Finite Group Databasebeta

← #11713583

Subgroups — #11713583

Order 768.

559 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 63 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
1C2×#5419C211yesmaximal, characteristic
2C2×#5419C211yesmaximal, characteristic
3C2×C4⋊#226C211yesmaximal, characteristic, complemented
4C2×C4×#226C211yesmaximal, characteristic, complemented
5C22⋊C4×SL(2,𝔽3)C211yesmaximal, characteristic
6#73062C211yesmaximal, characteristic, complemented
7#73061C211yesmaximal, characteristic, complemented
8#25161—31nomaximal, Sylow
9C23×SL(2,𝔽3)C2211yescharacteristic
10C22×#226C2211yescharacteristic
11C22×#226C2211yescharacteristic
12C2×C4×SL(2,𝔽3)C2211yescharacteristic
13C2×C4×SL(2,𝔽3)C2211yescharacteristic
14#5419—22no—
15#5419—22no—
16C2×#859C2211yescharacteristic
17C2×#859C2211yescharacteristic
18C4⋊#226—22no—
19C4×#226—22no—
20#4951—41nomaximal
21C2×#462—31no—
22C2×#451—31no—
23C2×#451—31no—
24C2×C4×Q16—31no—
25#2429—31no—
26Q8×C22⋊C4D311yescharacteristic
27#2068—31no—
28#2036—31no—
29#2036—31no—
30#2016—31no—
31#1969—31no—
32#1923—31no—
33#1878—31no—
34#1878—31no—
35#1863—31no—
36C22×SL(2,𝔽3)—21no—
37C22×SL(2,𝔽3)—21no—
38C22×SL(2,𝔽3)—21no—
39C22×SL(2,𝔽3)C2311yescharacteristic
40C2×#226—21no—
41C2×#226—21no—
42C2×#226—21no—
43C2×#226—21no—
44C2×#226—21no—
45C2×#226D412yes—
46C2×C3×C22⋊C4—41no—
47C2×#217—41no—
48C2×#217—41no—
49C2×C4⋊Dic3—41no—
50C2×C4×Dic3—41no—
51C4×SL(2,𝔽3)—21no—
52C4×SL(2,𝔽3)—21no—
53#859—21no—
54#859D412yes—
55C3⋊#95—41no—
56C3⋊#95—41no—
57C23×Q8C2×D311yescharacteristic
58C22×Q16—31no—
59C22×Q16—31no—
60C22×Q16—31no—
61C22×Q16—31no—
62C2×C4⋊Q8—31no—
63C2×C22⋊Q8—31no—
64C2×C22⋊Q8—31no—
65C2×C4×Q8—31no—
66C2×C4×Q8C2×D311yescharacteristic
67C2×C4×Q8C2×D311yescharacteristic
68#462—62no—
69#462—62no—
70#451—62no—
71#451—62no—
72#451—62no—
73#451—62no—
74C4×Q16—62no—
75C4×Q16—62no—
76C2×#107—31no—
77C2×#107—31no—
78C2×C4⋊C8—31no—
79C2×#103—31no—
80C2×#103—31no—
81C2×#103—31no—
82C2×#103—31no—
83C2×#103—31no—
84C2×#103—31no—
85C2×#103—31no—
86C2×#103—31no—
87C2×C22⋊C8—31no—
88C2×C22⋊C8—31no—
89C2×C4×C8—31no—
90#398—61no—
91#396—61no—
92#391—31no—
93#391—31no—
94#391—31no—
95#389—31no—
96#389—61no—
97#388—61no—
98#380—31no—
99C4×C22⋊C4—31no—
100#340—31no—
101#340—31no—
102#336—31no—
103#336—31no—
104C24×C3—41no—
105C22×C3×C4—41no—
106C22×C3×C4—41no—
107C22×Dic3—41no—
108C22×Dic3—41no—
109C22×Dic3—41no—
110C22×Dic3—41no—
111C2×SL(2,𝔽3)—41no—
112C2×SL(2,𝔽3)C2×D411yescharacteristic
113C2×SL(2,𝔽3)C2×D411yescharacteristic
114C2×SL(2,𝔽3)#4111yescharacteristic
115#226—22no—
116#226—41no—
117C3×C22⋊C4—44no—
118#217—82no—
119#217—82no—
120C4⋊Dic3—82no—
121C4×Dic3—82no—
122C22×Q8—21no—
123C22×Q8—21no—
124C22×Q8—21no—
125C22×Q8—31no—
126C22×Q8—31no—
127C22×Q8—31no—
128C22×Q8—61no—
129C22×Q8—61no—
130C22×Q8—61no—
131C22×Q8C22×D311yescharacteristic
132C23×C4—31no—
133C2×Q16—32no—
134C2×Q16—32no—
135C2×Q16—62no—
136C2×Q16—61no—
137C2×Q16—61no—
138C2×Q16—61no—
139C2×Q16—61no—
140C2×Q16—62no—
141C2×Q16—61no—
142C2×Q16—62no—
143C2×Q16—61no—
144C2×Q16—62no—
145C2×Q16—61no—
146C2×Q16—61no—
147C2×Q16—61no—
148C2×Q16—61no—
149C2×Q16—121no—
150C2×Q16—121no—
151C22×C8—31no—
152C22×C8—31no—
153C22×C8—31no—
154C22×C8—31no—
155C4⋊Q8—62no—
156C4⋊Q8—121no—
157C22⋊Q8—62no—
158C22⋊Q8—62no—
159C22⋊Q8—121no—
160C22⋊Q8—121no—
161C4×Q8—21no—
162C4×Q8—21no—
163C4×Q8—61no—
164C4×Q8—61no—
165C4×Q8—62no—
166C4×Q8—121no—
167C2×C4⋊C4—31no—
168C2×C4⋊C4—31no—
169C2×C4⋊C4—31no—
170C2×C4⋊C4—31no—
171C2×C4⋊C4—61no—
172C2×C4⋊C4—61no—
173C2×C4⋊C4—61no—
174C2×C4⋊C4—61no—
175C2×C4⋊C4—61no—
176C2×C4⋊C4—61no—
177C2×C22⋊C4—31no—
178C2×C22⋊C4—61no—
179C2×C22⋊C4—61no—
180C2×C22⋊C4S411yescharacteristic
181C2×C42—31no—
182C2×C42—31no—
183C2×C42—61no—
184#107—62no—
185#107—121no—
186C4⋊C8—62no—
187#103—32no—
188#103—32no—
189#103—62no—
190#103—62no—
191#103—62no—
192#103—62no—
193#103—61no—
194#103—62no—
195#103—61no—
196#103—121no—
197C22⋊C8—62no—
198C22⋊C8—62no—
199C4×C8—62no—
200#95—31no—
201#95—31no—
202#95—61no—
203#95—61no—
204#95—61no—
205#95—61no—
206C23×C3—42no—
207C23×C3—41no—
208C23×C3—81no—
209C23×C3—81no—
210C23×C3—81no—
211C23×C3—81no—
212C23×C3—81no—
213C23×C3—81no—
214C2×C3×C4—42no—
215C2×C3×C4—42no—
216C2×C3×C4—81no—
217C2×C3×C4—81no—
218C2×C3×C4—81no—
219C2×C3×C4—81no—
220C2×Dic3—42no—
221C2×Dic3—42no—
222C2×Dic3—81no—
223C2×Dic3—81no—
224C2×Dic3—81no—
225C2×Dic3—81no—
226C2×Dic3—81no—
227C2×Dic3—81no—
228C2×Dic3—81no—
229C2×Dic3—81no—
230C2×Dic3—81no—
231C2×Dic3—81no—
232SL(2,𝔽3)#12111yescharacteristic
233C24C2×S411yescharacteristic
234C2×Q8—31no—
235C2×Q8—31no—
236C2×Q8—31no—
237C2×Q8—32no—
238C2×Q8—32no—
239C2×Q8—41no—
240C2×Q8—61no—
241C2×Q8—61no—
242C2×Q8—61no—
243C2×Q8—61no—
244C2×Q8—61no—
245C2×Q8—61no—
246C2×Q8—61no—
247C2×Q8—61no—
248C2×Q8—61no—
249C2×Q8—61no—
250C2×Q8—61no—
251C2×Q8—61no—
252C2×Q8—61no—
253C2×Q8—61no—
254C2×Q8—61no—
255C2×Q8—61no—
256C2×Q8—122no—
257C2×Q8—121no—
258C2×Q8—121no—
259C2×Q8—121no—
260C2×Q8#23511yescharacteristic
261C2×Q8D3×D411yescharacteristic
262C2×Q8D3×D411yescharacteristic
263C22×C4—31no—
264C22×C4—31no—
265C22×C4—31no—
266C22×C4—31no—
267C22×C4—61no—
268C22×C4—61no—
269C22×C4—61no—
270C22×C4—61no—
271C22×C4—61no—
272C22×C4—61no—
273C22×C4—61no—
274C22×C4—61no—
275C22×C4—61no—
276C22×C4—61no—
277C22×C4C2×S411yescharacteristic
278C22×C4C2×S411yescharacteristic
279Q16—62no—
280Q16—62no—
281Q16—62no—
282Q16—62no—
283Q16—124no—
284Q16—121no—
285Q16—121no—
286Q16—121no—
287Q16—124no—
288Q16—121no—
289C2×C8—32no—
290C2×C8—32no—
291C2×C8—61no—
292C2×C8—61no—
293C2×C8—61no—
294C2×C8—61no—
295C2×C8—61no—
296C2×C8—61no—
297C2×C8—61no—
298C2×C8—61no—
299C2×C8—61no—
300C2×C8—61no—
301C4⋊C4—32no—
302C4⋊C4—61no—
303C4⋊C4—61no—
304C4⋊C4—61no—
305C4⋊C4—61no—
306C4⋊C4—61no—
307C4⋊C4—61no—
308C4⋊C4—61no—
309C4⋊C4—122no—
310C4⋊C4—122no—
311C4⋊C4—122no—
312C4⋊C4—122no—
313C4⋊C4—122no—
314C4⋊C4—122no—
315C22⋊C4—121no—
316C22⋊C4—122no—
317C22⋊C4—122no—
318C22⋊C4#22614yescomplemented
319C42—61no—
320C42—61no—
321C42—61no—
322C42—61no—
323C42—122no—
324C22×C3—41no—
325C22×C3—41no—
326C22×C3—41no—
327C22×C3—41no—
328C22×C3—41no—
329C22×C3—41no—
330C22×C3—41no—
331C22×C3—82no—
332C22×C3—82no—
333C22×C3—82no—
334C22×C3—161no—
335C22×C3—161no—
336C22×C3—161no—
337C22×C3—161no—
338C3×C4—82no—
339C3×C4—82no—
340Dic3—82no—
341Dic3—82no—
342Dic3—161no—
343Dic3—161no—
344C23—21no—
345C23—21no—
346C23—21no—
347C23—21no—
348C23—21no—
349C23—21no—
350C23C2×#22612yes—
351C23C22×S411yescharacteristic
352Q8—31no—
353Q8—31no—
354Q8—31no—
355Q8—61no—
356Q8—62no—
357Q8—62no—
358Q8—62no—
359Q8—62no—
360Q8—121no—
361Q8—121no—
362Q8—121no—
363Q8—121no—
364Q8—121no—
365Q8—121no—
366Q8—121no—
367Q8—121no—
368Q8#88411yescharacteristic
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—31no—
374C2×C4—31no—
375C2×C4—31no—
376C2×C4—31no—
377C2×C4—31no—
378C2×C4—31no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—62no—
383C2×C4—62no—
384C2×C4—61no—
385C2×C4—61no—
386C2×C4—61no—
387C2×C4—61no—
388C2×C4—61no—
389C2×C4—61no—
390C2×C4—121no—
391C2×C4—121no—
392C2×C4—121no—
393C2×C4—121no—
394C2×C4—121no—
395C2×C4—121no—
396C2×C4—121no—
397C2×C4—121no—
398C2×C4—121no—
399C2×C4—121no—
400C2×C4—121no—
401C2×C4—121no—
402C2×C4—121no—
403C2×C4—121no—
404C2×C4C2×#22612yes—
405C2×C4C2×#22612yes—
406C8—62no—
407C8—62no—
408C8—121no—
409C8—121no—
410C2×C3—41no—
411C2×C3—41no—
412C2×C3—41no—
413C2×C3—41no—
414C2×C3—41no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—162no—
418C22—22no—
419C22—22no—
420C22—22no—
421C22—41no—
422C22—41no—
423C22—41no—
424C22—41no—
425C22#591211yescharacteristic
426C22D4×S411yescharacteristic
427C22D4×S411yescharacteristic
428C22C22×#22611yescharacteristic
429C22#592311yescharacteristic
430C22#592711yescharacteristic
431C22#592711yescharacteristic
432C4—22no—
433C4—22no—
434C4—31no—
435C4—31no—
436C4—31no—
437C4—31no—
438C4—121no—
439C4—122no—
440C4—121no—
441C4—122no—
442C4—121no—
443C4—241no—
444C4—241no—
445C3—41noSylow
446C2—42no—
447C2#8529911yescharacteristic
448C2#8532211yescharacteristic
449C2#8535211yescharacteristic
450C2#8538311yescharacteristic
451C2#8538611yescharacteristic
452C2D4×#22611yescharacteristic
453C2D4×#22611yescharacteristic
subgroupG/H
center, Frattini, socleC23C22×S4
commutator, cosocleC22×SL(2,𝔽3)C23
FittingQ8×C22⋊C4D3
prime radicalC23×SL(2,𝔽3)C22
hypercenterC2×C22⋊C4S4
hypocenterSL(2,𝔽3)#121
LCSG ▷ C22×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C23 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C4⋊#226C2semidirect
4. C2×C4×#226C2semidirect
6. #73062C2semidirect
7. #73061C2semidirect
318. C22⋊C4#226semidirect

Maximal subgroups

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

subgroupindex
C2×#54192
C2×C4⋊#2262
C2×C4×#2262
C22⋊C4×SL(2,𝔽3)2
#730622
#730612
#251613
#49514