Finite Group Databasebeta

← #10697342

Subgroups — #10697342

Order 768.

598 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 107 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#83310C211yesmaximal, characteristic
2#83270C211yesmaximal, characteristic, complemented
3#78538C211yesmaximal, characteristic, complemented
4C3×#1980C211yesmaximal, characteristic, complemented
5#69490C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic, complemented
7#67668C211yesmaximal, characteristic, complemented
8#10204—31nomaximal, Sylow
9C3×#532C2211yescharacteristic
10C2×#909—22no—
11#5477C2211yescharacteristic
12C2×#869C2211yescharacteristic
13C2×C3×#104—22no—
14C3×C4×M16C2211yescharacteristic
15C3×C4×C4⋊C4C2211yescharacteristic
16#5087—21no—
17#5078—21no—
18#5072—21no—
19#5070—21no—
20#5038—22no—
21#5033—22no—
22#5015—21no—
23#5007—22no—
24#5007—21no—
25#4925C2211yescharacteristic
26C2×C4⋊(C3⋊C8)C2211yescharacteristic
27#4715—22no—
28#4699—21no—
29#4693—21no—
30#4529—22no—
31#4451—22no—
32#3245—31no—
33#3226—31no—
34#3152—31no—
35#1988—31no—
36#1980D311yescharacteristic, complemented
37#1502—31no—
38#1283—31no—
39C2×C3×#41—21no—
40C2×#235—21no—
41C22×Dic6—21no—
42C2×C3×M16D412yes—
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
53C2×#213—22no—
54#930—21no—
55#930—21no—
56C2×C4⋊Dic3—21no—
57C2×C4⋊Dic3—21no—
58C22×C3⋊C8D412yes—
59#909—44no—
60#909—44no—
61C2×Dic12—24no—
62C2×#204—24no—
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×#104—44no—
73C3×#97—21no—
74C3×C4×C8—21no—
75C3×#95—21no—
76#832—24no—
77#832—22no—
78#830D412yes—
79#809—22no—
80#807—24no—
81#807—22no—
82C4⋊(C3⋊C8)—21no—
83C4⋊(C3⋊C8)—21no—
84C4⋊(C3⋊C8)—21no—
85C4×C3⋊C8—21no—
86C2×#137—62no—
87#532—31no—
88#532C2×D311yescharacteristic
89C2×C4⋊Q8—31no—
90#500—61no—
91#497—62no—
92#494—61no—
93#492—61no—
94#486—61no—
95#480—62no—
96#474—62no—
97#474—61no—
98#474—61no—
99#466—62no—
100#459—61no—
101#459—61no—
102#431—31no—
103C2×C4⋊C8—31no—
104C2×#104—22no—
105C4×M16C2×D311yescharacteristic
106C4×C4⋊C4C2×D311yescharacteristic
107#340—62no—
108#327—62no—
109C3×#41—42no—
110C3×#41—42no—
111C2×C3×Q8—21no—
112C2×C3×D4—21no—
113C2×C3×D4—21no—
114C2×C3×D4—21no—
115C2×C3×D4—41no—
116C22×C3×C4—41no—
117C22×C3×C4—41no—
118C22×C3×C4C2×D411yescharacteristic
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C2×#67—41no—
122C22×Dic3—41no—
123#235—42no—
124#235—42no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×C4×D3—41no—
129C2×Dic6—22no—
130C2×Dic6—21no—
131C2×Dic6—42no—
132C2×Dic6—42no—
133C2×Dic6—41no—
134C3×M16—22no—
135C3×M16D814yescomplemented
136C2×C3×C8—22no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—21no—
140C3×C4⋊C4—41no—
141C3×C22⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42—42no—
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42#4111yescharacteristic
148#213—82no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151C4⋊Dic3—22no—
152C4⋊Dic3—21no—
153C4⋊Dic3—22no—
154C4⋊Dic3—21no—
155C4⋊Dic3—41no—
156C4⋊Dic3—41no—
157C2×C3⋊C8—22no—
158C2×C3⋊C8—22no—
159C2×C3⋊C8—42no—
160Dic12—48no—
161#204—48no—
162C2×#41—21no—
163C2×#41—61no—
164C22×Q8—61no—
165#137—124no—
166#137—124no—
167C2×Q16—64no—
168C2×SD16—64no—
169C2×D8—62no—
170C2×M16#6712yes—
171C22×C8—32no—
172C4⋊Q8—32no—
173C4⋊Q8—31no—
174C4⋊Q8—61no—
175C4⋊Q8—61no—
176C4⋊Q8—61no—
177C4⋊Q8D1211yescharacteristic
178#127—31no—
179#127D1211yescharacteristic
180#124—21no—
181#124—61no—
182#121—21no—
183#121—21no—
184#121—61no—
185#121—61no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—61no—
189C4×D4—61no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—61no—
192C2×C4⋊C4—61no—
193C2×C42—21no—
194C2×C42C22×D311yescharacteristic
195#107—62no—
196#107—64no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200#104—44no—
201#102—62no—
202#102—62no—
203#102—64no—
204C22⋊C8—32no—
205#97—21no—
206C4×C8—21no—
207C4×C8—61no—
208#95—21no—
209C23×C3—41no—
210C23×C3#12711yescharacteristic
211C22×D3—41no—
212C3×Q8—42no—
213C3×D4—42no—
214C3×D4—41no—
215C3×D4—41no—
216C3×D4—42no—
217C3×D4—81no—
218C3×D4—81no—
219C2×C3×C4—21no—
220C2×C3×C4—21no—
221C2×C3×C4—41no—
222C2×C3×C4—42no—
223C2×C3×C4—41no—
224C2×C3×C4—81no—
225C2×C3×C4—81no—
226C2×C3×C4—82no—
227C2×C3×C4#12011yescharacteristic
228C2×C3×C4#12011yescharacteristic
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12111yescharacteristic
231C2×C3×C4#12111yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4C2×D812yes—
234C2×C3×C4AGL(1,ℤ/8)12yes—
235#67—81no—
236#67—81no—
237C2×Dic3—42no—
238C2×Dic3—41no—
239C2×Dic3—81no—
240C2×Dic3—81no—
241D12—42no—
242D12—41no—
243D12—41no—
244D12—81no—
245C4×D3—82no—
246Dic6—44no—
247Dic6—42no—
248Dic6—82no—
249C3×C8—24no—
250C3⋊C8—42no—
251C3⋊C8—42no—
252#41—42no—
253#41—42no—
254#41—122no—
255#41—122no—
256C2×Q8—21no—
257C2×Q8—61no—
258C2×Q8—62no—
259C2×Q8—121no—
260C2×Q8—122no—
261C2×Q8—122no—
262C2×D4—21no—
263C2×D4—21no—
264C2×D4—21no—
265C2×D4—41no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—61no—
269C2×D4—121no—
270C22×C4—41no—
271C22×C4—41no—
272C22×C4—121no—
273C22×C4—121no—
274C22×C4D3×D411yescharacteristic
275C22×C4D3×D411yescharacteristic
276C22×C4C2×#6711yescharacteristic
277Q16—128no—
278SD16—128no—
279D8—242no—
280M16—22no—
281M16#21314yescomplemented
282C2×C8—22no—
283C2×C8—62no—
284C2×C8—62no—
285C2×C8—122no—
286C4⋊C4—22no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—62no—
291C4⋊C4—61no—
292C4⋊C4—62no—
293C4⋊C4—61no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C42—42no—
301C42C2×D1211yescharacteristic
302C42#23511yescharacteristic
303C42D3×D411yescharacteristic
304C42D3×D411yescharacteristic
305C22×C3—21no—
306C22×C3—21no—
307C22×C3—81no—
308C22×C3—81no—
309C22×C3—81no—
310C22×C3#39211yescharacteristic
311C22×C3#49311yescharacteristic
312C22×C3#49611yescharacteristic
313C2×D3—81no—
314C2×D3—81no—
315C2×D3—81no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—82no—
320C3×C4—81no—
321C3×C4#44712yes—
322C3×C4#45511yescharacteristic
323C3×C4#45611yescharacteristic
324C3×C4#45912yes—
325Dic3—82no—
326Dic3—81no—
327C23—41no—
328C23—121no—
329C23#94011yescharacteristic
330Q8—42no—
331Q8—124no—
332Q8—122no—
333Q8—242no—
334D4—42no—
335D4—42no—
336D4—41no—
337D4—41no—
338D4—81no—
339D4—81no—
340D4—122no—
341D4—121no—
342D4—121no—
343D4—241no—
344D4—241no—
345D4—241no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—41no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—121no—
355C2×C4—122no—
356C2×C4—241no—
357C2×C4—241no—
358C2×C4—242no—
359C2×C4#88211yescharacteristic
360C2×C4#88211yescharacteristic
361C2×C4#88411yescharacteristic
362C2×C4#88411yescharacteristic
363C2×C4#93011yescharacteristic
364C2×C4C2×#21312yes—
365C2×C4#93711yescharacteristic
366C2×C4#94912yes—
367C8—24no—
368C8—122no—
369C8—122no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3—161no—
373C2×C3#200711yescharacteristic
374C2×C3#200911yescharacteristic
375C2×C3#201111yescharacteristic
376D3—161no—
377C22—21no—
378C22—21no—
379C22—81no—
380C22—81no—
381C22—81no—
382C22—241no—
383C22—241no—
384C22—241no—
385C22#496011yescharacteristic
386C22#505811yescharacteristic
387C22#507311yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—82no—
392C4—81no—
393C4—241no—
394C4—242no—
395C4#482511yescharacteristic
396C4#482611yescharacteristic
397C4#501512yes—
398C4#503612yes—
399C3#1020411yescharacteristic, complemented, Sylow
400C2—21no—
401C2—21no—
402C2—161no—
403C2—481no—
404C2#6965811yescharacteristic
405C2#7039311yescharacteristic
406C2#7041211yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1980C2
socleC22×C3#392
prime radical#83310C2
hypercenter#1980D3
hypocenterC3#10204
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1980

Complements

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

NH ≅ G/Ntype
2. #83270C2semidirect
3. #78538C2semidirect
4. C3×#1980C2semidirect
5. #69490C2semidirect
6. #69469C2semidirect
7. #67668C2semidirect
36. #1980D3semidirect
135. C3×M16D8semidirect
281. M16#213semidirect
399. C3#10204semidirect

Maximal subgroups

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

subgroupindex
#833102
#832702
#785382
C3×#19802
#694902
#694692
#676682
#102043