Finite Group Databasebeta

← #10697339

Subgroups — #10697339

Order 768.

598 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 598 conjugacy classes of subgroups — too many to draw. Only its 109 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×#3152C211yesmaximal, characteristic, complemented
2#83310C211yesmaximal, characteristic
3#83215C211yesmaximal, characteristic, complemented
4#71811C211yesmaximal, characteristic, complemented
5#71780C211yesmaximal, characteristic, complemented
6#69867C211yesmaximal, characteristic, complemented
7#67668C211yesmaximal, characteristic, complemented
8#10204—31nomaximal, Sylow
9C2×C3×#137—22no—
10C3×#532C2211yescharacteristic
11C2×C3×C4⋊Q8C2211yescharacteristic
12#5477C2211yescharacteristic
13C3×#497—22no—
14C3×#494—21no—
15C3×#492—21no—
16C3×C4×M16C2211yescharacteristic
17C2×#825—22no—
18#5088—21no—
19#5087—22no—
20#5087—21no—
21#5078—21no—
22#5072—21no—
23#5038—22no—
24#5015—21no—
25#5007—21no—
26#4967—22no—
27C4×C4⋊Dic3C2211yescharacteristic
28#4925C2211yescharacteristic
29C2×C4⋊(C3⋊C8)C2211yescharacteristic
30#4537—22no—
31#4529—22no—
32#3245—31no—
33#3226—31no—
34#3152D311yescharacteristic, complemented
35#1988—31no—
36#1980—31no—
37#1502—31no—
38#1283—31no—
39C2×C3×#41—21no—
40C22×C3×Q8—21no—
41C2×#235—21no—
42C3×#137—44no—
43C3×#137—44no—
44C2×C3×Q16—24no—
45C2×C3×SD16—24no—
46C2×C3×M16D412yes—
47C3×C4⋊Q8—21no—
48C3×C4⋊Q8—21no—
49C3×C4⋊Q8—21no—
50C3×C4⋊Q8D412yes—
51C3×C4⋊Q8D411yescharacteristic
52C3×#127D411yescharacteristic
53C3×#124—21no—
54C3×#121—21no—
55C3×#121—21no—
56C3×C4×D4—21no—
57C3×C4×D4—21no—
58C2×C3×C4⋊C4—21no—
59C2×C3×C4⋊C4—21no—
60C2×C3×C42C2311yescharacteristic
61C2×#213—22no—
62#930—21no—
63#930—21no—
64C2×C4⋊Dic3—21no—
65C2×C4×Dic3—21no—
66C22×C3⋊C8D412yes—
67#875—21no—
68#874D411yescharacteristic
69C4×D12—21no—
70C4×D12—21no—
71#869D411yescharacteristic
72C3×#97—21no—
73C3×C4×C8—21no—
74#832—22no—
75C3⋊#95—21no—
76#830D412yes—
77#825—44no—
78#809—24no—
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—22no—
87#532—31no—
88#532C2×D311yescharacteristic
89C2×C4⋊Q8C2×D311yescharacteristic
90#500—61no—
91#497—22no—
92#494—21no—
93#492—21no—
94#486—61no—
95#480—62no—
96#474—61no—
97#474—62no—
98#474—61no—
99#466—62no—
100#459—61no—
101#459—61no—
102#431—31no—
103C2×C4⋊C8—31no—
104C2×#104—62no—
105C4×M16C2×D311yescharacteristic
106C4×C4⋊C4—31no—
107#340—62no—
108#327—62no—
109C3×#41—42no—
110C3×#41—42no—
111C2×C3×Q8—21no—
112C2×C3×Q8—22no—
113C2×C3×Q8—41no—
114C2×C3×Q8—42no—
115C2×C3×Q8—42no—
116C2×C3×D4—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—41no—
120C22×C3×C4—41no—
121C22×C3×C4—41no—
122C22×C3×C4C2×D411yescharacteristic
123C22×C3×C4C2×D411yescharacteristic
124C22×C3×C4C2×D411yescharacteristic
125C2×#67—41no—
126C22×Dic3—41no—
127#235—42no—
128#235—42no—
129C2×D12—21no—
130C2×D12—21no—
131C2×D12—21no—
132C2×C4×D3—41no—
133C2×Dic6—21no—
134C3×Q16—48no—
135C3×SD16—48no—
136C3×M16—22no—
137C3×M16D814yescomplemented
138C2×C3×C8—22no—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—21no—
141C3×C4⋊C4—22no—
142C3×C4⋊C4—21no—
143C3×C4⋊C4—41no—
144C3×C4⋊C4—41no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42#4111yescharacteristic
151#213—82no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—41no—
154C4⋊Dic3—22no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—41no—
158C4×Dic3—42no—
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161C2×C3⋊C8—42no—
162C2×#41—21no—
163C2×#41—61no—
164C22×Q8—21no—
165#137—44no—
166#137—44no—
167C2×Q16—24no—
168C2×SD16—24no—
169C2×D8—62no—
170C2×M16#6712yes—
171C22×C8—32no—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—31no—
176C4⋊Q8#6712yes—
177C4⋊Q8#6711yescharacteristic
178#127—31no—
179#127#6711yescharacteristic
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—21no—
192C2×C4⋊C4—61no—
193C2×C42—61no—
194C2×C42C22×D311yescharacteristic
195#107—62no—
196#107—64no—
197C4⋊C8—61no—
198C4⋊C8—61no—
199C4⋊C8—61no—
200#104—124no—
201#102—62no—
202#102—64no—
203#102—62no—
204C22⋊C8—32no—
205#97—21no—
206C4×C8—21no—
207C4×C8—61no—
208#95—61no—
209C23×C3—41no—
210C23×C3#12711yescharacteristic
211C22×D3—41no—
212C3×Q8—44no—
213C3×Q8—42no—
214C3×Q8—82no—
215C3×D4—42no—
216C3×D4—41no—
217C3×D4—41no—
218C3×D4—81no—
219C3×D4—81no—
220C3×D4—81no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—42no—
226C2×C3×C4—81no—
227C2×C3×C4—81no—
228C2×C3×C4—82no—
229C2×C3×C4#12011yescharacteristic
230C2×C3×C4#12011yescharacteristic
231C2×C3×C4#12011yescharacteristic
232C2×C3×C4#12111yescharacteristic
233C2×C3×C4#12111yescharacteristic
234C2×C3×C4#12111yescharacteristic
235C2×C3×C4C2×D812yes—
236C2×C3×C4AGL(1,ℤ/8)12yes—
237#67—81no—
238#67—81no—
239C2×Dic3—42no—
240C2×Dic3—41no—
241C2×Dic3—81no—
242C2×Dic3—81no—
243D12—42no—
244D12—41no—
245D12—41no—
246D12—42no—
247C4×D3—82no—
248Dic6—42no—
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—22no—
258C2×Q8—41no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8—61no—
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—48no—
278SD16—48no—
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—22no—
289C4⋊C4—21no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—62no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—121no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C42—122no—
301C42D3×D411yescharacteristic
302C42#23711yescharacteristic
303C42C2×#6711yescharacteristic
304C42C2×#6711yescharacteristic
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—81no—
320C3×C4—82no—
321C3×C4#44712yes—
322C3×C4#45511yescharacteristic
323C3×C4#45611yescharacteristic
324C3×C4#45912yes—
325Dic3—82no—
326Dic3—81no—
327C23—41no—
328C23—121no—
329C23#94011yescharacteristic
330Q8—44no—
331Q8—42no—
332Q8—82no—
333Q8—122no—
334D4—41no—
335D4—41no—
336D4—42no—
337D4—81no—
338D4—81no—
339D4—81no—
340D4—122no—
341D4—121no—
342D4—121no—
343D4—122no—
344D4—241no—
345D4—241no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—82no—
354C2×C4—122no—
355C2×C4—121no—
356C2×C4—241no—
357C2×C4—241no—
358C2×C4—242no—
359C2×C4C2×#21312yes—
360C2×C4#93212yes—
361C2×C4#93711yescharacteristic
362C2×C4#93711yescharacteristic
363C2×C4#93811yescharacteristic
364C2×C4#93911yescharacteristic
365C2×C4#93911yescharacteristic
366C2×C4#95311yescharacteristic
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#507311yescharacteristic
386C22#507411yescharacteristic
387C22#522311yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—81no—
392C4—82no—
393C4—242no—
394C4—241no—
395C4#507212yes—
396C4#520211yescharacteristic
397C4#520511yescharacteristic
398C4#521712yes—
399C3#1020411yescharacteristic, complemented, Sylow
400C2—21no—
401C2—21no—
402C2—161no—
403C2—481no—
404C2#7176711yescharacteristic
405C2#7211011yescharacteristic
406C2#7211111yescharacteristic
subgroupG/H
centerC22#5223
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3152C2
socleC22×C3#392
prime radical#83310C2
hypercenter#3152D3
hypocenterC3#10204
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3152

Complements

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

NH ≅ G/Ntype
1. C3×#3152C2semidirect
3. #83215C2semidirect
4. #71811C2semidirect
5. #71780C2semidirect
6. #69867C2semidirect
7. #67668C2semidirect
34. #3152D3semidirect
137. 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
C3×#31522
#833102
#832152
#718112
#717802
#698672
#676682
#102043