Finite Group Databasebeta

← #10697623

Subgroups — #10697623

Order 768.

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

normal subgroups only

This group has 536 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
1#83350C211yesmaximal, characteristic
2#83324C211yesmaximal, characteristic, complemented
3#73807C211yesmaximal, characteristic
4C3×#1503C211yesmaximal, characteristic, complemented
5#70780C211yesmaximal, characteristic, complemented
6#70757C211yesmaximal, characteristic
7#67620C211yesmaximal, characteristic, complemented
8#10249—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C2×#869C2211yescharacteristic
12C3×#431C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5087—21no—
15#5084—21no—
16#5079—21no—
17#5078—21no—
18#5066—22no—
19#5044—22no—
20#5021—22no—
21#5001—21no—
22#4995—21no—
23#4987—22no—
24#4939—21no—
25#4936—21no—
26#4925C2211yescharacteristic
27#4924C2211yescharacteristic
28#4673—22no—
29C3×#328—22no—
30#4482—22no—
31#4459—22no—
32#3230—31no—
33#3077—31no—
34#2563—31no—
35#1549—31no—
36#1503D311yescharacteristic, complemented
37#1478—31no—
38#1303—31no—
39C22×D12D411yescharacteristic
40C22×C4×D3—21no—
41C22×Dic6—21no—
42C3×C4⋊Q8D411yescharacteristic
43C3×C4⋊Q8D411yescharacteristic
44C3×#125—21no—
45C3×C22⋊Q8—21no—
46C3×C22⋊Q8—21no—
47C3×C4×Q8—21no—
48C3×C4×Q8—21no—
49C3×#117—21no—
50C2×C3×C4⋊C4—21no—
51C2×C3×C42—21no—
52C2×C3×C42C2311yescharacteristic
53C2×(C2×D3)⋊C4—21no—
54C2×(C2×D3)⋊C4—24no—
55C2×C4⋊Dic3—21no—
56C2×C4⋊Dic3D411yescharacteristic
57C2×C4⋊Dic3D412yes—
58C2×#208—21no—
59C4×D12—21no—
60C4×D12—21no—
61C4×D12—21no—
62C4×D12—21no—
63#869—21no—
64#869—21no—
65#869—21no—
66#869—21no—
67C3×C4⋊C8—21no—
68C3×C22⋊C8—21no—
69C3×C4×C8—21no—
70C3×#95D412yes—
71#835—24no—
72C3⋊#95—24no—
73#830—21no—
74#809—22no—
75#809—22no—
76#808—22no—
77#807—22no—
78#807—24no—
79C4⋊(C3⋊C8)—21no—
80C4⋊(C3⋊C8)D412yes—
81C4⋊(C3⋊C8)D412yes—
82C4×C3⋊C8—21no—
83#533C2×D311yescharacteristic
84C2×C4⋊Q8—31no—
85C2×C4×D4—31no—
86#500—61no—
87#487—61no—
88#484—62no—
89#478—62no—
90#477—62no—
91#476—61no—
92#474—61no—
93#431—31no—
94#431C2×D311yescharacteristic
95#428—61no—
96#423—31no—
97#418—61no—
98#393—62no—
99#390—61no—
100#382—61no—
101#380—62no—
102C4×C4⋊C4C2×D311yescharacteristic
103#332—62no—
104#328—22no—
105D4⋊C8—62no—
106C23×D3—21no—
107C2×C3×Q8—21no—
108C2×C3×Q8—21no—
109C22×C3×C4—22no—
110C22×C3×C4C2×D411yescharacteristic
111C22×C3×C4C2×D411yescharacteristic
112C22×C3×C4C2×D411yescharacteristic
113C22×Dic3—21no—
114C22×Dic3—22no—
115C2×D12—21no—
116C2×D12—21no—
117C2×D12—21no—
118C2×D12—21no—
119C2×D12—21no—
120C2×D12Q1612yes—
121C2×C4×D3—44no—
122C2×C4×D3—41no—
123C2×C4×D3—41no—
124C2×Dic6—41no—
125C2×Dic6—41no—
126C2×Dic6—82no—
127C2×C3×C8—41no—
128C3×C4⋊C4—21no—
129C3×C4⋊C4—21no—
130C3×C4⋊C4—21no—
131C3×C4⋊C4—21no—
132C3×C4⋊C4—42no—
133C3×C4⋊C4—41no—
134C3×C4⋊C4—41no—
135C3×C22⋊C4—41no—
136C3×C42—41no—
137C3×C42—41no—
138C3×C42—41no—
139C3×C42C2×D411yescharacteristic
140C3×C42C2×D411yescharacteristic
141C3×C42C2×D411yescharacteristic
142C3×C42#4111yescharacteristic
143(C2×D3)⋊C4—41no—
144(C2×D3)⋊C4—41no—
145(C2×D3)⋊C4—84no—
146C4⋊Dic3—21no—
147C4⋊Dic3—22no—
148C4⋊Dic3—22no—
149C4⋊Dic3—21no—
150C4⋊Dic3—41no—
151C4⋊Dic3—41no—
152#208—41no—
153#208—41no—
154C2×C3⋊C8—22no—
155C2×C3⋊C8—22no—
156C22×Q8—61no—
157C22×D4—31no—
158C23×C4—61no—
159C2×M16—61no—
160C4⋊Q8—61no—
161C4⋊Q8—61no—
162C4⋊Q8—61no—
163C4⋊Q8—61no—
164C4⋊Q8D1211yescharacteristic
165C4⋊Q8D1211yescharacteristic
166#125—21no—
167C22⋊Q8—21no—
168C22⋊Q8—21no—
169C4×Q8—21no—
170C4×Q8—21no—
171C4×D4—61no—
172C4×D4—61no—
173C4×D4—61no—
174C4×D4—61no—
175#117—21no—
176C2×C4⋊C4—21no—
177C2×C4⋊C4—31no—
178C2×C4⋊C4—32no—
179C2×C4⋊C4—61no—
180C2×C22⋊C4—61no—
181C2×C22⋊C4—64no—
182C2×C42—21no—
183C2×C42C22×D311yescharacteristic
184#107—62no—
185#107—64no—
186#106—62no—
187C4⋊C8—21no—
188C4⋊C8—32no—
189C4⋊C8—32no—
190C4⋊C8—61no—
191#103—64no—
192#102—62no—
193#102—62no—
194C22⋊C8—21no—
195C22⋊C8—61no—
196C4×C8—21no—
197C4×C8—61no—
198#95—64no—
199#95#6712yes—
200C23×C3#12011yescharacteristic
201C22×D3—22no—
202C22×D3—42no—
203C22×D3—42no—
204C22×D3—41no—
205C22×D3—41no—
206C3×Q8—42no—
207C3×Q8—81no—
208C2×C3×C4—22no—
209C2×C3×C4—41no—
210C2×C3×C4—42no—
211C2×C3×C4—42no—
212C2×C3×C4—41no—
213C2×C3×C4—41no—
214C2×C3×C4—42no—
215C2×C3×C4#12011yescharacteristic
216C2×C3×C4#12011yescharacteristic
217C2×C3×C4#12111yescharacteristic
218C2×C3×C4#12111yescharacteristic
219C2×C3×C4#12111yescharacteristic
220C2×C3×C4#12711yescharacteristic
221C2×C3×C4C2×Q1611yescharacteristic
222C2×C3×C4#13511yescharacteristic
223C2×C3×C4#13711yescharacteristic
224C2×C3×C4#13711yescharacteristic
225C2×Dic3—44no—
226C2×Dic3—42no—
227C2×Dic3—42no—
228C2×Dic3—41no—
229D12—22no—
230D12—22no—
231D12—41no—
232D12—41no—
233C4×D3—82no—
234C4×D3—82no—
235Dic6—82no—
236Dic6—82no—
237C3×C8—81no—
238C3⋊C8—42no—
239C3⋊C8—42no—
240C24—61no—
241C2×Q8—21no—
242C2×Q8—21no—
243C2×Q8—121no—
244C2×Q8—121no—
245C2×Q8—242no—
246C2×D4—32no—
247C2×D4—61no—
248C2×D4—61no—
249C2×D4—61no—
250C2×D4—61no—
251C2×D4—61no—
252C22×C4—22no—
253C22×C4—61no—
254C22×C4—62no—
255C22×C4—124no—
256C22×C4—121no—
257C22×C4—121no—
258C22×C4D3×D411yescharacteristic
259C22×C4D3×D411yescharacteristic
260C22×C4C2×#6711yescharacteristic
261M16—121no—
262M16—121no—
263C2×C8—41no—
264C2×C8—62no—
265C2×C8—62no—
266C4⋊C4—21no—
267C4⋊C4—21no—
268C4⋊C4—21no—
269C4⋊C4—21no—
270C4⋊C4—42no—
271C4⋊C4—41no—
272C4⋊C4—41no—
273C4⋊C4—62no—
274C4⋊C4—61no—
275C4⋊C4—61no—
276C4⋊C4—62no—
277C4⋊C4—121no—
278C4⋊C4—121no—
279C22⋊C4—41no—
280C22⋊C4—121no—
281C22⋊C4—121no—
282C22⋊C4—244no—
283C42—41no—
284C42—41no—
285C42—41no—
286C42C2×D1211yescharacteristic
287C42#23511yescharacteristic
288C42D3×D411yescharacteristic
289C42D3×D411yescharacteristic
290C22×C3—21no—
291C22×C3—21no—
292C22×C3#39211yescharacteristic
293C22×C3#45111yescharacteristic
294C22×C3#45211yescharacteristic
295C2×D3—44no—
296C2×D3—42no—
297C2×D3—82no—
298C2×D3—81no—
299C2×D3—81no—
300C3×C4—21no—
301C3×C4—21no—
302C3×C4—21no—
303C3×C4—82no—
304C3×C4—81no—
305C3×C4—81no—
306C3×C4#45111yescharacteristic
307C3×C4#45211yescharacteristic
308C3×C4#45412yes—
309C3×C4#46711yescharacteristic
310C3×C4#47011yescharacteristic
311Dic3—81no—
312Dic3—82no—
313C23—62no—
314C23—122no—
315C23—122no—
316C23—121no—
317C23—121no—
318C23#93711yescharacteristic
319Q8—42no—
320Q8—81no—
321Q8—242no—
322Q8—242no—
323D4—62no—
324D4—62no—
325D4—121no—
326D4—121no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—42no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—124no—
335C2×C4—122no—
336C2×C4—121no—
337C2×C4—122no—
338C2×C4—242no—
339C2×C4—242no—
340C2×C4#88211yescharacteristic
341C2×C4#88211yescharacteristic
342C2×C4#88411yescharacteristic
343C2×C4#88411yescharacteristic
344C2×C4#91511yescharacteristic
345C2×C4#91611yescharacteristic
346C2×C4D3×Q1611yescharacteristic
347C2×C4#91811yescharacteristic
348C2×C4#93011yescharacteristic
349C2×C4#94011yescharacteristic
350C8—81no—
351C8—122no—
352C8—122no—
353C2×C3—21no—
354C2×C3—21no—
355C2×C3#200111yescharacteristic
356C2×C3#200311yescharacteristic
357C2×C3#200811yescharacteristic
358D3—82no—
359C22—21no—
360C22—21no—
361C22—124no—
362C22—122no—
363C22—242no—
364C22—241no—
365C22—241no—
366C22#496011yescharacteristic
367C22#517311yescharacteristic
368C22#518611yescharacteristic
369C4—21no—
370C4—21no—
371C4—21no—
372C4—82no—
373C4—81no—
374C4—81no—
375C4—242no—
376C4—241no—
377C4#480811yescharacteristic
378C4#480911yescharacteristic
379C4#481311yescharacteristic
380C4#481611yescharacteristic
381C4#482712yes—
382C3#1024911yescharacteristic, complemented, Sylow
383C2—21no—
384C2—21no—
385C2—242no—
386C2#7005011yescharacteristic
387C2#7027311yescharacteristic
388C2#7038811yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1503C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1503D3
hypocenterC3#10249
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
2. #83324C2semidirect
4. C3×#1503C2semidirect
5. #70780C2semidirect
7. #67620C2semidirect
36. #1503D3semidirect
382. C3#10249semidirect

Maximal subgroups

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

subgroupindex
#833502
#833242
#738072
C3×#15032
#707802
#707572
#676202
#102493