Finite Group Databasebeta

← #10697340

Subgroups — #10697340

Order 768.

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

normal subgroups only

This group has 598 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
1#83215C211yesmaximal, characteristic, complemented
2#83203C211yesmaximal, characteristic, complemented
3#79026C211yesmaximal, characteristic
4C3×#1988C211yesmaximal, characteristic, complemented
5#69492C211yesmaximal, characteristic, complemented
6#69469C211yesmaximal, characteristic, complemented
7#67706C211yesmaximal, characteristic, complemented
8#10204—31nomaximal, Sylow
9C2×C3×C4⋊Q8C2211yescharacteristic
10C2×#942—22no—
11#5477C2211yescharacteristic
12#5477C2211yescharacteristic
13C2×C3×C4⋊C8C2211yescharacteristic
14C3×C4×C4⋊C4C2211yescharacteristic
15#5115—22no—
16#5092—21no—
17#5088—21no—
18#5087—22no—
19#5086—22no—
20#5085—21no—
21#5078—21no—
22#4967—22no—
23C2×#805—22no—
24#4925C2211yescharacteristic
25C4×#208C2211yescharacteristic
26#4846—21no—
27#4846—21no—
28#4843—21no—
29#4843—21no—
30C3×#340—22no—
31#4451—22no—
32#3245—31no—
33#3226—31no—
34#3152—31no—
35#1988D311yescharacteristic, complemented
36#1980—31no—
37#1502—31no—
38#1283—31no—
39C22×C3×Q8—21no—
40C2×#235—21no—
41C2×#235—21no—
42C22×C3×C8D412yes—
43C3×C4⋊Q8—21no—
44C3×C4⋊Q8—21no—
45C3×C4⋊Q8—21no—
46C3×C4⋊Q8D412yes—
47C2×C3×C4⋊C4—21no—
48C2×C3×C4⋊C4—21no—
49C2×C3×C4⋊C4—21no—
50C2×C3×C42—21no—
51C2×C3×C42C2311yescharacteristic
52C2×#216—24no—
53#942—44no—
54#942—44no—
55C2×#215—24no—
56#930—21no—
57#930—21no—
58#930—21no—
59#930—21no—
60C2×#208D412yes—
61C2×D24—22no—
62#875—21no—
63#875—21no—
64#874D411yescharacteristic
65#874D411yescharacteristic
66C4×D12—21no—
67C4×D12—21no—
68C4×D12—21no—
69C4×D12—21no—
70#869D411yescharacteristic
71#869D411yescharacteristic
72C3×C4⋊C8—21no—
73C3×C4⋊C8—21no—
74C3×#95—21no—
75#830D412yes—
76#821—22no—
77#821—22no—
78#818—22no—
79#809—24no—
80#807—24no—
81#805—44no—
82C4⋊(C3⋊C8)—21no—
83#803—21no—
84C4×C3⋊C8—21no—
85C4×C3⋊C8—21no—
86C2×#137—62no—
87#532—31no—
88#532—31no—
89C2×C4⋊Q8C2×D311yescharacteristic
90#500—61no—
91#497—62no—
92#494—61no—
93#492—61no—
94#486—61no—
95#480—62no—
96#474—61no—
97#474—61no—
98#474—62no—
99#466—62no—
100#459—61no—
101#459—61no—
102#431—31no—
103C2×C4⋊C8C2×D311yescharacteristic
104C2×#104—62no—
105C4×M16—31no—
106C4×C4⋊C4C2×D311yescharacteristic
107#340—22no—
108#327—62no—
109C2×C3×Q8—22no—
110C2×C3×Q8—41no—
111C2×C3×Q8—42no—
112C2×C3×Q8—42no—
113C22×C3×C4—41no—
114C22×C3×C4—41no—
115C22×C3×C4C2×D411yescharacteristic
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4C2×D411yescharacteristic
118C2×#67—41no—
119C2×#67—41no—
120#235—42no—
121#235—42no—
122#235—42no—
123#235—42no—
124C2×D12—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—21no—
128C2×D12—21no—
129C2×D12—21no—
130C2×C4×D3—41no—
131C2×C4×D3—41no—
132C2×Dic6—21no—
133C2×Dic6—21no—
134C2×C3×C8—22no—
135C2×C3×C8—42no—
136C3×C4⋊C4—22no—
137C3×C4⋊C4—22no—
138C3×C4⋊C4—22no—
139C3×C4⋊C4—41no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C42—42no—
143C3×C42C2×D411yescharacteristic
144C3×C42C2×D411yescharacteristic
145C3×C42C2×D411yescharacteristic
146C3×C42#4111yescharacteristic
147#216—48no—
148#215—48no—
149(C2×D3)⋊C4—41no—
150(C2×D3)⋊C4—41no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153C4⋊Dic3—21no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157#208—22no—
158#208D814yescomplemented
159C2×C3⋊C8—22no—
160C2×C3⋊C8—22no—
161D24—82no—
162C2×#41—61no—
163C2×#41—61no—
164C22×Q8—21no—
165#137—124no—
166#137—124no—
167C2×Q16—64no—
168C2×SD16—64no—
169C2×D8—62no—
170C2×M16—32no—
171C22×C8#6712yes—
172C4⋊Q8—21no—
173C4⋊Q8—21no—
174C4⋊Q8—21no—
175C4⋊Q8—31no—
176C4⋊Q8—31no—
177C4⋊Q8D1212yes—
178#127—31no—
179#127—31no—
180#124—61no—
181#124—61no—
182#121—61no—
183#121—61no—
184#121—61no—
185#121—61no—
186C4×D4—61no—
187C4×D4—61no—
188C4×D4—61no—
189C4×D4—61no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C42—21no—
194C2×C42C22×D311yescharacteristic
195#107—64no—
196#107—62no—
197C4⋊C8—21no—
198C4⋊C8—21no—
199C4⋊C8—61no—
200#104—124no—
201#102—64no—
202#102—62no—
203#102—62no—
204C22⋊C8—32no—
205#97—61no—
206C4×C8—61no—
207C4×C8—61no—
208#95—21no—
209C23×C3#12711yescharacteristic
210C22×D3—41no—
211C22×D3—41no—
212C3×Q8—44no—
213C3×Q8—82no—
214C2×C3×C4—21no—
215C2×C3×C4—21no—
216C2×C3×C4—41no—
217C2×C3×C4—42no—
218C2×C3×C4—42no—
219C2×C3×C4—81no—
220C2×C3×C4—81no—
221C2×C3×C4—81no—
222C2×C3×C4—81no—
223C2×C3×C4#12011yescharacteristic
224C2×C3×C4#12011yescharacteristic
225C2×C3×C4#12011yescharacteristic
226C2×C3×C4#12111yescharacteristic
227C2×C3×C4#12111yescharacteristic
228C2×C3×C4#12111yescharacteristic
229C2×C3×C4C2×D812yes—
230C2×C3×C4AGL(1,ℤ/8)12yes—
231#67—81no—
232#67—81no—
233#67—81no—
234#67—81no—
235C2×Dic3—41no—
236C2×Dic3—41no—
237D12—42no—
238D12—42no—
239D12—41no—
240D12—41no—
241D12—41no—
242D12—41no—
243D12—42no—
244D12—81no—
245C4×D3—82no—
246C4×D3—82no—
247Dic6—42no—
248Dic6—42no—
249C3×C8—42no—
250C3⋊C8—24no—
251C3⋊C8—42no—
252#41—122no—
253#41—122no—
254#41—122no—
255#41—122no—
256C2×Q8—22no—
257C2×Q8—41no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—61no—
261C2×Q8—61no—
262C2×D4—61no—
263C2×D4—61no—
264C2×D4—61no—
265C2×D4—61no—
266C2×D4—61no—
267C2×D4—61no—
268C2×D4—121no—
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—34no—
281M16—62no—
282C2×C8—22no—
283C2×C8—42no—
284C2×C8—62no—
285C2×C8—62no—
286C4⋊C4—22no—
287C4⋊C4—22no—
288C4⋊C4—22no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C22⋊C4—121no—
297C22⋊C4—121no—
298C22⋊C4—121no—
299C22⋊C4—121no—
300C42—42no—
301C42C2×D1211yescharacteristic
302C42#23511yescharacteristic
303C42D3×D411yescharacteristic
304C42D3×D411yescharacteristic
305C22×C3—21no—
306C22×C3—21no—
307C22×C3#39211yescharacteristic
308C22×C3#49311yescharacteristic
309C22×C3#49611yescharacteristic
310C2×D3—81no—
311C2×D3—81no—
312C2×D3—81no—
313C2×D3—81no—
314C2×D3—81no—
315C2×D3—81no—
316C3×C4—21no—
317C3×C4—21no—
318C3×C4—21no—
319C3×C4—82no—
320C3×C4—82no—
321C3×C4#44712yes—
322C3×C4#45511yescharacteristic
323C3×C4#45611yescharacteristic
324C3×C4#45912yes—
325Dic3—81no—
326Dic3—81no—
327C23—121no—
328C23—121no—
329C23#94011yescharacteristic
330Q8—44no—
331Q8—82no—
332Q8—122no—
333Q8—122no—
334D4—122no—
335D4—122no—
336D4—122no—
337D4—121no—
338D4—121no—
339D4—121no—
340D4—121no—
341D4—241no—
342D4—241no—
343D4—241no—
344D4—241no—
345D4—241no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—41no—
349C2×C4—42no—
350C2×C4—42no—
351C2×C4—81no—
352C2×C4—81no—
353C2×C4—81no—
354C2×C4—81no—
355C2×C4—121no—
356C2×C4—121no—
357C2×C4—242no—
358C2×C4—242no—
359C2×C4#88211yescharacteristic
360C2×C4#88211yescharacteristic
361C2×C4#88411yescharacteristic
362C2×C4#88411yescharacteristic
363C2×C4D3×D812yes—
364C2×C4#91412yes—
365C2×C4#93011yescharacteristic
366C2×C4#93711yescharacteristic
367C8—42no—
368C8—64no—
369C8—122no—
370C2×C3—21no—
371C2×C3—21no—
372C2×C3#200711yescharacteristic
373C2×C3#200911yescharacteristic
374C2×C3#201111yescharacteristic
375D3—161no—
376D3—161no—
377C22—21no—
378C22—21no—
379C22—241no—
380C22—241no—
381C22—241no—
382C22—241no—
383C22—241no—
384C22—241no—
385C22#496011yescharacteristic
386C22#515111yescharacteristic
387C22#517611yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—82no—
392C4—82no—
393C4—241no—
394C4—241no—
395C4#477312yes—
396C4#478612yes—
397C4#506111yescharacteristic
398C4#509511yescharacteristic
399C3#1020411yescharacteristic, complemented, Sylow
400C2—21no—
401C2—21no—
402C2—481no—
403C2—481no—
404C2#6969411yescharacteristic
405C2#7020911yescharacteristic
406C2#7035111yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1988C2
socleC22×C3#392
prime radical#79026C2
hypercenter#1988D3
hypocenterC3#10204
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1988

Complements

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

NH ≅ G/Ntype
1. #83215C2semidirect
2. #83203C2semidirect
4. C3×#1988C2semidirect
5. #69492C2semidirect
6. #69469C2semidirect
7. #67706C2semidirect
35. #1988D3semidirect
158. #208D8semidirect
399. C3#10204semidirect

Maximal subgroups

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

subgroupindex
#832152
#832032
#790262
C3×#19882
#694922
#694692
#677062
#102043