Finite Group Databasebeta

← #10696159

Subgroups — #10696159

Order 768.

550 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 95 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#82314C211yesmaximal, characteristic, complemented
2#82308C211yesmaximal, characteristic
3#78497C211yesmaximal, characteristic, complemented
4C3×#1955C211yesmaximal, characteristic, complemented
5#70429C211yesmaximal, characteristic, complemented
6#70426C211yesmaximal, characteristic, complemented
7#67660C211yesmaximal, characteristic, complemented
8#10033—31nomaximal, Sylow
9C2×C3×C4×D4C2211yescharacteristic
10#5477C2211yescharacteristic
11#5470C2211yescharacteristic
12C2×C3×C4⋊C8C2211yescharacteristic
13C2×C3×C22⋊C8—22no—
14C2×C3×#97C2211yescharacteristic
15#5218—21no—
16#5217—21no—
17#5113—22no—
18#5019—21no—
19C4×#214—22no—
20#5015—21no—
21C4×#213—22no—
22#5008—21no—
23#5007—21no—
24#4971—21no—
25#4967—21no—
26#4925C2211yescharacteristic
27#4925C2211yescharacteristic
28#4714—21no—
29#4714—21no—
30#4713—21no—
31#4703—21no—
32#4484—22no—
33#4483—22no—
34#3135—31no—
35#3101—31no—
36#3100—31no—
37#1955D311yescharacteristic, complemented
38#1481—31no—
39#1480—31no—
40#1275—31no—
41C22×C3×D4D411yescharacteristic
42C23×C3×C4—21no—
43C2×#235—21no—
44C22×C3×C8—21no—
45C22×C3×C8D412yes—
46C3×C4×D4—21no—
47C3×C4×D4—21no—
48C3×C4×D4—21no—
49C3×C4×D4—21no—
50C2×C3×C4⋊C4D411yescharacteristic
51C2×C3×C22⋊C4—21no—
52C2×C3×C42C2311yescharacteristic
53C2×#214—22no—
54C2×#213—22no—
55#930—21no—
56#930—21no—
57#926—21no—
58#924—21no—
59#924—21no—
60#875—21no—
61#874D411yescharacteristic
62C4×D12—21no—
63C4×D12—21no—
64#870—21no—
65#869D411yescharacteristic
66#869D411yescharacteristic
67#869D411yescharacteristic
68C4×Dic6—21no—
69C4×Dic6—21no—
70C3×C4⋊C8—21no—
71C3×C4⋊C8—21no—
72C3×C22⋊C8—44no—
73C3×#97—21no—
74C3×#97—21no—
75#832—22no—
76#832—22no—
77#830—21no—
78#830—21no—
79#821—22no—
80#821—22no—
81#816—22no—
82#816—22no—
83#810—22no—
84#809—22no—
85#808—22no—
86#807—22no—
87C4⋊(C3⋊C8)—21no—
88C4⋊(C3⋊C8)—21no—
89C4×C3⋊C8D412yes—
90C4×C3⋊C8D412yes—
91#533—31no—
92#532—31no—
93C2×C4×D4C2×D311yescharacteristic
94#491—61no—
95#490—61no—
96#489—61no—
97#489—61no—
98#483—61no—
99#480—61no—
100#476—61no—
101#474—61no—
102#461—61no—
103#459—61no—
104#450—61no—
105#447—61no—
106C4×SD16—62no—
107C4×D8—62no—
108#431—31no—
109#431—31no—
110C2×C4⋊C8C2×D311yescharacteristic
111#416—62no—
112C2×C22⋊C8—22no—
113C2×#97C2×D311yescharacteristic
114Q8⋊C8—62no—
115D4⋊C8—62no—
116C24×C3—21no—
117C2×C3×D4—21no—
118C2×C3×D4—21no—
119C2×C3×D4—42no—
120C22×C3×C4—21no—
121C22×C3×C4—41no—
122C22×C3×C4—41no—
123C22×C3×C4—42no—
124C22×C3×C4—42no—
125C22×C3×C4C2×D411yescharacteristic
126C22×C3×C4C2×D411yescharacteristic
127C22×C3×C4#4111yescharacteristic
128C2×#67—41no—
129#235—42no—
130#235—42no—
131C2×D12—21no—
132C2×D12—21no—
133C2×D12—21no—
134C2×C4×D3—41no—
135C2×Dic6—21no—
136C2×Dic6—21no—
137C2×Dic6—21no—
138C2×C3×C8—22no—
139C2×C3×C8—22no—
140C2×C3×C8—22no—
141C2×C3×C8—41no—
142C2×C3×C8—42no—
143C3×C4⋊C4—21no—
144C3×C4⋊C4—21no—
145C3×C22⋊C4—41no—
146C3×C22⋊C4—41no—
147C3×C42C2×D411yescharacteristic
148C3×C42C2×D411yescharacteristic
149C3×C42C2×D411yescharacteristic
150C3×C42C2×D411yescharacteristic
151#217—41no—
152#214—44no—
153#213—44no—
154(C2×D3)⋊C4—41no—
155(C2×D3)⋊C4—41no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160C4⋊Dic3—21no—
161C4⋊Dic3—21no—
162Dic3⋊C4—41no—
163Dic3⋊C4—41no—
164C4×Dic3—41no—
165C2×C3⋊C8—22no—
166C2×C3⋊C8—22no—
167C2×#41—61no—
168C22×D4D1211yescharacteristic
169C23×C4—21no—
170C2×SD16—62no—
171C2×D8—62no—
172C22×C8—21no—
173C22×C8#6712yes—
174C4⋊Q8—31no—
175C4⋊Q8—31no—
176C4⋊Q8—31no—
177#127—31no—
178#125—61no—
179#124—61no—
180C22⋊Q8—61no—
181C22⋊Q8—61no—
182#121—61no—
183#121—61no—
184C4×Q8—61no—
185C4×Q8—61no—
186C4×D4—21no—
187C4×D4—21no—
188C4×D4—21no—
189C4×D4—21no—
190C4×D4—61no—
191C4×D4—61no—
192#117—61no—
193C2×C4⋊C4D1211yescharacteristic
194C2×C22⋊C4—21no—
195C2×C42C22×D311yescharacteristic
196#107—62no—
197#106—62no—
198C4⋊C8—21no—
199C4⋊C8—21no—
200C4⋊C8—61no—
201C4⋊C8—61no—
202#103—62no—
203#103—62no—
204#103—62no—
205#102—62no—
206#102—62no—
207#102—62no—
208#102—62no—
209#102—62no—
210C22⋊C8—44no—
211C22⋊C8—61no—
212C22⋊C8—61no—
213#97—21no—
214#97—21no—
215C4×C8—32no—
216C4×C8—32no—
217C23×C3—22no—
218C23×C3—41no—
219C23×C3—42no—
220C23×C3—41no—
221C23×C3—42no—
222C23×C3#12111yescharacteristic
223C22×D3—41no—
224C3×D4—22no—
225C3×D4—22no—
226C3×D4—41no—
227C3×D4—41no—
228C2×C3×C4—41no—
229C2×C3×C4—41no—
230C2×C3×C4—41no—
231C2×C3×C4—42no—
232C2×C3×C4—82no—
233C2×C3×C4—81no—
234C2×C3×C4—81no—
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12011yescharacteristic
237C2×C3×C4#12011yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12111yescharacteristic
240C2×C3×C4#12711yescharacteristic
241C2×C3×C4#13512yes—
242C2×C3×C4AGL(1,ℤ/8)12yes—
243#67—82no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247D12—42no—
248D12—42no—
249D12—42no—
250C4×D3—81no—
251C4×D3—81no—
252Dic6—42no—
253Dic6—42no—
254Dic6—42no—
255C3×C8—42no—
256C3×C8—81no—
257C3⋊C8—42no—
258C3⋊C8—42no—
259C24—21no—
260#41—122no—
261#41—122no—
262C2×Q8—61no—
263C2×Q8—61no—
264C2×Q8—61no—
265C2×D4—21no—
266C2×D4—21no—
267C2×D4—42no—
268C2×D4—61no—
269C2×D4—61no—
270C2×D4—61no—
271C2×D4—121no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—41no—
275C22×C4—41no—
276C22×C4—42no—
277C22×C4—121no—
278C22×C4C2×D1211yescharacteristic
279C22×C4#23511yescharacteristic
280C22×C4C2×#6711yescharacteristic
281SD16—124no—
282D8—124no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8—22no—
286C2×C8—42no—
287C2×C8—41no—
288C2×C8—62no—
289C2×C8—62no—
290C4⋊C4—21no—
291C4⋊C4—21no—
292C4⋊C4—61no—
293C4⋊C4—61no—
294C4⋊C4—61no—
295C4⋊C4—61no—
296C4⋊C4—61no—
297C4⋊C4—61no—
298C4⋊C4—121no—
299C4⋊C4—121no—
300C22⋊C4—41no—
301C22⋊C4—41no—
302C22⋊C4—121no—
303C22⋊C4—121no—
304C22⋊C4—121no—
305C42—121no—
306C42D3×D411yescharacteristic
307C42D3×D411yescharacteristic
308C42D3×D411yescharacteristic
309C42D3×D411yescharacteristic
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—44no—
313C22×C3—42no—
314C22×C3—82no—
315C22×C3—81no—
316C22×C3—81no—
317C22×C3#39211yescharacteristic
318C22×C3#46312yes—
319C2×D3—81no—
320C2×D3—81no—
321C2×D3—81no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—21no—
326C3×C4—81no—
327C3×C4#44912yes—
328C3×C4#49511yescharacteristic
329C3×C4#49611yescharacteristic
330Dic3—81no—
331Dic3—81no—
332Dic3—81no—
333C23—22no—
334C23—41no—
335C23—42no—
336C23—41no—
337C23—42no—
338C23—121no—
339C23#93011yescharacteristic
340Q8—122no—
341Q8—122no—
342Q8—122no—
343D4—22no—
344D4—22no—
345D4—41no—
346D4—41no—
347D4—122no—
348D4—122no—
349D4—122no—
350D4—242no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—82no—
356C2×C4—81no—
357C2×C4—81no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—241no—
362C2×C4—241no—
363C2×C4#88211yescharacteristic
364C2×C4#88211yescharacteristic
365C2×C4#88411yescharacteristic
366C2×C4#88411yescharacteristic
367C2×C4#93212yes—
368C2×C4#93711yescharacteristic
369C2×C4#94011yescharacteristic
370C2×C4#95012yes—
371C8—42no—
372C8—81no—
373C8—122no—
374C8—122no—
375C2×C3—22no—
376C2×C3—82no—
377C2×C3#201011yescharacteristic
378C2×C3#201211yescharacteristic
379C2×C3#201711yescharacteristic
380D3—161no—
381C22—21no—
382C22—21no—
383C22—44no—
384C22—42no—
385C22—82no—
386C22—81no—
387C22—81no—
388C22—241no—
389C22—241no—
390C22—241no—
391C22#496011yescharacteristic
392C22#501612yes—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—81no—
398C4—241no—
399C4—241no—
400C4—241no—
401C4#503712yes—
402C4#505911yescharacteristic
403C4#507411yescharacteristic
404C3#1003311yescharacteristic, complemented, Sylow
405C2—22no—
406C2—82no—
407C2—481no—
408C2#6965911yescharacteristic
409C2#6999311yescharacteristic
410C2#7001311yescharacteristic
subgroupG/H
centerC2×C4#940
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1955C2
socleC22×C3#392
prime radical#82308C2
hypercenter#1955D3
hypocenterC3#10033
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1955

Complements

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

NH ≅ G/Ntype
1. #82314C2semidirect
3. #78497C2semidirect
4. C3×#1955C2semidirect
5. #70429C2semidirect
6. #70426C2semidirect
7. #67660C2semidirect
37. #1955D3semidirect
404. C3#10033semidirect

Maximal subgroups

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

subgroupindex
#823142
#823082
#784972
C3×#19552
#704292
#704262
#676602
#100333