C2×C4⋊(C3⋊C8)C2211yescharacteristic
30C2×C4×C3⋊C8C2211yescharacteristic
31D4⋊(C3⋊C8)—22no—
32#4524—22no—
33#4484—22no—
34#3246—31no—
35#3078—31no—
36#3037D311yescharacteristic
37#1984—31no—
38#1493—31no—
39#1492—31no—
40#1292—31no—
41C2×C3×#41—21no—
42C22×C3×Q8D411yescharacteristic
43C3×#137—42no—
44C3×#137—42no—
45C2×C3×Q16—22no—
46C2×C3×SD16—22no—
47C2×C3×M16—21no—
48C3×C4⋊Q8D411yescharacteristic
49C3×#127D411yescharacteristic
50C3×#124—21no—
51C3×#121—21no—
52C3×#121—21no—
53C3×C4×Q8—21no—
54C3×C4×Q8—21no—
55C3×C4×Q8—21no—
56C3×C4×Q8—21no—
57C3×C4×D4—21no—
58C3×C4×D4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C4⋊C4D411yescharacteristic
61C2×C3×C42—21no—
62C2×C3×C42C2311yescharacteristic
63C2×#214—22no—
64#926—21no—
65#924—21no—
66#924—21no—
67C2×#208—21no—
68C22×C3⋊C8—21no—
69C22×C3⋊C8D412yes—
70#870—21no—
71#869D411yescharacteristic
72#869D411yescharacteristic
73C4×Dic6—21no—
74C4×Dic6—21no—
75C3×C4⋊C8—21no—
76C3×C4⋊C8D412yes—
77C3×#103—22no—
78C3×#102—22no—
79#835—22no—
80#835—22no—
81#832—22no—
82#810—22no—
83#808—22no—
84#808—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4⋊(C3⋊C8)D412yes—
90C4×C3⋊C8—21no—
91C4×C3⋊C8—21no—
92C2×#137—21no—
93#533—31no—
94#532C2×D311yescharacteristic
95C2×C4×Q8C2×D311yescharacteristic
96#479—62no—
97#477—61no—
98#476—61no—
99#475—61no—
100#475—61no—
101#465—62no—
102#464—22no—
103#462—21no—
104#461—61no—
105#460—21no—
106#460—61no—
107#428—61no—
108#423—31no—
109#423C2×D311yescharacteristic
110C2×C4⋊C8—31no—
111#419—61no—
112#417—21no—
113C2×C4×C8—31no—
114#336—62no—
115Q8⋊C8—62no—
116D4⋊C8—62no—
117C3×#41—41no—
118C3×#41—41no—
119C3×#41—41no—
120C3×#41—41no—
121C2×C3×Q8—21no—
122C2×C3×Q8—21no—
123C2×C3×Q8—21no—
124C2×C3×Q8—21no—
125C2×C3×Q8—21no—
126C2×C3×Q8—21no—
127C2×C3×Q8SD1612yescomplemented
128C2×C3×D4—21no—
129C2×C3×D4—21no—
130C2×C3×D4—21no—
131C2×C3×D4—41no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4—41no—
135C22×C3×C4C2×D411yescharacteristic
136C22×C3×C4C2×D411yescharacteristic
137C22×C3×C4#4111yescharacteristic
138C2×Dic6—21no—
139C2×Dic6—21no—
140C3×Q16—44no—
141C3×SD16—44no—
142C3×M16—41no—
143C3×M16—41no—
144C2×C3×C8—22no—
145C3×C4⋊C4—21no—
146C3×C4⋊C4—21no—
147C3×C4⋊C4—21no—
148C3×C4⋊C4—21no—
149C3×C4⋊C4—41no—
150C3×C4⋊C4—41no—
151C3×C22⋊C4—41no—
152C3×C22⋊C4—41no—
153C3×C42—41no—
154C3×C42—41no—
155C3×C42C2×D411yescharacteristic
156C3×C42C2×D411yescharacteristic
157C3×C42C2×D411yescharacteristic
158C3×C42C2×D411yescharacteristic
159#217—41no—
160#214—82no—
161C4⋊Dic3—21no—
162C4⋊Dic3—21no—
163C4⋊Dic3—21no—
164C4⋊Dic3—21no—
165Dic3⋊C4—41no—
166Dic3⋊C4—41no—
167C4×Dic3—41no—
168#208—41no—
169#208—41no—
170C2×C3⋊C8—22no—
171C2×C3⋊C8—22no—
172C2×C3⋊C8—22no—
173C2×C3⋊C8—41no—
174C2×C3⋊C8—41no—
175C2×C3⋊C8—42no—
176C2×#41—21no—
177C22×Q8#6711yescharacteristic
178#137—42no—
179#137—42no—
180C2×Q16—22no—
181C2×SD16—22no—
182C2×SD16—62no—
183C2×M16—21no—
184C2×M16—61no—
185C22×C8—32no—
186C22×C8—61no—
187C4⋊Q8—31no—
188C4⋊Q8—31no—
189C4⋊Q8#6711yescharacteristic
190#127#6711yescharacteristic
191#125—61no—
192#124—21no—
193C22⋊Q8—61no—
194C22⋊Q8—61no—
195#121—21no—
196#121—21no—
197C4×Q8—21no—
198C4×Q8—21no—
199C4×Q8—21no—
200C4×Q8—21no—
201C4×Q8—61no—
202C4×Q8—61no—
203C4×D4—21no—
204C4×D4—21no—
205#117—61no—
206C2×C4⋊C4—21no—
207C2×C4⋊C4#6711yescharacteristic
208C2×C42—21no—
209C2×C42C22×D311yescharacteristic
210#107—62no—
211#106—62no—
212#106—62no—
213C4⋊C8—21no—
214C4⋊C8—32no—
215C4⋊C8—61no—
216C4⋊C8—61no—
217C4⋊C8—61no—
218C4⋊C8#6712yes—
219#103—22no—
220#103—62no—
221#103—62no—
222#103—62no—
223#102—22no—
224#102—62no—
225C4×C8—61no—
226C4×C8—61no—
227C23×C3—41no—
228C23×C3#12111yescharacteristic
229C3×Q8—22no—
230C3×Q8—22no—
231C3×Q8—41no—
232C3×Q8—41no—
233C3×Q8—41no—
234C3×Q8—41no—
235C3×D4—42no—
236C3×D4—41no—
237C3×D4—41no—
238C3×D4—41no—
239C3×D4—41no—
240C3×D4—81no—
241C3×D4—81no—
242C2×C3×C4—21no—
243C2×C3×C4—21no—
244C2×C3×C4—22no—
245C2×C3×C4—41no—
246C2×C3×C4—41no—
247C2×C3×C4—41no—
248C2×C3×C4—42no—
249C2×C3×C4—41no—
250C2×C3×C4—41no—
251C2×C3×C4—81no—
252C2×C3×C4—81no—
253C2×C3×C4#12011yescharacteristic
254C2×C3×C4#12011yescharacteristic
255C2×C3×C4#12011yescharacteristic
256C2×C3×C4#12111yescharacteristic
257C2×C3×C4#12111yescharacteristic
258C2×C3×C4#12711yescharacteristic
259C2×C3×C4C2×SD1611yescharacteristic
260C2×C3×C4#13511yescharacteristic
261C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
262C2×C3×C4#13711yescharacteristic
263C2×Dic3—41no—
264C2×Dic3—41no—
265Dic6—42no—
266Dic6—41no—
267Dic6—41no—
268C3×C8—42no—
269C3⋊C8—42no—
270C3⋊C8—42no—
271C3⋊C8—42no—
272#41—41no—
273#41—41no—
274#41—41no—
275#41—41no—
276C2×Q8—21no—
277C2×Q8—21no—
278C2×Q8—21no—
279C2×Q8—21no—
280C2×Q8—21no—
281C2×Q8—21no—
282C2×Q8—61no—
283C2×Q8—61no—
284C2×Q8#21412yescomplemented
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—41no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—41no—
292C22×C4D3×D411yescharacteristic
293C22×C4#23711yescharacteristic
294C22×C4C2×#6711yescharacteristic
295Q16—44no—
296SD16—44no—
297SD16—242no—
298M16—41no—
299M16—41no—
300M16—121no—
301M16—121no—
302C2×C8—22no—
303C2×C8—62no—
304C2×C8—62no—
305C2×C8—62no—
306C2×C8—121no—
307C2×C8—121no—
308C2×C8—122no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—61no—
316C4⋊C4—61no—
317C4⋊C4—61no—
318C4⋊C4—61no—
319C4⋊C4—121no—
320C4⋊C4—121no—
321C22⋊C4—41no—
322C22⋊C4—41no—
323C22⋊C4—121no—
324C42—41no—
325C42—41no—
326C42—121no—
327C42D3×D411yescharacteristic
328C42D3×D411yescharacteristic
329C42C2×#6711yescharacteristic
330C42C2×#6711yescharacteristic
331C22×C3—21no—
332C22×C3—21no—
333C22×C3—81no—
334C22×C3—81no—
335C22×C3—81no—
336C22×C3#39211yescharacteristic
337C22×C3#46511yescharacteristic
338C22×C3#46811yescharacteristic
339C3×C4—21no—
340C3×C4—21no—
341C3×C4—21no—
342C3×C4—42no—
343C3×C4—81no—
344C3×C4—81no—
345C3×C4#44811yescharacteristic
346C3×C4#44911yescharacteristic
347C3×C4#45011yescharacteristic
348C3×C4#45211yescharacteristic
349C3×C4#45511yescharacteristic
350C3×C4#45611yescharacteristic
351Dic3—81no—
352Dic3—81no—
353C23—41no—
354C23#93911yescharacteristic
355Q8—22no—
356Q8—22no—
357Q8—41no—
358Q8—41no—
359Q8—41no—
360Q8—41no—
361Q8—122no—
362Q8—121no—
363Q8—121no—
364D4—42no—
365D4—41no—
366D4—41no—
367D4—41no—
368D4—41no—
369D4—81no—
370D4—81no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—22no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—81no—
381C2×C4—81no—
382C2×C4—121no—
383C2×C4—121no—
384C2×C4#93211yescharacteristic
385C2×C4C2×#21411yescharacteristic
386C2×C4#93711yescharacteristic
387C2×C4#93711yescharacteristic
388C2×C4#93811yescharacteristic
389C2×C4#93911yescharacteristic
390C2×C4#94011yescharacteristic
391C2×C4#95011yescharacteristic
392C2×C4#95111yescharacteristic
393C2×C4#95311yescharacteristic
394C8—42no—
395C8—122no—
396C8—122no—
397C8—122no—
398C2×C3—21no—
399C2×C3—21no—
400C2×C3—161no—
401C2×C3#200011yescharacteristic
402C2×C3#200711yescharacteristic
403C2×C3#201711yescharacteristic
404C22—21no—
405C22—21no—
406C22—81no—
407C22—81no—
408C22—81no—
409C22#504111yescharacteristic
410C22#504211yescharacteristic
411C22#522311yescharacteristic
412C4—21no—
413C4—21no—
414C4—21no—
415C4—42no—
416C4—81no—
417C4—81no—
418C4—241no—
419C4—241no—
420C4#503711yescharacteristic
421C4#504011yescharacteristic
422C4#520211yescharacteristic
423C4#520511yescharacteristic
424C4#521811yescharacteristic
425C4#523911yescharacteristic
426C3#1025711yescharacteristic, complemented, Sylow
427C2—21no—
428C2—21no—
429C2—161no—
430C2#7168511yescharacteristic
431C2#7211011yescharacteristic
432C2#7212111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3037C2
socleC22×C3#392
prime radicalC3×#532C22
hypercenter#3037D3
hypocenterC3#10257
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
3. #82431C2semidirect
5. #71634C2semidirect
6. #70501C2semidirect
7. #67711C2semidirect
127. C2×C3×Q8SD16semidirect
284. C2×Q8#214semidirect
426. C3#10257semidirect

Maximal subgroups

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

subgroupindex
C3×#30372
#833202
#824312
#718022
#716342
#705012
#677112
#102573