Finite Group Databasebeta

← #10696007

Subgroups — #10696007

Order 768.

528 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 528 conjugacy classes of subgroups — too many to draw. Only its 83 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#83353C211yesmaximal, characteristic
2#83326C211yesmaximal, characteristic, complemented
3#73806C211yesmaximal, characteristic
4C3×#1986C211yesmaximal, characteristic, complemented
5#69863C211yesmaximal, characteristic, complemented
6#69855C211yesmaximal, characteristic
7#67621C211yesmaximal, characteristic, complemented
8#10009—31nomaximal, Sylow
9C3×#533C2211yescharacteristic
10C2×C4×D12C2211yescharacteristic
11C2×#870C2211yescharacteristic
12C3×C4×M16C2211yescharacteristic
13C3×C4×C4⋊C4C2211yescharacteristic
14#5087—21no—
15#5083—21no—
16#5082—21no—
17#5081—21no—
18#5046—21no—
19#5044—21no—
20#5044—21no—
21#5043—21no—
22#5026—22no—
23#5022—22no—
24#4987—21no—
25#4987—21no—
26#4940—22no—
27#4926C2211yescharacteristic
28#4926C2211yescharacteristic
29#4672—21no—
30#4659—21no—
31C3×#339—22no—
32#4465—22no—
33#4459—22no—
34#3236—31no—
35#3110—31no—
36#2562—31no—
37#1986D311yescharacteristic, complemented
38#1528—31no—
39#1490—31no—
40#1285—31no—
41C22×D12D411yescharacteristic
42C22×C4×D3—21no—
43C2×C3×M16—21no—
44C3×C4⋊Q8D411yescharacteristic
45C3×C4⋊Q8D411yescharacteristic
46C3×#125—21no—
47C3×C22⋊Q8—21no—
48C3×C22⋊Q8—21no—
49C3×C4×Q8—21no—
50C3×C4×Q8—21no—
51C3×#117—21no—
52C2×C3×C4⋊C4—21no—
53C2×C3×C42D412yes—
54C2×C3×C42C2311yescharacteristic
55C2×#215—22no—
56C2×(C2×D3)⋊C4—22no—
57C2×(C2×D3)⋊C4—21no—
58C2×(C2×D3)⋊C4—22no—
59C2×C4⋊Dic3D411yescharacteristic
60C2×C4⋊Dic3D411yescharacteristic
61C2×C4⋊Dic3D411yescharacteristic
62C2×Dic3⋊C4—21no—
63C2×Dic3⋊C4—21no—
64C4×D12—21no—
65C4×D12—21no—
66C4×D12—21no—
67C4×D12—21no—
68#870—21no—
69#870—21no—
70#870—21no—
71#870—21no—
72C3×#97—21no—
73C3×C4×C8—21no—
74C3×#95—21no—
75#835—22no—
76#835—22no—
77#835—22no—
78C3⋊#95—22no—
79C3⋊#95—22no—
80#830—21no—
81#830—21no—
82#809—22no—
83#808—22no—
84#807—22no—
85#807—22no—
86C4⋊(C3⋊C8)—21no—
87C4⋊(C3⋊C8)D412yes—
88#803—21no—
89#803D412yes—
90#533C2×D311yescharacteristic
91C2×#125—31no—
92C2×C4×D4—31no—
93#501—61no—
94#491—61no—
95#484—61no—
96#484—61no—
97#483—61no—
98#479—62no—
99#474—61no—
100#461—61no—
101#450—61no—
102#440—62no—
103#432—31no—
104#432—31no—
105C4×M16C2×D311yescharacteristic
106#400—61no—
107#394—61no—
108#388—62no—
109#380—61no—
110#380—61no—
111C4×C4⋊C4C2×D311yescharacteristic
112#339—22no—
113#330—62no—
114D4⋊C8—62no—
115C23×D3—21no—
116C2×C3×Q8—21no—
117C2×C3×Q8—21no—
118C22×C3×C4—22no—
119C22×C3×C4C2×D411yescharacteristic
120C22×C3×C4C2×D411yescharacteristic
121C22×C3×C4C2×D411yescharacteristic
122C22×Dic3—21no—
123C22×Dic3—21no—
124C22×Dic3—21no—
125C2×D12—21no—
126C2×D12—21no—
127C2×D12—42no—
128C2×C4×D3—42no—
129C2×C4×D3—42no—
130C2×C4×D3—41no—
131C2×C4×D3—41no—
132C3×M16—22no—
133C3×M16—41no—
134C2×C3×C8—41no—
135C3×C4⋊C4—21no—
136C3×C4⋊C4—21no—
137C3×C4⋊C4—21no—
138C3×C4⋊C4—21no—
139C3×C4⋊C4—42no—
140C3×C4⋊C4—41no—
141C3×C4⋊C4—41no—
142C3×C22⋊C4—41no—
143C3×C42—42no—
144C3×C42—41no—
145C3×C42C2×D411yescharacteristic
146C3×C42C2×D411yescharacteristic
147C3×C42C2×D411yescharacteristic
148C3×C42#4111yescharacteristic
149#215—44no—
150(C2×D3)⋊C4—44no—
151(C2×D3)⋊C4—41no—
152(C2×D3)⋊C4—41no—
153(C2×D3)⋊C4—82no—
154C4⋊Dic3—21no—
155C4⋊Dic3—21no—
156C4⋊Dic3—21no—
157C4⋊Dic3—21no—
158C4⋊Dic3—21no—
159C4⋊Dic3—21no—
160Dic3⋊C4—41no—
161Dic3⋊C4—41no—
162Dic3⋊C4—41no—
163Dic3⋊C4—41no—
164C2×C3⋊C8—22no—
165C2×C3⋊C8—22no—
166C22×D4—31no—
167C23×C4—61no—
168C2×SD16—62no—
169C2×M16—21no—
170C4⋊Q8D1211yescharacteristic
171C4⋊Q8D1211yescharacteristic
172#125—21no—
173#125—61no—
174#125—61no—
175#125—61no—
176#125—61no—
177C22⋊Q8—21no—
178C22⋊Q8—21no—
179C4×Q8—21no—
180C4×Q8—21no—
181C4×D4—61no—
182C4×D4—61no—
183C4×D4—61no—
184C4×D4—61no—
185#117—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—31no—
188C2×C4⋊C4—31no—
189C2×C4⋊C4—31no—
190C2×C4⋊C4—61no—
191C2×C4⋊C4—61no—
192C2×C22⋊C4—62no—
193C2×C22⋊C4—61no—
194C2×C22⋊C4—62no—
195C2×C42#6712yes—
196C2×C42C22×D311yescharacteristic
197#107—62no—
198#107—62no—
199#106—62no—
200C4⋊C8—32no—
201C4⋊C8—61no—
202#103—62no—
203#103—62no—
204#103—62no—
205#102—62no—
206C22⋊C8—61no—
207C22⋊C8—61no—
208#97—21no—
209#97—32no—
210#97—61no—
211C4×C8—21no—
212#95—21no—
213#95—62no—
214#95—62no—
215C23×C3#12011yescharacteristic
216C22×D3—22no—
217C22×D3—41no—
218C22×D3—41no—
219C22×D3—41no—
220C22×D3—41no—
221C22×D3—41no—
222C22×D3—41no—
223C3×Q8—42no—
224C3×Q8—42no—
225C2×C3×C4—21no—
226C2×C3×C4—21no—
227C2×C3×C4—22no—
228C2×C3×C4—41no—
229C2×C3×C4—42no—
230C2×C3×C4—42no—
231C2×C3×C4—41no—
232C2×C3×C4—41no—
233C2×C3×C4—42no—
234C2×C3×C4#12011yescharacteristic
235C2×C3×C4#12011yescharacteristic
236C2×C3×C4#12111yescharacteristic
237C2×C3×C4#12111yescharacteristic
238C2×C3×C4#12111yescharacteristic
239C2×C3×C4#12711yescharacteristic
240C2×Dic3—42no—
241C2×Dic3—41no—
242C2×Dic3—41no—
243C2×Dic3—41no—
244C2×Dic3—41no—
245C2×Dic3—41no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248D12—22no—
249D12—22no—
250D12—41no—
251D12—41no—
252C4×D3—82no—
253C4×D3—82no—
254C3×C8—42no—
255C3⋊C8—42no—
256C3⋊C8—42no—
257C24—61no—
258C2×Q8—21no—
259C2×Q8—21no—
260C2×D4—61no—
261C2×D4—61no—
262C2×D4—122no—
263C22×C4—22no—
264C22×C4—61no—
265C22×C4—61no—
266C22×C4—61no—
267C22×C4—122no—
268C22×C4—122no—
269C22×C4—121no—
270C22×C4—121no—
271C22×C4D3×D411yescharacteristic
272C22×C4D3×D411yescharacteristic
273C22×C4C2×#6711yescharacteristic
274SD16—124no—
275M16—22no—
276M16—41no—
277C2×C8—41no—
278C2×C8—62no—
279C2×C8—62no—
280C4⋊C4—21no—
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—21no—
284C4⋊C4—42no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—61no—
288C4⋊C4—61no—
289C4⋊C4—61no—
290C4⋊C4—61no—
291C4⋊C4—61no—
292C4⋊C4—61no—
293C4⋊C4—121no—
294C4⋊C4—121no—
295C4⋊C4—121no—
296C4⋊C4—121no—
297C22⋊C4—41no—
298C22⋊C4—124no—
299C22⋊C4—121no—
300C22⋊C4—121no—
301C22⋊C4—242no—
302C42—42no—
303C42—41no—
304C42C2×D1211yescharacteristic
305C42#23511yescharacteristic
306C42D3×D411yescharacteristic
307C42D3×D411yescharacteristic
308C22×C3—21no—
309C22×C3—21no—
310C22×C3#39211yescharacteristic
311C22×C3#45511yescharacteristic
312C22×C3#45611yescharacteristic
313C2×D3—42no—
314C2×D3—42no—
315C2×D3—42no—
316C2×D3—81no—
317C2×D3—81no—
318C2×D3—81no—
319C2×D3—81no—
320C3×C4—21no—
321C3×C4—21no—
322C3×C4—21no—
323C3×C4—21no—
324C3×C4—21no—
325C3×C4—82no—
326C3×C4—81no—
327C3×C4—81no—
328C3×C4#45412yes—
329Dic3—81no—
330Dic3—81no—
331Dic3—81no—
332C23—62no—
333C23—121no—
334C23—121no—
335C23—121no—
336C23—121no—
337C23—121no—
338C23—121no—
339C23#93711yescharacteristic
340Q8—42no—
341Q8—42no—
342D4—62no—
343D4—62no—
344D4—121no—
345D4—121no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—22no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—42no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—42no—
355C2×C4—122no—
356C2×C4—121no—
357C2×C4—121no—
358C2×C4—121no—
359C2×C4—121no—
360C2×C4—121no—
361C2×C4—121no—
362C2×C4—121no—
363C2×C4—242no—
364C2×C4—242no—
365C2×C4#88211yescharacteristic
366C2×C4#88211yescharacteristic
367C2×C4#88411yescharacteristic
368C2×C4#88411yescharacteristic
369C2×C4#93011yescharacteristic
370C2×C4#94011yescharacteristic
371C8—42no—
372C8—122no—
373C8—122no—
374C2×C3—21no—
375C2×C3—21no—
376C2×C3#200311yescharacteristic
377C2×C3#200711yescharacteristic
378C2×C3#201011yescharacteristic
379D3—82no—
380C22—21no—
381C22—21no—
382C22—122no—
383C22—122no—
384C22—122no—
385C22—241no—
386C22—241no—
387C22—241no—
388C22—241no—
389C22#496011yescharacteristic
390C22#506111yescharacteristic
391C22#509511yescharacteristic
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—21no—
397C4—82no—
398C4—81no—
399C4—81no—
400C4—241no—
401C4—241no—
402C4—241no—
403C4#482712yes—
404C3#1000911yescharacteristic, complemented, Sylow
405C2—21no—
406C2—21no—
407C2—242no—
408C2#6969411yescharacteristic
409C2#7038811yescharacteristic
410C2#7040611yescharacteristic
subgroupG/H
centerC22#4960
commutator, cosocleC2×C3×C42C23
FrattiniC2×C42C22×D3
FittingC3×#1986C2
socleC22×C3#392
prime radicalC22×D12D4
hypercenter#1986D3
hypocenterC3#10009
LCSG ▷ C2×C3×C42 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1986

Complements

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

NH ≅ G/Ntype
2. #83326C2semidirect
4. C3×#1986C2semidirect
5. #69863C2semidirect
7. #67621C2semidirect
37. #1986D3semidirect
404. C3#10009semidirect

Maximal subgroups

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

subgroupindex
#833532
#833262
#738062
C3×#19862
#698632
#698552
#676212
#100093