Finite Group Databasebeta

← #10698883

Subgroups — #10698883

Order 768.

428 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 101 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×#3037C211yesmaximal, characteristic
2#71806C211yesmaximal, characteristic
3#71802C211yesmaximal, characteristic
4#71637C211yesmaximal, characteristic, complemented
5C2×#4535C211yesmaximal, characteristic, complemented
6#70490C211yesmaximal, characteristic, complemented
7#67711C211yesmaximal, characteristic, complemented
8#10459—31nomaximal, Sylow
9C2×C3×#137—21no—
10C3×#532C2211yescharacteristic
11C2×C3×C4×Q8C2211yescharacteristic
12C3×#464—22no—
13C3×#462—21no—
14C3×#460—21no—
15C3×#423C2211yescharacteristic
16C3×#417—21no—
17C2×#836—21no—
18#4924C2211yescharacteristic
19#4924C2211yescharacteristic
20C2×C4⋊(C3⋊C8)C2211yescharacteristic
21C2×C4×C3⋊C8C2211yescharacteristic
22#4547—21no—
23#4544—21no—
24#4535—21no—
25#4535—21no—
26D4⋊(C3⋊C8)—21no—
27D4⋊(C3⋊C8)—21no—
28#4530—21no—
29#4480—22no—
30#3037D311yescharacteristic
31#1801—31no—
32#1566—31no—
33#1547—31no—
34#1492—31no—
35C2×Q8⋊C8—31no—
36#1292—31no—
37C2×C3×#41—21no—
38C22×C3×Q8D411yescharacteristic
39C3×#137—42no—
40C3×#137—42no—
41C2×C3×Q16—22no—
42C2×C3×SD16—22no—
43C2×C3×M16—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×#127D411yescharacteristic
46C3×#124—21no—
47C3×#121—21no—
48C3×#121—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×C4×Q8—21no—
52C3×C4×Q8—21no—
53C3×C4×D4—21no—
54C3×C4×D4—21no—
55C2×C3×C4⋊C4—21no—
56C2×C3×C4⋊C4D411yescharacteristic
57C2×C3×C42—21no—
58C2×C3×C42C2311yescharacteristic
59C2×#208—21no—
60C2×#208—21no—
61C22×C3⋊C8—21no—
62C22×C3⋊C8—21no—
63C3×C4⋊C8—21no—
64C3×C4⋊C8D412yes—
65C3×#103—22no—
66C3×#102—22no—
67#836—42no—
68C4⋊(C3⋊C8)—21no—
69C4⋊(C3⋊C8)—21no—
70C4⋊(C3⋊C8)—21no—
71C4⋊(C3⋊C8)—21no—
72C4⋊(C3⋊C8)—21no—
73C4⋊(C3⋊C8)—21no—
74C4×C3⋊C8—21no—
75C4×C3⋊C8D411yescharacteristic
76C4×C3⋊C8D411yescharacteristic
77C2×#137—21no—
78#532C2×D311yescharacteristic
79C2×C4×Q8C2×D311yescharacteristic
80#464—22no—
81#462—21no—
82#460—21no—
83#423—31no—
84#423—31no—
85#423C2×D311yescharacteristic
86C2×C4⋊C8—31no—
87#417—21no—
88C2×#101—61no—
89C2×C4×C8—31no—
90#343—61no—
91#334—62no—
92#332—61no—
93#331—61no—
94Q8⋊C8—61no—
95Q8⋊C8—61no—
96D4⋊C8—61no—
97D4⋊C8—61no—
98C3×#41—41no—
99C3×#41—41no—
100C3×#41—41no—
101C3×#41—41no—
102C2×C3×Q8—21no—
103C2×C3×Q8—21no—
104C2×C3×Q8—21no—
105C2×C3×Q8—21no—
106C2×C3×Q8—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8SD1612yes—
109C2×C3×D4—21no—
110C2×C3×D4—21no—
111C2×C3×D4—21no—
112C2×C3×D4—41no—
113C22×C3×C4—21no—
114C22×C3×C4—21no—
115C22×C3×C4—41no—
116C22×C3×C4C2×D411yescharacteristic
117C22×C3×C4#4111yescharacteristic
118C22×C3×C4#4111yescharacteristic
119C3×Q16—44no—
120C3×SD16—44no—
121C3×M16—41no—
122C3×M16—41no—
123C2×C3×C8—22no—
124C3×C4⋊C4—21no—
125C3×C4⋊C4—21no—
126C3×C4⋊C4—21no—
127C3×C4⋊C4—21no—
128C3×C4⋊C4—41no—
129C3×C4⋊C4—41no—
130C3×C22⋊C4—41no—
131C3×C22⋊C4—41no—
132C3×C42—41no—
133C3×C42—41no—
134C3×C42C2×D411yescharacteristic
135C3×C42C2×D411yescharacteristic
136C3×C42C2×D411yescharacteristic
137C3×C42#4111yescharacteristic
138#208—41no—
139#208—41no—
140#208—41no—
141#208—41no—
142C2×C3⋊C8—21no—
143C2×C3⋊C8—21no—
144C2×C3⋊C8—42no—
145C2×C3⋊C8—41no—
146C2×C3⋊C8—41no—
147C2×C3⋊C8—41no—
148C2×C3⋊C8—81no—
149C2×#41—21no—
150C22×Q8#6711yescharacteristic
151#137—42no—
152#137—42no—
153C2×Q16—22no—
154C2×SD16—22no—
155C2×M16—21no—
156C2×M16—61no—
157C2×M16—61no—
158C22×C8—61no—
159C22×C8—61no—
160C4⋊Q8#6711yescharacteristic
161#127#6711yescharacteristic
162#124—21no—
163#121—21no—
164#121—21no—
165C4×Q8—21no—
166C4×Q8—21no—
167C4×Q8—21no—
168C4×Q8—21no—
169C4×D4—21no—
170C4×D4—21no—
171C2×C4⋊C4—21no—
172C2×C4⋊C4#6711yescharacteristic
173C2×C42—21no—
174C2×C42C22×D311yescharacteristic
175C4⋊C8—21no—
176C4⋊C8—61no—
177C4⋊C8—61no—
178C4⋊C8—61no—
179C4⋊C8—61no—
180C4⋊C8—61no—
181C4⋊C8—61no—
182C4⋊C8#6712yes—
183#103—22no—
184#102—22no—
185#101—122no—
186C4×C8—31no—
187C4×C8—31no—
188C4×C8—61no—
189C23×C3—41no—
190C23×C3#12311yescharacteristic
191C3×Q8—22no—
192C3×Q8—22no—
193C3×Q8—41no—
194C3×Q8—41no—
195C3×Q8—41no—
196C3×Q8—41no—
197C3×D4—41no—
198C3×D4—41no—
199C3×D4—41no—
200C3×D4—41no—
201C3×D4—81no—
202C3×D4—81no—
203C3×D4—81no—
204C2×C3×C4—21no—
205C2×C3×C4—21no—
206C2×C3×C4—22no—
207C2×C3×C4—41no—
208C2×C3×C4—41no—
209C2×C3×C4—41no—
210C2×C3×C4—41no—
211C2×C3×C4—41no—
212C2×C3×C4—81no—
213C2×C3×C4—81no—
214C2×C3×C4—81no—
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12111yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12311yescharacteristic
220C2×C3×C4#12411yescharacteristic
221C2×C3×C4C2×SD1611yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
224C2×C3×C4AGL(1,ℤ/8)11yescharacteristic
225C3×C8—42no—
226C3⋊C8—22no—
227C3⋊C8—41no—
228C3⋊C8—81no—
229C3⋊C8—81no—
230C3⋊C8—81no—
231#41—41no—
232#41—41no—
233#41—41no—
234#41—41no—
235C2×Q8—21no—
236C2×Q8—21no—
237C2×Q8—21no—
238C2×Q8—21no—
239C2×Q8—21no—
240C2×Q8—21no—
241C2×Q8#21412yes—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4—41no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—41no—
249C22×C4#23711yescharacteristic
250C22×C4#23711yescharacteristic
251C22×C4C2×#6711yescharacteristic
252Q16—44no—
253SD16—44no—
254M16—41no—
255M16—41no—
256M16—121no—
257M16—121no—
258M16—121no—
259M16—121no—
260C2×C8—22no—
261C2×C8—61no—
262C2×C8—61no—
263C2×C8—122no—
264C2×C8—121no—
265C2×C8—121no—
266C2×C8—121no—
267C2×C8—241no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—21no—
271C4⋊C4—21no—
272C4⋊C4—41no—
273C4⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C42—41no—
277C42—41no—
278C42D3×D411yescharacteristic
279C42#23711yescharacteristic
280C42C2×#6711yescharacteristic
281C42C2×#6711yescharacteristic
282C22×C3—21no—
283C22×C3—21no—
284C22×C3—81no—
285C22×C3—81no—
286C22×C3—81no—
287C22×C3#39411yescharacteristic
288C22×C3#48111yescharacteristic
289C22×C3#48211yescharacteristic
290C3×C4—21no—
291C3×C4—21no—
292C3×C4—21no—
293C3×C4—42no—
294C3×C4—81no—
295C3×C4—81no—
296C3×C4#44911yescharacteristic
297C3×C4#45011yescharacteristic
298C3×C4#46511yescharacteristic
299C3×C4#46911yescharacteristic
300C3×C4#47111yescharacteristic
301C3×C4#47311yescharacteristic
302C23—41no—
303C23#93511yescharacteristic
304Q8—22no—
305Q8—22no—
306Q8—41no—
307Q8—41no—
308Q8—41no—
309Q8—41no—
310D4—41no—
311D4—41no—
312D4—41no—
313D4—41no—
314D4—81no—
315D4—81no—
316D4—81no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—22no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—41no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—81no—
326C2×C4—81no—
327C2×C4—81no—
328C2×C4#93211yescharacteristic
329C2×C4C2×#21411yescharacteristic
330C2×C4#93511yescharacteristic
331C2×C4#93611yescharacteristic
332C2×C4#93811yescharacteristic
333C2×C4#93911yescharacteristic
334C2×C4#93911yescharacteristic
335C2×C4#94911yescharacteristic
336C2×C4#95011yescharacteristic
337C2×C4#95311yescharacteristic
338C8—42no—
339C8—62no—
340C8—121no—
341C8—241no—
342C8—241no—
343C8—241no—
344C2×C3—21no—
345C2×C3—21no—
346C2×C3—161no—
347C2×C3#203411yescharacteristic
348C2×C3#205011yescharacteristic
349C2×C3#205211yescharacteristic
350C22—21no—
351C22—21no—
352C22—81no—
353C22—81no—
354C22—81no—
355C22#503411yescharacteristic
356C22#503511yescharacteristic
357C22#522211yescharacteristic
358C4—21no—
359C4—21no—
360C4—21no—
361C4—42no—
362C4—81no—
363C4—81no—
364C4#503911yescharacteristic
365C4#504111yescharacteristic
366C4#520311yescharacteristic
367C4#520411yescharacteristic
368C4#521811yescharacteristic
369C4#523811yescharacteristic
370C3#1045911yescharacteristic, complemented, Sylow
371C2—21no—
372C2—21no—
373C2—161no—
374C2#7168011yescharacteristic
375C2#7210511yescharacteristic
376C2#7211711yescharacteristic
subgroupG/H
centerC22#5222
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#3037C2
socleC22×C3#394
prime radicalC3×#532C22
hypercenter#3037D3
hypocenterC3#10459
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3037

Complements

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

NH ≅ G/Ntype
4. #71637C2semidirect
5. C2×#4535C2semidirect
6. #70490C2semidirect
7. #67711C2semidirect
370. C3#10459semidirect

Maximal subgroups

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

subgroupindex
C3×#30372
#718062
#718022
#716372
C2×#45352
#704902
#677112
#104593