Finite Group Databasebeta

← #10697210

Subgroups — #10697210

Order 768.

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

normal subgroups only

This group has 550 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
1C2×#5031C211yesmaximal, characteristic, complemented
2#82441C211yesmaximal, characteristic
3#79268C211yesmaximal, characteristic
4C3×#1461C211yesmaximal, characteristic, complemented
5#70778C211yesmaximal, characteristic, complemented
6#70758C211yesmaximal, characteristic
7#67745C211yesmaximal, characteristic, complemented
8#10185—31nomaximal, Sylow
9C2×C3×C4×Q8C2211yescharacteristic
10C22×#216—21no—
11C2×#942—21no—
12#5477C2211yescharacteristic
13C2×#869C2211yescharacteristic
14C3×#431C2211yescharacteristic
15C3×#431C2211yescharacteristic
16#5031—21no—
17#5031—22no—
18#5031—22no—
19#5028—22no—
20#5027—21no—
21C2×#810—21no—
22#4966—21no—
23#4924C2211yescharacteristic
24C2×C4⋊(C3⋊C8)C2211yescharacteristic
25#4846—22no—
26#4736—22no—
27#4694—21no—
28#4682—21no—
29C3×#324—22no—
30#4486—22no—
31#4459—22no—
32#3226—31no—
33#3037—31no—
34C2×#462—31no—
35#1546—31no—
36#1480—31no—
37#1461D311yescharacteristic, complemented
38#1301—31no—
39C22×C3×Q8D411yescharacteristic
40C2×#235—21no—
41C22×Dic6—21no—
42C3×C4×Q8—21no—
43C3×C4×Q8—21no—
44C3×C4×Q8—21no—
45C3×C4×Q8—21no—
46C2×C3×C4⋊C4—21no—
47C2×C3×C4⋊C4D411yescharacteristic
48C2×C3×C42—21no—
49C2×C3×C42C2311yescharacteristic
50C2×#216—22no—
51C2×#216—24no—
52C2×#216—42no—
53C2×#216—42no—
54#942—42no—
55#942—42no—
56C2×#215—22no—
57#930—21no—
58#930—21no—
59C2×C4⋊Dic3—21no—
60C2×C4⋊Dic3—21no—
61C2×#208—21no—
62C22×C3⋊C8—21no—
63#875—21no—
64#874D411yescharacteristic
65C4×D12—21no—
66C4×D12—21no—
67#869—21no—
68#869—21no—
69#869—21no—
70#869D411yescharacteristic
71#869D412yes—
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×C22⋊C8—21no—
75C3×C22⋊C8D412yes—
76C3×C4×C8—21no—
77C3×C4×C8—21no—
78#821—24no—
79#818—24no—
80#810—22no—
81#810—24no—
82#809—22no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4⋊(C3⋊C8)D412yes—
86C4⋊(C3⋊C8)D412yes—
87C2×#137—61no—
88C22×Q16—61no—
89#532—31no—
90C2×C4⋊Q8—31no—
91C2×C4×Q8C2×D311yescharacteristic
92#500—61no—
93#486—61no—
94#480—62no—
95#474—62no—
96#464—62no—
97#462—62no—
98#462—62no—
99#462—61no—
100#460—61no—
101#431C2×D311yescharacteristic
102#431C2×D311yescharacteristic
103#423—31no—
104C2×C4⋊C8—31no—
105#417—61no—
106C2×#103—61no—
107#332—62no—
108D4⋊C8—62no—
109#324—22no—
110C2×C3×Q8—21no—
111C2×C3×Q8—21no—
112C2×C3×Q8—21no—
113C2×C3×Q8—21no—
114C2×C3×Q8—21no—
115C2×C3×Q8D812yes—
116C22×C3×C4—21no—
117C22×C3×C4—21no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4#4111yescharacteristic
121C2×#67—41no—
122C22×Dic3—41no—
123#235—41no—
124#235—41no—
125#235—41no—
126#235—41no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×C4×D3—41no—
131C2×Dic6—22no—
132C2×Dic6—21no—
133C2×Dic6—41no—
134C2×Dic6—41no—
135C2×Dic6—41no—
136C2×Dic6—41no—
137C2×Dic6—41no—
138C2×C3×C8—22no—
139C2×C3×C8—41no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—21no—
142C3×C4⋊C4—41no—
143C3×C4⋊C4—41no—
144C3×C42—41no—
145C3×C42—41no—
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150#216—44no—
151#216—48no—
152#215—44no—
153(C2×D3)⋊C4—41no—
154(C2×D3)⋊C4—41no—
155C4⋊Dic3—21no—
156C4⋊Dic3—22no—
157C4⋊Dic3—22no—
158C4⋊Dic3—21no—
159C4⋊Dic3—41no—
160C4⋊Dic3—41no—
161#208—41no—
162#208—41no—
163C2×C3⋊C8—22no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—41no—
166C2×C3⋊C8—41no—
167C2×#41—61no—
168C22×Q8—61no—
169C22×Q8D1211yescharacteristic
170#137—122no—
171#137—122no—
172C2×Q16—64no—
173C2×Q16—62no—
174C2×Q16—122no—
175C2×Q16—122no—
176C2×SD16—62no—
177C2×M16—61no—
178C22×C8—61no—
179C4⋊Q8—32no—
180C4⋊Q8—31no—
181C4⋊Q8—61no—
182C4⋊Q8—61no—
183C4⋊Q8—61no—
184#127—31no—
185#124—61no—
186#121—61no—
187#121—61no—
188C4×Q8—21no—
189C4×Q8—21no—
190C4×Q8—21no—
191C4×Q8—21no—
192C4×D4—61no—
193C4×D4—61no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—61no—
196C2×C4⋊C4—61no—
197C2×C4⋊C4D1211yescharacteristic
198C2×C42—21no—
199C2×C42C22×D311yescharacteristic
200#107—64no—
201C4⋊C8—21no—
202C4⋊C8—21no—
203C4⋊C8—32no—
204C4⋊C8—32no—
205C4⋊C8—61no—
206C4⋊C8—61no—
207#103—64no—
208#103—62no—
209#102—62no—
210#102—64no—
211C22⋊C8—21no—
212C22⋊C8#6712yes—
213C4×C8—21no—
214C4×C8—21no—
215C23×C3#12111yescharacteristic
216C22×D3—41no—
217C3×Q8—22no—
218C3×Q8—22no—
219C3×Q8—41no—
220C3×Q8—41no—
221C2×C3×C4—22no—
222C2×C3×C4—22no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—81no—
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12711yescharacteristic
234C2×C3×C4C2×D811yescharacteristic
235C2×C3×C4#13511yescharacteristic
236C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
237C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
238#67—81no—
239#67—81no—
240C2×Dic3—42no—
241C2×Dic3—41no—
242C2×Dic3—81no—
243C2×Dic3—81no—
244D12—42no—
245D12—41no—
246D12—41no—
247D12—42no—
248C4×D3—81no—
249C4×D3—81no—
250Dic6—42no—
251Dic6—41no—
252Dic6—42no—
253Dic6—41no—
254Dic6—81no—
255Dic6—81no—
256C3×C8—42no—
257C3×C8—81no—
258C3⋊C8—42no—
259C3⋊C8—42no—
260#41—121no—
261#41—121no—
262#41—121no—
263#41—121no—
264C2×Q8—21no—
265C2×Q8—21no—
266C2×Q8—21no—
267C2×Q8—21no—
268C2×Q8—21no—
269C2×Q8—61no—
270C2×Q8—62no—
271C2×Q8—121no—
272C2×Q8—121no—
273C2×Q8—121no—
274C2×Q8—121no—
275C2×Q8—121no—
276C2×Q8D2412yes—
277C2×D4—61no—
278C2×D4—61no—
279C2×D4—61no—
280C2×D4—121no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—121no—
284C22×C4—121no—
285C22×C4C2×D1211yescharacteristic
286C22×C4#23511yescharacteristic
287C22×C4C2×#6711yescharacteristic
288Q16—128no—
289Q16—124no—
290SD16—124no—
291M16—121no—
292M16—121no—
293C2×C8—22no—
294C2×C8—41no—
295C2×C8—62no—
296C2×C8—62no—
297C2×C8—121no—
298C2×C8—121no—
299C4⋊C4—21no—
300C4⋊C4—21no—
301C4⋊C4—41no—
302C4⋊C4—41no—
303C4⋊C4—62no—
304C4⋊C4—61no—
305C4⋊C4—62no—
306C4⋊C4—61no—
307C4⋊C4—121no—
308C4⋊C4—121no—
309C22⋊C4—121no—
310C22⋊C4—121no—
311C42—41no—
312C42—41no—
313C42D3×D411yescharacteristic
314C42D3×D411yescharacteristic
315C42D3×D411yescharacteristic
316C42D3×D411yescharacteristic
317C22×C3—21no—
318C22×C3—21no—
319C22×C3#39211yescharacteristic
320C22×C3#46611yescharacteristic
321C22×C3#46911yescharacteristic
322C2×D3—81no—
323C2×D3—81no—
324C2×D3—81no—
325C3×C4—21no—
326C3×C4—21no—
327C3×C4—21no—
328C3×C4—42no—
329C3×C4—81no—
330C3×C4#44711yescharacteristic
331C3×C4#44711yescharacteristic
332C3×C4#44911yescharacteristic
333C3×C4#44911yescharacteristic
334C3×C4#47312yes—
335Dic3—82no—
336Dic3—81no—
337C23—121no—
338C23#93011yescharacteristic
339Q8—22no—
340Q8—22no—
341Q8—41no—
342Q8—41no—
343Q8—122no—
344Q8—121no—
345Q8—122no—
346Q8—121no—
347Q8—241no—
348Q8—241no—
349D4—122no—
350D4—121no—
351D4—121no—
352D4—122no—
353D4—241no—
354D4—241no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—81no—
362C2×C4—121no—
363C2×C4—122no—
364C2×C4—241no—
365C2×C4—241no—
366C2×C4—241no—
367C2×C4—241no—
368C2×C4#88211yescharacteristic
369C2×C4#88211yescharacteristic
370C2×C4#88411yescharacteristic
371C2×C4#88411yescharacteristic
372C2×C4C2×D2411yescharacteristic
373C2×C4#90411yescharacteristic
374C2×C4#90811yescharacteristic
375C2×C4#90811yescharacteristic
376C2×C4#93711yescharacteristic
377C2×C4#94011yescharacteristic
378C8—42no—
379C8—81no—
380C8—122no—
381C8—122no—
382C2×C3—21no—
383C2×C3—21no—
384C2×C3#199911yescharacteristic
385C2×C3#200911yescharacteristic
386C2×C3#202011yescharacteristic
387D3—161no—
388C22—21no—
389C22—21no—
390C22—241no—
391C22—241no—
392C22—241no—
393C22#496011yescharacteristic
394C22#511511yescharacteristic
395C22#513511yescharacteristic
396C4—21no—
397C4—21no—
398C4—21no—
399C4—42no—
400C4—81no—
401C4—241no—
402C4—242no—
403C4#473211yescharacteristic
404C4#473211yescharacteristic
405C4#473411yescharacteristic
406C4#473411yescharacteristic
407C4#475512yes—
408C3#1018511yescharacteristic, complemented, Sylow
409C2—21no—
410C2—21no—
411C2—481no—
412C2#6996111yescharacteristic
413C2#7005311yescharacteristic
414C2#7012511yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1461C2
socleC22×C3#392
prime radical#5477C22
hypercenter#1461D3
hypocenterC3#10185
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1461

Complements

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

NH ≅ G/Ntype
1. C2×#5031C2semidirect
4. C3×#1461C2semidirect
5. #70778C2semidirect
7. #67745C2semidirect
37. #1461D3semidirect
408. C3#10185semidirect

Maximal subgroups

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

subgroupindex
C2×#50312
#824412
#792682
C3×#14612
#707782
#707582
#677452
#101853